**The Planck Power – A Numerical Coincidence or a Fundamental Number in Cosmology?**

Jack Denur

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/61642

#### **Abstract**

The Planck system of units has been recognized as the most fundamental such system in physics ever since Dr. Max Planck first derived it in 1899. The Planck system of units in general, and especially the Planck power in particular, suggest a simple and interesting cosmological model. Perhaps this model may at least to some degree represent the real Universe; even if it does not, it seems interesting conceptually. The Planck power equals the Planck energy divided by the Planck time, or equivalently the Planck mass times *c*<sup>2</sup> divided by the Planck time. We show that the nongravitational mass-energy of our local region (L-region) of the Universe is, at least approximately, to within a numerical factor on the order of 2, equal to the Planck power times the elapsed cosmic time since the Big Bang. This result is shown to be consistent, to within a numerical factor on the order of 2, with results obtained via alternative derivations. We justify employing primarily L-regions within an observer's cosmological *event* horizon, rather than O-regions (observable regions) within an observer's cosmological *particle* horizon. Perhaps this might imply that as nongravitational mass-energy leaves the cosmological event horizon of our L-region via the Hubble flow, it is replaced at the rate of the Planck power and at the expense of negative gravitational energy. Thus the total mass-energy of our L-region, and likewise of all L-regions, is conserved at the value zero. Some questions concerning the Second Law of Thermodynamics and possible thwarting of the heat death of the Universe predicted thereby, whether via Planck-power input or via some other agency, are discussed.

**Keywords**: Planck system of units, L-regions (local regions), O-regions (observable regions), comoving frame, Second Law of Thermodynamics, heat death, Planck power versus heat death, low-entropy boundary conditions versus heat death, kinetic versus thermodynamic control, kinetic control versus heat death, minimal Boltzmann brains, extraordinary observers.

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### **1. Introduction**

In Sect. 2 we define and distinguish between local regions (L-regions) within an observer's cosmological *event* horizon and observable regions (O-regions) within an observer's cosmological *particle* horizon, of the Universe, and justify primarily employing L-regions. In Sect. 3 we discuss the importance of the Planck system of units, which has been recognized as the most fundamental such system in physics ever since Dr. Max Planck first derived it in 1899. We then consider a possibly important role of the Planck system of units, especially of the Planck power, in cosmology. Perhaps the ensuing cosmological model may at least to some degree represent the real Universe; even if it does not, it seems interesting conceptually. The Planck power equals the Planck energy divided by the Planck time, or equivalently the Planck mass times *c*<sup>2</sup> divided by the Planck time. In Sect. 3 we show that the nongravitational mass-energy of our local region (L-region) of the Universe is, at least approximately, to within a numerical factor on the order of 2, equal to the Planck power times the elapsed cosmic time since the Big Bang. This result is shown to be consistent, to within a numerical factor on the order of 2, with results obtained via alternative derivations. We consider the possible inference that as nongravitational mass-energy leaves the cosmological event horizon of our L-region via the Hubble flow, it is replaced at the rate of the Planck power and at the expense of negative gravitational energy. The problem of consistency with astronomical and astrophysical observations is discussed in Sect. 4. In Sects. 3 and 4 we consider only nonoscillating cosmologies (except for brief parenthetical mentions of oscillating ones in the second-to-last paragraph of Sect. 4). In Sects. 5–8 we consider both nonoscillating and oscillating cosmologies. Some questions concerning the Second Law of Thermodynamics and possible thwarting of the heat death predicted thereby are discussed with respect to the Planck power in Sect. 4, with respect to cosmology in general and minimal Boltzmann brains in particular in Sect. 5, with respect inflation to in Sect. 6, and with respect to kinetic versus thermodynamic control in Sects. 4 and 7. (We discuss possible thwarting of the heat death with respect to kinetic versus thermodynamic control mainly as regards the Planck power in particular in Sect. 4 but more generally in Sect. 7.) A brief review concerning the Multiverse, and some alternative viewpoints, are given in Sect. 8.

#### **2. L-regions and O-regions**

In this chapter we will consider primarily *local* regions or L-regions of the Universe rather than *observable* regions or O-regions [1] thereof, although we will also consider O-regions as necessary [1].<sup>1</sup> We now define and distinguish between L-regions and O-regions, and justify primarily employing L-regions, as opposed to O-regions only occasionally, as necessary [1]. Let *R* be the radial ruler distance or proper distance [2] to the boundary of our L-region, that is to our cosmological *event* horizon [3], where the Hubble flow is at *c*, the speed of light in vacuum; beyond this horizon it exceeds *c*. Thus if the Hubble constant *H* (*τ*) does not vary with cosmic time [4,5] *τ* and is always equal to its present value *H*0, then light emitted at the *present* cosmic time [4,5] *τ*<sup>0</sup> by sources beyond our cosmological event horizon [2,3] and hence beyond our L-region can never reach us. Likewise, light emitted at the *present* cosmic time [4,5] *τ*<sup>0</sup> by us can never reach them. Also, if the Hubble constant *H* (*τ*) does not vary with cosmic time [4,5] *τ* and is always equal to its present value *H*0, then our cosmological event horizon [2,3] is always at fixed ruler distance *R*<sup>0</sup> = *c*/*H*<sup>0</sup> away and hence our L-region

<sup>1</sup> (Re: Entry [1], Ref. [1]) In Ref. [1] observable regions of the Universe are referred to as O-regions for short. We have followed this notation with respect to both O-regions and local regions (L-regions) in this chapter, with L-regions being of primary interest to us.

of the Universe is always of fixed size. [We denote the value of a given quantity *Q today* (at the *present* cosmic time *τ*0) by *Q*<sup>0</sup> and its value at *general* cosmic time *τ* by *Q* (*τ*).]

**1. Introduction**

224 Recent Advances in Thermo and Fluid Dynamics

and some alternative viewpoints, are given in Sect. 8.

**2. L-regions and O-regions**

being of primary interest to us.

In Sect. 2 we define and distinguish between local regions (L-regions) within an observer's cosmological *event* horizon and observable regions (O-regions) within an observer's cosmological *particle* horizon, of the Universe, and justify primarily employing L-regions. In Sect. 3 we discuss the importance of the Planck system of units, which has been recognized as the most fundamental such system in physics ever since Dr. Max Planck first derived it in 1899. We then consider a possibly important role of the Planck system of units, especially of the Planck power, in cosmology. Perhaps the ensuing cosmological model may at least to some degree represent the real Universe; even if it does not, it seems interesting conceptually. The Planck power equals the Planck energy divided by the Planck time, or equivalently the Planck mass times *c*<sup>2</sup> divided by the Planck time. In Sect. 3 we show that the nongravitational mass-energy of our local region (L-region) of the Universe is, at least approximately, to within a numerical factor on the order of 2, equal to the Planck power times the elapsed cosmic time since the Big Bang. This result is shown to be consistent, to within a numerical factor on the order of 2, with results obtained via alternative derivations. We consider the possible inference that as nongravitational mass-energy leaves the cosmological event horizon of our L-region via the Hubble flow, it is replaced at the rate of the Planck power and at the expense of negative gravitational energy. The problem of consistency with astronomical and astrophysical observations is discussed in Sect. 4. In Sects. 3 and 4 we consider only nonoscillating cosmologies (except for brief parenthetical mentions of oscillating ones in the second-to-last paragraph of Sect. 4). In Sects. 5–8 we consider both nonoscillating and oscillating cosmologies. Some questions concerning the Second Law of Thermodynamics and possible thwarting of the heat death predicted thereby are discussed with respect to the Planck power in Sect. 4, with respect to cosmology in general and minimal Boltzmann brains in particular in Sect. 5, with respect inflation to in Sect. 6, and with respect to kinetic versus thermodynamic control in Sects. 4 and 7. (We discuss possible thwarting of the heat death with respect to kinetic versus thermodynamic control mainly as regards the Planck power in particular in Sect. 4 but more generally in Sect. 7.) A brief review concerning the Multiverse,

In this chapter we will consider primarily *local* regions or L-regions of the Universe rather than *observable* regions or O-regions [1] thereof, although we will also consider O-regions as necessary [1].<sup>1</sup> We now define and distinguish between L-regions and O-regions, and justify primarily employing L-regions, as opposed to O-regions only occasionally, as necessary [1]. Let *R* be the radial ruler distance or proper distance [2] to the boundary of our L-region, that is to our cosmological *event* horizon [3], where the Hubble flow is at *c*, the speed of light in vacuum; beyond this horizon it exceeds *c*. Thus if the Hubble constant *H* (*τ*) does not vary with cosmic time [4,5] *τ* and is always equal to its present value *H*0, then light emitted at the *present* cosmic time [4,5] *τ*<sup>0</sup> by sources beyond our cosmological event horizon [2,3] and hence beyond our L-region can never reach us. Likewise, light emitted at the *present* cosmic time [4,5] *τ*<sup>0</sup> by us can never reach them. Also, if the Hubble constant *H* (*τ*) does not vary with cosmic time [4,5] *τ* and is always equal to its present value *H*0, then our cosmological event horizon [2,3] is always at fixed ruler distance *R*<sup>0</sup> = *c*/*H*<sup>0</sup> away and hence our L-region

<sup>1</sup> (Re: Entry [1], Ref. [1]) In Ref. [1] observable regions of the Universe are referred to as O-regions for short. We have followed this notation with respect to both O-regions and local regions (L-regions) in this chapter, with L-regions Light emitted at *past* cosmic times *τ < τ*<sup>0</sup> (but not too far in the past) by sources now beyond (but not too far beyond) our cosmological event horizon [*R*<sup>0</sup> = *c*/*H*<sup>0</sup> always if *H* (*τ*) = *H*<sup>0</sup> always] and hence beyond our L-region but still within our O-region [1–3] can reach us, because when this light was emitted these sources were still within our L-region. Likewise, light emitted in the *past τ < τ*<sup>0</sup> (but not too far in the past) by us can reach them. The boundary of our O-region of the Universe is our cosmological *particle* horizon [1–3]. The boundary of our O-region (our cosmological particle horizon) is further away than the boundary of our L-region (our cosmological particle horizon) [1–3]. If *H* (*τ*) = *H*<sup>0</sup> always, not only is the boundary of our O-region currently at ruler distance R<sup>0</sup> *> R*<sup>0</sup> = *c*/*H*<sup>0</sup> but R (*τ*) gets further away with increasing cosmic time *τ* [4,5], while the boundary of our L-region *R* (*τ*) always remains fixed at *R*<sup>0</sup> = *c*/*H*0. The fixed size of our (or any) L-region given constant *H* (*τ*) = *H*<sup>0</sup> simplifies our discussions. More importantly, all parts of our (or any) L-region are *always* in casual contact, while outer parts of our (or any) O-region beyond the limit of the corresponding L-region *were* but no longer *are* in causal contact. Hence we will primarily employ L-regions rather than O-regions.

Hubble flow exceeding *c* may seem to violate Special Relativity. But General Relativity not Special Relativity — is applicable in cosmology [6]. Special Relativity is applicable only within local inertial frames, and any given observer is not — indeed cannot be — in the same local inertial frame as this observer's cosmological event horizon [1–6] (and even less so as this observer's cosmological particle horizon [1–6]). Thus Hubble flow exceeding *c* does not violate General Relativity [6]. It should also be noted that the Hubble flow is motion with space rather than through space — every object in the Hubble flow is at rest in the comoving frame [7]. An object's motion, if any, relative to the comoving frame [7] is its *peculiar* motion.<sup>2</sup>

At the 27th Texas Symposium on Relativistic Astrophysics [8], values of the Hubble constant today *H*<sup>0</sup> from the upper 60s to the low 70s (km / s) / Mpc were given [8], so *H*<sup>0</sup> ≈ 70 (km / s) / Mpc splits the difference [8]. These values were essentially unchanged from those obtained shortly preceding this Symposium [9,10]. The *Planck* 2015 results [11] state a value of *H*<sup>0</sup> = 68 (km / s) / Mpc [11], but this *Planck* 2015 work [11] also cites other recent results that range from the low 60s (km / s) / Mpc to the low 70s (km / s) / Mpc. Thus the value *H*<sup>0</sup> = 68 (km / s) / Mpc [11] not only is the most reliable and most recent one as of this writing, but it also splits the difference of the range of other recent results cited in this *Planck* 2015 work [11]. Hence we take the Hubble constant today to be *<sup>H</sup>*<sup>0</sup> . <sup>=</sup> <sup>68</sup> (km / s) / Mpc <sup>≈</sup> 2.2 <sup>×</sup> <sup>10</sup>−<sup>18</sup> (km / s) / km <sup>=</sup> 2.2 <sup>×</sup> <sup>10</sup>−<sup>18</sup> <sup>s</sup>−<sup>1</sup> [11].3

<sup>2</sup> (Re: Entry [7], Ref. [2]) An observer in the comoving frame (ideally in intergalactic space as far removed as possible from local gravitational fields such as those of galaxies, stars, etc.) sees the 2.7 K cosmic background radiation as isotropic (apart from fluctuations of fractional magnitude F ≈ <sup>10</sup>−5, which can be "smoothed out" via, say, computer processing to yield a uniform background). But even Earth is a fairly good approximation to the comoving frame: Earth's peculiar motion ≈ 380 km / s *c* (see p. 352 of Ref. [2]) with respect to the cosmic background radiation is fairly slow, and local gravitational fields are fairly weak (*v*escape *c*).

<sup>3</sup> (Re: Entries [8]–[11], Refs. [2], [8], [10], and [11]) As per Entries [8]–[11], results for the Hubble constant have improved with time, asymptotically converging onto those provided by Ref. [11]. The results for the Hubble constant as per Ref. [8] are in essential agreement with Entry [9]. The history of values of the Hubble constant also is briefly discussed in Entry [9] and Ref. [10]. Reference [10] surveys the history of values of the Hubble constant determined via work done through 2012. Reference [10] was for sale at the 27th Texas Symposium on Relativistic Astrophysics, held at the Fairmont Hotel in Dallas, Texas, December 8–13, 2013.

#### **3. The Planck power in cosmology**

The Planck system of units has been recognized as the most fundamental such system in physics ever since Dr. Max Planck first derived it in 1899 [12–15]. It is based on Planck's reduced constant ¯*h* ≡ *h*/2*π* (or Planck's original constant *h*), the speed of light in vacuum *c*, and the universal gravitational constant *G*, with Boltzmann's constant *k* usually also included. These four fundamental physical constants are seen by *everything*, corresponding to the Planck system of units encompassing *universal* domain. By contrast, for example, the fundamental electric charge is seen only by electrically-charged particles.<sup>4</sup>

The Planck system of units in general, and especially the Planck power in particular, suggest a simple and interesting cosmological model. Perhaps this model may at least to some degree represent the real Universe; even if it does not, it seems interesting conceptually.

Multiply the Planck mass *m*Planck = (*hc*¯ /*G*) 1/2 by *<sup>c</sup>*<sup>2</sup> to obtain the Planck energy *<sup>E</sup>*Planck = *hc*¯ 5/*G* 1/2 [12–15]. Divide the Planck energy by the Planck time *<sup>t</sup>*Planck <sup>=</sup> *hG*¯ /*c*<sup>5</sup>1/2 to obtain the Planck power *<sup>P</sup>*Planck <sup>=</sup> *<sup>c</sup>*5/*<sup>G</sup>* . <sup>=</sup> 3.64 <sup>×</sup> <sup>1052</sup> <sup>W</sup> ⇐⇒ *<sup>P</sup>*Planck/*c*<sup>2</sup> <sup>=</sup> *<sup>c</sup>*3/*<sup>G</sup>* . = 4.05 <sup>×</sup> <sup>1035</sup> kg / s [12–15]. [The dot-equal sign ( . =) means "very nearly equal to."] Note that — unlike the Planck length, mass, energy, time, and temperature *T*Planck = *E*Planck/*k* = *hc*¯ 5/*G* 1/2 /*k*, indeed unlike most if not all other Planck units (at least most if not all other useful ones except the Planck speed *l*Planck/*t*Planck = *c*) — the Planck power (whether or not divided by *c*2) does *not* contain ¯*h*, but only *G* and *c*. Thus — unlike the Planck length *l*Planck, Planck mass, Planck energy, Planck time, and Planck temperature, indeed unlike most if not all other Planck units (at least most if not all other useful ones except the Planck speed *l*Planck/*t*Planck = *c*) — is the Planck power a *classical* quantity *in*dependent of quantum effects, if not absolutely then at least via opposing quantum effects canceling out, as ¯*h* cancels out in the division *P*Planck = *E*Planck/*t*Planck? With respect to the Planck speed *l*Planck/*t*Planck = *c* note that *c* is the fundamental speed in the classical (nonquantum) theories of Special and General Relativity.

Now multiply *P*Planck/*c*<sup>2</sup> by the age of the Universe, the elapsed cosmic time [4,5] since the Big Bang, *<sup>τ</sup>*<sup>0</sup> <sup>≈</sup> 4.5 <sup>×</sup> 1017 <sup>s</sup> <sup>≈</sup> 1.4 <sup>×</sup> 1010 y [11]. This yields an estimate of

$$M\_0 \approx \frac{P\_{\text{Planck}} \tau\_0}{c^2} \approx 1.8 \times 10^{53} \,\text{kg} \tag{1}$$

for the mass of our L-region (not considering the negative gravitational energy). But *<sup>M</sup>*<sup>0</sup> <sup>≈</sup> 1.8 <sup>×</sup> 1053 kg is of order-of-magnitude agreement with an estimate of *<sup>M</sup>*<sup>0</sup> assuming that the mass-energy density of of our L-region of the Universe [1,3] equals the critical density *ρ*crit [16], as seems to be the case if not exactly then at least to within a very close approximation. The density critical density *ρ*crit corresponds to the borderline between ever-expanding and oscillating Universes given vanishing cosmological constant, i.e., Λ = 0,

<sup>4</sup> (Re: Entries [12] and [15], Refs. [12] and [15]) A concise listing of Planck units and other useful data, entitled "Some Useful Numbers in Conventional and Geometrized Units," is provided in the back endcover of Ref. [12]. In this back endcover of Ref. [12] the Planck length is referred to as the Planck distance (elsewhere in Ref. [12] it is referred to as the Planck length) and the Planck power is referred to as the emission factor. Reference [12] cites Ref. [13] as the most important work in the derivation of the Planck system of units. Reference [15], like Ref. [12], cites Ref. [13]. Additionally, in Sect. 31.1, Ref. [15] gives a brief historical survey of works deriving the Planck system of units. Reference [15] extends the Planck system of units to also include Boltzmann's constant *k*.

and to spacetime being flat, and hence space Euclidean, on the largest scales, i.e., to the spatial curvature index being 0 rather than <sup>+</sup>1 or <sup>−</sup>1, given *any* value of <sup>Λ</sup> [16–20].5 The critical density is

**3. The Planck power in cosmology**

226 Recent Advances in Thermo and Fluid Dynamics

Multiply the Planck mass *m*Planck = (*hc*¯ /*G*)

obtain the Planck power *<sup>P</sup>*Planck <sup>=</sup> *<sup>c</sup>*5/*<sup>G</sup>* .

4.05 <sup>×</sup> <sup>1035</sup> kg / s [12–15]. [The dot-equal sign ( .

 *hc*¯ 5/*G*

 *hc*¯ 5/*G*

General Relativity.

The Planck system of units has been recognized as the most fundamental such system in physics ever since Dr. Max Planck first derived it in 1899 [12–15]. It is based on Planck's reduced constant ¯*h* ≡ *h*/2*π* (or Planck's original constant *h*), the speed of light in vacuum *c*, and the universal gravitational constant *G*, with Boltzmann's constant *k* usually also included. These four fundamental physical constants are seen by *everything*, corresponding to the Planck system of units encompassing *universal* domain. By contrast, for example, the

The Planck system of units in general, and especially the Planck power in particular, suggest a simple and interesting cosmological model. Perhaps this model may at least to some degree

— unlike the Planck length, mass, energy, time, and temperature *T*Planck = *E*Planck/*k* =

useful ones except the Planck speed *l*Planck/*t*Planck = *c*) — the Planck power (whether or not divided by *c*2) does *not* contain ¯*h*, but only *G* and *c*. Thus — unlike the Planck length *l*Planck, Planck mass, Planck energy, Planck time, and Planck temperature, indeed unlike most if not all other Planck units (at least most if not all other useful ones except the Planck speed *l*Planck/*t*Planck = *c*) — is the Planck power a *classical* quantity *in*dependent of quantum effects, if not absolutely then at least via opposing quantum effects canceling out, as ¯*h* cancels out in the division *P*Planck = *E*Planck/*t*Planck? With respect to the Planck speed *l*Planck/*t*Planck = *c* note that *c* is the fundamental speed in the classical (nonquantum) theories of Special and

Now multiply *P*Planck/*c*<sup>2</sup> by the age of the Universe, the elapsed cosmic time [4,5] since the

for the mass of our L-region (not considering the negative gravitational energy). But *<sup>M</sup>*<sup>0</sup> <sup>≈</sup> 1.8 <sup>×</sup> 1053 kg is of order-of-magnitude agreement with an estimate of *<sup>M</sup>*<sup>0</sup> assuming that the mass-energy density of of our L-region of the Universe [1,3] equals the critical density *ρ*crit [16], as seems to be the case if not exactly then at least to within a very close approximation. The density critical density *ρ*crit corresponds to the borderline between ever-expanding and oscillating Universes given vanishing cosmological constant, i.e., Λ = 0,

<sup>4</sup> (Re: Entries [12] and [15], Refs. [12] and [15]) A concise listing of Planck units and other useful data, entitled "Some Useful Numbers in Conventional and Geometrized Units," is provided in the back endcover of Ref. [12]. In this back endcover of Ref. [12] the Planck length is referred to as the Planck distance (elsewhere in Ref. [12] it is referred to as the Planck length) and the Planck power is referred to as the emission factor. Reference [12] cites Ref. [13] as the most important work in the derivation of the Planck system of units. Reference [15], like Ref. [12], cites Ref. [13]. Additionally, in Sect. 31.1, Ref. [15] gives a brief historical survey of works deriving the Planck system of units.

Big Bang, *<sup>τ</sup>*<sup>0</sup> <sup>≈</sup> 4.5 <sup>×</sup> 1017 <sup>s</sup> <sup>≈</sup> 1.4 <sup>×</sup> 1010 y [11]. This yields an estimate of

*<sup>M</sup>*<sup>0</sup> <sup>≈</sup> *<sup>P</sup>*Planck*τ*<sup>0</sup>

Reference [15] extends the Planck system of units to also include Boltzmann's constant *k*.

1/2 /*k*, indeed unlike most if not all other Planck units (at least most if not all other

1/2 by *<sup>c</sup>*<sup>2</sup> to obtain the Planck energy *<sup>E</sup>*Planck =

<sup>=</sup> 3.64 <sup>×</sup> <sup>1052</sup> <sup>W</sup> ⇐⇒ *<sup>P</sup>*Planck/*c*<sup>2</sup> <sup>=</sup> *<sup>c</sup>*3/*<sup>G</sup>* .

*<sup>c</sup>*<sup>2</sup> <sup>≈</sup> 1.8 <sup>×</sup> <sup>1053</sup> kg (1)

=) means "very nearly equal to."] Note that

*hG*¯ /*c*<sup>5</sup>1/2 to

=

fundamental electric charge is seen only by electrically-charged particles.<sup>4</sup>

represent the real Universe; even if it does not, it seems interesting conceptually.

1/2 [12–15]. Divide the Planck energy by the Planck time *<sup>t</sup>*Planck <sup>=</sup>

$$
\rho\_{\rm crit} = \frac{3H\_0^2}{8\pi G} \approx 8.65 \times 10^{-27} \frac{\rm kg}{\rm m^3}.\tag{2}
$$

Applying the most recent and best result for *H*0, namely *H*<sup>0</sup> ≈ 68 (km / s) / Mpc ≈ 2.2 × <sup>10</sup>−<sup>18</sup> (km / s) / km <sup>=</sup> 2.2 <sup>×</sup> <sup>10</sup>−<sup>18</sup> <sup>s</sup>−<sup>1</sup> [11], yields as an estimate of *<sup>M</sup>*<sup>0</sup>

$$M\_0 \approx \frac{4\pi}{3} \rho\_{\rm crit} R\_0^3 = \frac{4\pi}{3} \frac{3H\_0^2}{8\pi G} \left(\frac{c}{H\_0}\right)^3 = \frac{c^3}{2G H\_0} \approx 9.2 \times 10^{52} \,\mathrm{kg}.\tag{3}$$

In Eq. (3) we assume that the volume of our L-region is given by the Euclidean value 4*πR*<sup>3</sup> <sup>0</sup>/3. But since astronomical observations indicate that spacetime is flat, and hence space is Euclidean, on the largest scales, i.e., that the spatial curvature index is 0 rather than +1 or −1, this assumption seems justified [11,16–20]. Is the order-of-magnitude agreement between Eqs. (1) and (3) merely a numerical coincidence? Or does it suggest that the Planck power plays a fundamental role in cosmology — entailing a link between the smallest (Planck-length and Planck-time) and largest (cosmological) scales?

While there is order–of magnitude agreement between Eqs. (1) and (3), there is a discrepancy between them by a factor of ≈ 2. That is, Planck-power input as per Eq. (1) seems to imply *ρ* ≈ 2*ρ*crit. Since in this era of precision cosmology all quantities in Eqs. (1)–(3) are known far more accurately than to within a factor of 2, it seems that this factor of ≈ 2 can*not* simply be dismissed. But we admit that we have no explanation for this factor of ≈ 2. Furthermore, we will see that Eqs. (5)–(7) seem to imply a discrepancy with Eq. (1) by a factor of ≈ 3/2 in the opposite direction, i.e., that Planck-power input as per Eq. (1) seems to imply *ρ* ≈ 2*ρ*crit/3. Such discrepancies by numerical factors on the order of 2 may prove our Planck-power hypothesis to be wrong. At the very least they prove that even if it is right *in general* it is only an *introductory* hypothesis whose *details* still need to be understood. Then again, perhaps because there *is* consistency to within a small numerical factors of *O* ∼ 2, our Planck-power hypothesis may be correct *in general* as an *introductory* hypothesis, even though, even if correct *in general*, its *details* still need to be understood.

Do our considerations so far in this Sect. 3 suggest that, even though the Universe certainly began with the Big Bang, there has been since the Big Bang mass-energy input, at least on the average, at the Planck power, into our L-region of the Universe? We list several alternative proposals for such input (this list probably is not exhaustive): (a) steady-state-theory mass-energy input *ex nihilo* [21–23], (b) mass-energy input *ex nihilo* via other means [24,25], (c) mass-energy input at the expense of negative gravitational energy [26–32] rather than *ex nihilo*, or (d) mass-energy input at the expense of nongravitational negative energy, for example, at the expense of the negative-energy C field in some versions of the steady-state theory [33–35]. If at the expense of negative gravitational energy as per proposal (c), then *forever* the total (mass plus gravitational) energy of our L-region, and likewise of any L-region, of the Universe, and hence of the Universe as a whole, is conserved at the value

<sup>5</sup> (Re: Entry [20], Ref. [14]) The critical density and density parameter are employed on various occasions throughout Chap. 29 on cosmology in Ref. [14].

zero [26–32]. (There are "certain 'positivity' theorems ... which tell us that the total energy of a system, including the 'negative gravitational potential energy contributions' ..., cannot be negative [32]." But positivity theorems do seem to allow the total energy of a system, including the negative gravitational energy, to be strictly zero. Also, perhaps positivity theorems need necessarily apply only for isolated sources in asymptotically-flat spacetime.) In this chapter we will mainly presume proposal (c) from the immediately preceding list, for the following reasons: (i) Unlike proposals (a) and (b), proposal (c) entails no violation of the First Law of Thermodynamics (conservation of mass-energy). (ii) Negative gravitational energy is *known* to exist, unlike the negative-energy C field of proposal (d), which was perhaps introduced at least partially *ad hoc* to render the steady-state theory consistent with the First Law of Thermodynamics (conservation of mass-energy). Moreover, unlike gravity, the C field not only has never been observed, but also entails difficulties of its own [34,35]. (iii) We will show that proposal (c) need not be inconsistent with the observed features of the Universe.

The Universe clearly shows evolutionary rather than steady-state [21–23,33–35] behavior since the Big Bang. But it could stabilize to a steady state in the future. It could already now be thus stabilizing or even thus stabilized in the very recent past with as yet no or at most very limited observational evidence that might be suggestive of such stabilization. Thus even if there is steady-state-type creation of mass-energy since the Big Bang at the rate of the Planck power (we presume, in light of the immediately preceding paragraph, most likely at the expense of the Universe's negative gravitational energy), perhaps this might be compatible with the observed evolutionary behavior of the Universe since the Big Bang. (This point and related ones will be discussed in more detail in Sect. 4.)

Although General Relativity is required for an accurate consideration of the Universe's gravity, the following Newtonian approximation may be valid as an order-of-magnitude estimate [26–32]. Such an estimate is suggestive in favor of Planck-power input at the expense of negative gravitational energy [26–32], which does not require a violation of the First Law of Thermodynamics (conservation of mass-energy) [26–32], as opposed to Planck-power input *ex nihilo* [21–25], which would require such a violation, or via C-field input, the C field never having been observed and also entailing its own difficulties [34,35]. In accordance with the last paragraph of Sect. 2, we take the Hubble constant today to be *<sup>H</sup>*<sup>0</sup> . <sup>=</sup> <sup>68</sup> (km / s) / Mpc <sup>≈</sup> 2.2 <sup>×</sup> <sup>10</sup>−<sup>18</sup> (km / s) / km <sup>=</sup> 2.2 <sup>×</sup> <sup>10</sup>−<sup>18</sup> <sup>s</sup>−<sup>1</sup> [8–11]. Thus neglecting any variation of *<sup>H</sup>* (*τ*) with *<sup>τ</sup>*, *<sup>τ</sup>*<sup>0</sup> <sup>=</sup> 1/*H*<sup>0</sup> <sup>≈</sup> 4.5 <sup>×</sup> <sup>10</sup><sup>17</sup> s consistently with the previously given value, and the ruler radius of our L-region of the Universe is *R*<sup>0</sup> = *<sup>c</sup>τ*<sup>0</sup> <sup>=</sup> *<sup>c</sup>*/*H*<sup>0</sup> <sup>≈</sup> 1.4 <sup>×</sup> 1023 km <sup>=</sup> 1.4 <sup>×</sup> 1026 m. The positive mass-energy of our L-region of the Universe within our cosmological event horizon is *M*0*c*<sup>2</sup> and the negative Newtonian gravitational energy of our L-region is ≈ −*GM*<sup>2</sup> <sup>0</sup>/*R*0. Hence in the Newtonian approximation setting the total energy equal to zero yields [26–32]

$$E\_{\text{total}} = E\_{\text{mass}} + E\_{\text{gravitational}} = 0$$

$$\implies M\_0 c^2 - \frac{GM\_0^2}{R\_0} = 0$$

$$\implies \frac{M\_0}{R\_0} = \frac{c^2}{G}. \tag{4}$$

Applying our previously derived values of *<sup>M</sup>*<sup>0</sup> and *<sup>R</sup>*<sup>0</sup> yields *<sup>M</sup>*0/*R*<sup>0</sup> <sup>≈</sup> 1.8 <sup>×</sup> <sup>10</sup><sup>53</sup> kg /1.36 <sup>×</sup> <sup>1026</sup> <sup>m</sup> <sup>≈</sup> 1.32 <sup>×</sup> <sup>1027</sup> kg / m. We have *<sup>c</sup>*2/*<sup>G</sup>* . <sup>=</sup> 1.35 <sup>×</sup> <sup>10</sup><sup>27</sup> kg / m. Thus Eq. (4) is fulfilled as closely as we can expect, especially given that our Newtonian approximation should be expected to provide only order-of-magnitude estimates, and also perhaps because (even after an initial fast inflationary stage) *H* (*τ*) may not be strictly constant.

zero [26–32]. (There are "certain 'positivity' theorems ... which tell us that the total energy of a system, including the 'negative gravitational potential energy contributions' ..., cannot be negative [32]." But positivity theorems do seem to allow the total energy of a system, including the negative gravitational energy, to be strictly zero. Also, perhaps positivity theorems need necessarily apply only for isolated sources in asymptotically-flat spacetime.) In this chapter we will mainly presume proposal (c) from the immediately preceding list, for the following reasons: (i) Unlike proposals (a) and (b), proposal (c) entails no violation of the First Law of Thermodynamics (conservation of mass-energy). (ii) Negative gravitational energy is *known* to exist, unlike the negative-energy C field of proposal (d), which was perhaps introduced at least partially *ad hoc* to render the steady-state theory consistent with the First Law of Thermodynamics (conservation of mass-energy). Moreover, unlike gravity, the C field not only has never been observed, but also entails difficulties of its own [34,35]. (iii) We will show that proposal (c) need not be inconsistent with the observed features of the

The Universe clearly shows evolutionary rather than steady-state [21–23,33–35] behavior since the Big Bang. But it could stabilize to a steady state in the future. It could already now be thus stabilizing or even thus stabilized in the very recent past with as yet no or at most very limited observational evidence that might be suggestive of such stabilization. Thus even if there is steady-state-type creation of mass-energy since the Big Bang at the rate of the Planck power (we presume, in light of the immediately preceding paragraph, most likely at the expense of the Universe's negative gravitational energy), perhaps this might be compatible with the observed evolutionary behavior of the Universe since the Big Bang. (This

Although General Relativity is required for an accurate consideration of the Universe's gravity, the following Newtonian approximation may be valid as an order-of-magnitude estimate [26–32]. Such an estimate is suggestive in favor of Planck-power input at the expense of negative gravitational energy [26–32], which does not require a violation of the First Law of Thermodynamics (conservation of mass-energy) [26–32], as opposed to Planck-power input *ex nihilo* [21–25], which would require such a violation, or via C-field input, the C field never having been observed and also entailing its own difficulties [34,35]. In accordance with the last paragraph of Sect. 2, we take the Hubble constant today to

<sup>=</sup> <sup>68</sup> (km / s) / Mpc <sup>≈</sup> 2.2 <sup>×</sup> <sup>10</sup>−<sup>18</sup> (km / s) / km <sup>=</sup> 2.2 <sup>×</sup> <sup>10</sup>−<sup>18</sup> <sup>s</sup>−<sup>1</sup> [8–11]. Thus

<sup>0</sup>/*R*0. Hence in the Newtonian approximation

*<sup>G</sup>* . (4)

neglecting any variation of *<sup>H</sup>* (*τ*) with *<sup>τ</sup>*, *<sup>τ</sup>*<sup>0</sup> <sup>=</sup> 1/*H*<sup>0</sup> <sup>≈</sup> 4.5 <sup>×</sup> <sup>10</sup><sup>17</sup> s consistently with the previously given value, and the ruler radius of our L-region of the Universe is *R*<sup>0</sup> = *<sup>c</sup>τ*<sup>0</sup> <sup>=</sup> *<sup>c</sup>*/*H*<sup>0</sup> <sup>≈</sup> 1.4 <sup>×</sup> 1023 km <sup>=</sup> 1.4 <sup>×</sup> 1026 m. The positive mass-energy of our L-region of the Universe within our cosmological event horizon is *M*0*c*<sup>2</sup> and the negative Newtonian

*E*total = *E*mass + *E*gavitational = 0

<sup>=</sup><sup>⇒</sup> *<sup>M</sup>*0*c*<sup>2</sup> <sup>−</sup> *GM*<sup>2</sup>

<sup>=</sup> *<sup>c</sup>*<sup>2</sup>

*M*<sup>0</sup> *R*0

=⇒

0 *R*0

= 0

point and related ones will be discussed in more detail in Sect. 4.)

gravitational energy of our L-region is ≈ −*GM*<sup>2</sup>

setting the total energy equal to zero yields [26–32]

Universe.

228 Recent Advances in Thermo and Fluid Dynamics

be *<sup>H</sup>*<sup>0</sup> .

There is yet another order-of-magnitude result that is consistent with our Planck-power hypothesis. Applying Eq. (1), rate of Planck-power mass input into our L-region is

$$\left(\frac{dM}{d\tau}\right)\_{\text{in}} \approx \frac{M\_0}{\tau\_0} \approx \frac{P\_{\text{Planck}}}{c^2} = \frac{\frac{c^3}{G}}{c^2} = \frac{c^3}{G}.\tag{5}$$

Letting *ρ* be the average density of our L-region, the rate of Hubble-flow mass-export from our L-region is

$$
\left(\frac{dM}{d\tau}\right)\_{\text{out}} = 4\pi R\_0^2 \rho c = 4\pi \left(\frac{c}{H\_0}\right)^2 \rho c = \frac{4\pi \rho c^3}{H\_0^2}.\tag{6}
$$

In Eq. (6) we assume that the surface area bounding our L-region is given by the Euclidean value 4*πR*<sup>2</sup> <sup>0</sup>. But since astronomical observations indicate that spacetime is flat, and hence space is Euclidean, on the largest scales, i.e., that the spatial curvature index is 0 rather than +1 or −1, this assumption seems justified [11,16–20]. For steady-state to obtain we must have

$$
\begin{split}
\left(\frac{dM}{d\tau}\right)\_{\text{net}} &= \left(\frac{dM}{d\tau}\right)\_{\text{in}} - \left(\frac{dM}{d\tau}\right)\_{\text{out}} = 0 \\
\implies \frac{c^3}{G} - 4\pi R\_0^2 \rho c &= \frac{c^3}{G} - \frac{4\pi \rho c^3}{H\_0^2} = 0 \\
\implies \frac{1}{G} - \frac{4\pi \rho}{H\_0^2} = 0 \\
\implies \rho &= \frac{c^2}{4\pi G R\_0^2} = \frac{H\_0^2}{4\pi G} \approx 5.8 \times 10^{-27} \frac{\text{kg}}{\text{m}^3}.\tag{7}
\end{split}
$$

The numerical value for *ρ* obtained in the last line of Eq. (7) is in order-of-magnitude agreement with *ρ*crit as per Eq. (2), as well as in order-of-magnitude agreement with observations.

Recalling the second paragraph following that containing Eqs. (1)–(3), note that Eqs. (5)–(7) seem to imply a discrepancy with Eq. (1) by a factor of ≈ 3/2 in the opposite direction from the discrepancy with Eq. (1) by a factor of ≈ 2 implied by Eq. (3). Planck-power input as per Eq. (1) seems to imply *ρ* ≈ 2*ρ*crit, while Eqs. (5)–(7) seem to imply *ρ* ≈ 2*ρ*crit/3. Since in this era of precision cosmology all quantities in Eqs. (1)–(3) and (5)–(7) are known far more accurately than to within a factor of 2, such discrepancies by numerical factors of *O* ∼ 2 may prove our Planck-power hypothesis to be wrong. At the very least they prove that even if it is right *in general* it is only an *introductory* hypothesis whose *details* still need to be understood. Then again, perhaps because there *is* consistency to within small numerical factors of *O* ∼ 2, our Planck-power hypothesis may be correct *in general* as an *introductory* hypothesis, even though, even if correct *in general*, its *details* still need to be understood.

While, even accepting discrepancies by a factor of *O* ∼ 2, the fulfillment of Eqs. (1)–(7) does not constitute proof of Planck-power input, it at least seems suggestive. Could Planck-power input, if it exists, be a classical process *in*dependent of quantum effects, if not absolutely then at least via opposing quantum effects canceling out, as ¯*h* cancels out in the division *P*Planck = *E*Planck/*t*Planck? Note that perhaps similar canceling out obtains with respect to the Planck speed *l*Planck/*t*Planck = *c*: *c* is the fundamental speed in the classical (nonquantum) theories of Special and General Relativity.

According to most current cosmological models, there probably have been one-time initial mass-energy inputs, for example associated with phase transitions ending fast-inflationary stages during the very early history of the Universe [36–40]. We should note that while the majority opinion is certainly in favor of inflation [36–50], there is some dissent [36–50]. (The difficulty in squaring inflation with the Second Law of Thermodynamics, and a possible resolution of this difficulty, will be discussed in Sect. 6.) Observational evidence that in early 2014 initially seemed convincing for inflation in general [41,42], albeit possibly ruling out a few specific types of inflation [41,42,47], has been questioned [43–50], but not disproved [43–50].<sup>6</sup> Moreover, even if an inflationary model is correct, recent observational findings disfavor simple models of inflation, such as quadratic and natural inflation [47]. Even if such one-time initial mass-energy inputs [36–40] occurred, could sustained mass-energy input then continue indefinitely such that the Planck power is at least a *floor* below which the *average* rate of mass-energy input into our L-region of the Universe cannot fall? It at least *appears* not to have fallen below this floor [16]. By the cosmological principle [51], if this is true of our L-region of the Universe then it must be true of *any* L-region thereof.

Thus our Planck-power hypothesis at least *appears* to entail a link between the smallest (Planck mass and Planck time) and largest (cosmological) scales, rather than being merely a numerical coincidence.

#### **4. Planck power and kinetic control versus heat death: Big-Bang-initiated evolution merging into steady state?**

In the simplest ever-expanding cosmologies, the Universe begins with a Big Bang and expands forever, with flat geometry on the largest scales, and with the Hubble constant *H* (*τ*) not varying with cosmic time [4,5] *τ* and always equal to its present value *H*0. As the Universe expands the Hubble flow carries mass-energy past the cosmological event horizon of our L-region of the Universe. But this "loss" is replaced by new positive mass-energy continually created within our L-region of the Universe *forever* at the rate of the Planck power and at the expense of our L-region's negative gravitational energy. This compensates

<sup>6</sup> (Re: Entry [48], Ref. [48]) Reference [48] shows that previous measurements of the acceleration of the Universe's expansion may require reconsideration, owing to discrepancies between visible-light and UV observations of type 1a supernovae.

for "losses" streaming past the cosmological event horizon of our L-region of the Universe via the Hubble flow — and does so consistently with the First Law of Thermodynamics (conservation of mass-energy) [26–32]. Because by the cosmological principle [51] our L-region is nothing special, the same is true of *any* L-region [3] of the Universe. Thus *forever* the total (mass plus gravitational) energy of our L-region, and likewise of any L-region, of the Universe, and hence of the Universe as a whole, is conserved at the value zero [26–32].

right *in general* it is only an *introductory* hypothesis whose *details* still need to be understood. Then again, perhaps because there *is* consistency to within small numerical factors of *O* ∼ 2, our Planck-power hypothesis may be correct *in general* as an *introductory* hypothesis, even

While, even accepting discrepancies by a factor of *O* ∼ 2, the fulfillment of Eqs. (1)–(7) does not constitute proof of Planck-power input, it at least seems suggestive. Could Planck-power input, if it exists, be a classical process *in*dependent of quantum effects, if not absolutely then at least via opposing quantum effects canceling out, as ¯*h* cancels out in the division *P*Planck = *E*Planck/*t*Planck? Note that perhaps similar canceling out obtains with respect to the Planck speed *l*Planck/*t*Planck = *c*: *c* is the fundamental speed in the classical (nonquantum)

According to most current cosmological models, there probably have been one-time initial mass-energy inputs, for example associated with phase transitions ending fast-inflationary stages during the very early history of the Universe [36–40]. We should note that while the majority opinion is certainly in favor of inflation [36–50], there is some dissent [36–50]. (The difficulty in squaring inflation with the Second Law of Thermodynamics, and a possible resolution of this difficulty, will be discussed in Sect. 6.) Observational evidence that in early 2014 initially seemed convincing for inflation in general [41,42], albeit possibly ruling out a few specific types of inflation [41,42,47], has been questioned [43–50], but not disproved [43–50].<sup>6</sup> Moreover, even if an inflationary model is correct, recent observational findings disfavor simple models of inflation, such as quadratic and natural inflation [47]. Even if such one-time initial mass-energy inputs [36–40] occurred, could sustained mass-energy input then continue indefinitely such that the Planck power is at least a *floor* below which the *average* rate of mass-energy input into our L-region of the Universe cannot fall? It at least *appears* not to have fallen below this floor [16]. By the cosmological principle [51], if this is true of our L-region of the Universe then it must be true of *any*

Thus our Planck-power hypothesis at least *appears* to entail a link between the smallest (Planck mass and Planck time) and largest (cosmological) scales, rather than being merely a

**4. Planck power and kinetic control versus heat death: Big-Bang-initiated**

In the simplest ever-expanding cosmologies, the Universe begins with a Big Bang and expands forever, with flat geometry on the largest scales, and with the Hubble constant *H* (*τ*) not varying with cosmic time [4,5] *τ* and always equal to its present value *H*0. As the Universe expands the Hubble flow carries mass-energy past the cosmological event horizon of our L-region of the Universe. But this "loss" is replaced by new positive mass-energy continually created within our L-region of the Universe *forever* at the rate of the Planck power and at the expense of our L-region's negative gravitational energy. This compensates

<sup>6</sup> (Re: Entry [48], Ref. [48]) Reference [48] shows that previous measurements of the acceleration of the Universe's expansion may require reconsideration, owing to discrepancies between visible-light and UV observations of type 1a

though, even if correct *in general*, its *details* still need to be understood.

theories of Special and General Relativity.

230 Recent Advances in Thermo and Fluid Dynamics

L-region thereof.

supernovae.

numerical coincidence.

**evolution merging into steady state?**

There is needed a mechanism whereby a sufficiently large fraction *f* of the Planck-power mass-energy input within the cosmological event horizon of our L-region, and of any L-region, of the Universe is produced in the form of hydrogen [52–57] (as in the original steady-state theory [21–23,33–35]), so that there will be fuel for stars [52–57]. Then there will *always* be stars [52–57], planets, and life — not only at the periphery but *even at the center* [58] of our island Universe [1] and likewise of every other island Universe [1] in the Multiverse [52–58]. Then the heat death predicted by the Second Law of Thermodynamics [59–63] will be thwarted not only at the periphery but *even at the center* [58] of our island Universe and likewise of every other island Universe in the Multiverse.7 The required sufficiently large fraction *f* is actually quite small. An order-of-magnitude estimate of the total number of stars within the cosmological event horizon [3] of our L-region of the Universe is <sup>∼</sup> <sup>10</sup><sup>22</sup> [64]. By the cosmological principle [51] our L-region of the Universe is nothing special, so there is no reason to suspect a substantially different total number of stars in other L-regions. The Sun's luminosity is *<sup>L</sup>*Sun <sup>=</sup> 3.828 <sup>×</sup> 1026 <sup>W</sup> <sup>≈</sup> <sup>10</sup>−26*P*Planck [56,57]. Thus if the average star were as luminous as the Sun then the total luminosity of <sup>∼</sup> 1022 stars would be *<sup>L</sup>*total <sup>∼</sup> <sup>10</sup>−26*P*Planck <sup>×</sup> 1022 <sup>∼</sup> <sup>10</sup>−4*P*Planck, implying that we require *<sup>f</sup>* <sup>∼</sup> *<sup>L</sup>*total/*P*Planck <sup>∼</sup> <sup>10</sup>−<sup>4</sup> [56,57,64]. But the average star is considerably less luminous than the Sun [56,57], so the best order-of-magnitude estimate is perhaps *<sup>L</sup>*total <sup>∼</sup> <sup>10</sup>−5*P*Planck, implying that we require only *<sup>f</sup>* <sup>∼</sup> <sup>10</sup>−<sup>5</sup> [56,57,64]. (Properties of the Sun, including its luminosity, are given in both conventional and geometrized units in the inside back cover of Ref. [12], and in conventional units in Appendix A in the inside front cover of Ref. [14] and in Table 8.1 on p. 219 of Ref. [10].) This small value *<sup>f</sup>* <sup>∼</sup> <sup>10</sup>−<sup>5</sup> is sufficient to sustain star formation *forever* not only at the periphery but *even at the center* [58] of our island Universe [1] and likewise of every other island Universe [1] in the Multiverse [52–58]. The remainder of the Planck-power input would be in forms other than hydrogen (perhaps traces of heavier elements, elementary particles of normal and/or dark matter, dark energy, etc.?).

But perhaps the *simplest* mode of Planck-power input is *initially* in the form of the *simplest* possible type of dark energy, corresponding to *positive constant* Λ — a *positive* cosmological *constant*. *Constancy* of Λ is required for *constancy* of Planck-power input initially in the form of Λ. *Positivity* of Λ seems to be required for *positivity* of Planck-power input being initially in the form of Λ, because negative Λ corresponds to contraction of space and hence to diminution of Λ-mass-energy. Thus the *simplest* possible type of dark energy, corresponding to *positive constant* Λ — a *positive* cosmological *constant* — is perhaps the type of dark energy that is most easily reconcilable with Planck-power input, in particular with

<sup>7</sup> (Re: Entry [63], Ref. [63]) Reference [63] considers various aspects of the Second Law of Thermodynamics and its relation to the arrow of time and to cosmology. Reference [63] was for sale at the 27th Texas Symposium on Relativistic Astrophysics, held at the Fairmont Hotel in Dallas, Texas, December 8–13, 2013.

*constancy* of Planck-power input. Moreover, *constant* Λ — a cosmological *constant* — is the *only, unique,* choice for Λ that can be put on the left-hand (geometry) side of Einstein's field equations without altering their symmetric and divergence-free form [65–67], "belonging to the field equations much as an additive constant belongs to an indefinite integral [65–67].8 Nevertheless the current trend is to put Λ on the right-hand (mass-energy-stress) side of Einstein's field equations, which allows more freedom [66]. But if Λ is put on the right-hand side "the rationale for its uniqueness then disappears: it no longer needs to be a divergence-free 'geometric' tensor, built solely from the *gµν* ... the geometric view of Λ ... is undoubtedly the simplest [66]". Thus we might speculate about a link between constancy of Λ as a (positive) cosmological constant [65–67] and constancy of (positive) Planck-power input: Perhaps Planck-power input occurs *initially* as (positive) cosmological-constant Λ, with *<sup>f</sup>* <sup>∼</sup> <sup>10</sup>−<sup>5</sup> thereof, then hopefully, somehow, via an as-yet-unknown mechanism, being transformed into hydrogen. It is important to note that — unlike equilibrium blackbody radiation — (positive) cosmological-constant-Λ dark energy seems to be at less than, indeed at far less than, maximum entropy. Thus there seems to be more than enough entropic "room" for *<sup>f</sup>* <sup>∼</sup> <sup>10</sup>−<sup>5</sup> of positive-cosmological-constant-<sup>Λ</sup> dark energy to decay into hydrogen, without requiring decay all the way to iron. Positive-cosmological-constant-Λ Planck-power input thus seems to offer the benefits but not the liabilities of the steady-state theory [21–23,33–35] [violation of mass-energy conservation without the C field (which has never been observed) and which also entails other difficulties with it [34,35] — recall the paragraph immediately following that containing Eq. (1)]. Positive cosmological-*constant* Λ also implies, or at least is consistent with, *constant H* (*τ*) = *H*<sup>0</sup> at all cosmic times *τ*, and hence a fixed size of our L-region, with its boundary (event horizon [2,3]) *R* (*τ*) always fixed at *R*<sup>0</sup> = *c*/*H*0. Thus to sum up this paragraph, the *simplest* model *overall* seems to entail (a) positive-cosmological-constant Λ, (b) Planck-power input *initially* as positive-cosmological-constant Λ at the expense of negative gravitational energy, with (c) *<sup>f</sup>* <sup>∼</sup> <sup>10</sup>−<sup>5</sup> of Planck-power input, then hopefully, somehow, via an as-yet-unknown mechanism, being transformed into hydrogen. We note that the most reliable and most recent astronomical and astrophysical observations and measurements as of this writing are consistent with *positive* cosmological-*constant*-Λ dark energy [68,69], indeed possibly or even probably *more* consistent with *positive* cosmological-*constant*-Λ dark energy than with any other alternative [68,69]. But, of course, this issue is far from being definitely decided [68,69]. Even though our main point in this chapter most naturally based on positive constant Λ, in Sects. 5–7 some other possibilities for Λ will be qualitatively considered.

We cannot help but notice that temperature fluctuations in the cosmic background radiation have a typical fractional magnitude of F ≈ <sup>10</sup>−<sup>5</sup> [70,71]. The *observed and measured* value F ≈ <sup>10</sup>−<sup>5</sup> [70,71] is obviously far more certain than the *speculated* value *<sup>f</sup>* <sup>∼</sup> <sup>10</sup>−5; hence the distinction between the ≈ symbol as opposed to the ∼ symbol. Although it is unlikely that there is a connection between F ≈ <sup>10</sup>−<sup>5</sup> [70,71] and *<sup>f</sup>* <sup>∼</sup> <sup>10</sup>−5, it doesn't seem to hurt if we at least mention this numerical concurrence — just in case there might be a connection.

But the following question arises: Even if there is Planck-power input, why is not *all* of it in a thermodynamically-most-probable maximum-entropy form such as (iron + equilibrium

<sup>8</sup> (Re: Entry [67], Ref. [67]) Reference [67] is cited in the passage from Ref. [2] that we cite in Entry [65].

blackbody radiation) and *none* of it as hydrogen — why is not *f* = 0 [52–57]? If this were the case then the heat death predicted by the Second Law of Thermodynamics [59–63] would *not* be thwarted even *with* Planck-power input. While we are not sure of an answer to this question, we can venture what *prima facie* at least seems to be a reasonable guess: (a) Planck-power input (if it exists) generates equal nonzero quantities of both positive mass-energy and negative gravitational energy starting from (zero positive energy + zero negative energy = zero total energy), and the entropy of (zero positive energy + zero negative energy = zero total energy) is *perforce* zero. (b) Planck-power input is a steady-state but nonequilibrium process that does not allow enough time for complete thermalization of the input from the initial value of zero entropy of (zero positive energy + zero negative energy = zero total energy) to the maximum possible positive entropy of (nonzero positive energy + nonzero negative energy = zero total energy) in a form such as (iron + equilibrium blackbody radiation). That is, Planck-power input is *kinetically rather than thermodynamically controlled* [72–77].9

*constancy* of Planck-power input. Moreover, *constant* Λ — a cosmological *constant* — is the *only, unique,* choice for Λ that can be put on the left-hand (geometry) side of Einstein's field equations without altering their symmetric and divergence-free form [65–67], "belonging to the field equations much as an additive constant belongs to an indefinite integral [65–67].8 Nevertheless the current trend is to put Λ on the right-hand (mass-energy-stress) side of Einstein's field equations, which allows more freedom [66]. But if Λ is put on the right-hand side "the rationale for its uniqueness then disappears: it no longer needs to be a divergence-free 'geometric' tensor, built solely from the *gµν* ... the geometric view of Λ ... is undoubtedly the simplest [66]". Thus we might speculate about a link between constancy of Λ as a (positive) cosmological constant [65–67] and constancy of (positive) Planck-power input: Perhaps Planck-power input occurs *initially* as (positive) cosmological-constant Λ, with *<sup>f</sup>* <sup>∼</sup> <sup>10</sup>−<sup>5</sup> thereof, then hopefully, somehow, via an as-yet-unknown mechanism, being transformed into hydrogen. It is important to note that — unlike equilibrium blackbody radiation — (positive) cosmological-constant-Λ dark energy seems to be at less than, indeed at far less than, maximum entropy. Thus there seems to be more than enough entropic "room" for *<sup>f</sup>* <sup>∼</sup> <sup>10</sup>−<sup>5</sup> of positive-cosmological-constant-<sup>Λ</sup> dark energy to decay into hydrogen, without requiring decay all the way to iron. Positive-cosmological-constant-Λ Planck-power input thus seems to offer the benefits but not the liabilities of the steady-state theory [21–23,33–35] [violation of mass-energy conservation without the C field (which has never been observed) and which also entails other difficulties with it [34,35] — recall the paragraph immediately following that containing Eq. (1)]. Positive cosmological-*constant* Λ also implies, or at least is consistent with, *constant H* (*τ*) = *H*<sup>0</sup> at all cosmic times *τ*, and hence a fixed size of our L-region, with its boundary (event horizon [2,3]) *R* (*τ*) always fixed at *R*<sup>0</sup> = *c*/*H*0. Thus to sum up this paragraph, the *simplest* model *overall* seems to entail (a) positive-cosmological-constant Λ, (b) Planck-power input *initially* as positive-cosmological-constant Λ at the expense of negative gravitational energy, with (c) *<sup>f</sup>* <sup>∼</sup> <sup>10</sup>−<sup>5</sup> of Planck-power input, then hopefully, somehow, via an as-yet-unknown mechanism, being transformed into hydrogen. We note that the most reliable and most recent astronomical and astrophysical observations and measurements as of this writing are consistent with *positive* cosmological-*constant*-Λ dark energy [68,69], indeed possibly or even probably *more* consistent with *positive* cosmological-*constant*-Λ dark energy than with any other alternative [68,69]. But, of course, this issue is far from being definitely decided [68,69]. Even though our main point in this chapter most naturally based on positive constant Λ, in

232 Recent Advances in Thermo and Fluid Dynamics

Sects. 5–7 some other possibilities for Λ will be qualitatively considered.

We cannot help but notice that temperature fluctuations in the cosmic background radiation have a typical fractional magnitude of F ≈ <sup>10</sup>−<sup>5</sup> [70,71]. The *observed and measured* value F ≈ <sup>10</sup>−<sup>5</sup> [70,71] is obviously far more certain than the *speculated* value *<sup>f</sup>* <sup>∼</sup> <sup>10</sup>−5; hence the distinction between the ≈ symbol as opposed to the ∼ symbol. Although it is unlikely that there is a connection between F ≈ <sup>10</sup>−<sup>5</sup> [70,71] and *<sup>f</sup>* <sup>∼</sup> <sup>10</sup>−5, it doesn't seem to hurt if we at least mention this numerical concurrence — just in case there might be a connection.

But the following question arises: Even if there is Planck-power input, why is not *all* of it in a thermodynamically-most-probable maximum-entropy form such as (iron + equilibrium

<sup>8</sup> (Re: Entry [67], Ref. [67]) Reference [67] is cited in the passage from Ref. [2] that we cite in Entry [65].

Thus even though, *thermodynamically*, Planck-power input should be in a maximum-entropy form such as (iron + equilibrium blackbody radiation), *kinetically* the reaction

zero positive energy + zero negative energy = zero total energy −→ nonzero positive energy + nonzero negative energy = zero total energy. (8)

occurs too quickly to allow thermodynamic equilibrium = maximum entropy to be attained. Yet even Planck-power input initially as positive-cosmological-constant Λ, with a fraction *f* ∼ 10−<sup>5</sup> of Planck-power input hopefully, somehow, via an as-yet-unknown mechanism, being transformed into hydrogen, entails *some* entropy increase. The entropy increase ∆*S* that it *does* entail is sufficient to render the probability of its reversal as per Boltzmann's relation between entropy and probability, Prob (∆*S*) = exp (−∆*S*/*k*), equal to zero for all practical purposes. Thus we are justified in placing only a forward arrow (no reverse arrow) at the beginning of the second line of Eq. (8). Thus Planck-power input entails enough entropy increase to stabilize it and prevent its reversal. But it occurs quickly enough to allow *kinetic control* [72–77] to prevent it from entailing *maximal* entropy increase.

To recapitulate out considerations thus far in Sect. 4: Perhaps the simplest possible Planck-power input is *initially* as positive-cosmological-constant Λ. *Positivity* of Λ is required for *positivity* of Planck-power input, and *constancy* of Λ is required for *constancy* of Planck-power input. *Constancy* of Λ is requisite for Λ to be most simply encompassed within Einstein's field equations [65–67], besides correlating with constancy of Planck-power input. Positive-cosmological-*constant* Λ also implies, or at least is consistent with, *constant H* (*τ*) = *H*<sup>0</sup> at all cosmic times *τ*, and hence a fixed size of our L-region, with its boundary

<sup>9</sup> (Re: Entries [72–77], Refs. [72–77]) Kinetic versus thermodynamic control is specifically discussed on pp. 41–42 of Ref. [72], p. 43 of Ref. [73], p. 35 of Ref. [74], pp. 311–312 of Ref. [76], and p. 438 of Ref. [77]. Kinetic versus thermodynamic control are contrasted in Sects. 2.18 and 19.11 of Ref. [75]. Helpful auxiliary material is provided in Sects. 2.15–2.17, 2.19–2.21, and 19.16 of Ref. [75]. Reference [73] does not render Ref. [72] obsolete, because Ref. [72] discusses aspects not discussed in Ref. [73], and vice versa. Likewise, Reference [77] does not render Ref. [76] obsolete, because Ref. [76] discusses aspects not discussed in Ref. [77], and vice versa.

(event horizon [2,3]) *<sup>R</sup>* (*τ*) always fixed at *<sup>R</sup>*<sup>0</sup> <sup>=</sup> *<sup>c</sup>*/*H*0. But we wish for a fraction *<sup>f</sup>* <sup>∼</sup> <sup>10</sup>−<sup>5</sup> of Planck-power input hopefully, somehow, via an as-yet-unknown mechanism, being transformed into *hydrogen*. *Hydrogen*, so that stars can have fuel. But why hydrogen? Why not a thermodynamically dead form such as (iron + equilibrium blackbody radiation)? Because kinetically, it would be much more difficult for positive-cosmological-constant Λ to be transformed into a complex atom such as iron than into the simplest one — hydrogen. Thus while *thermodynamic* control would favor iron, if *kinetic* control wins then hydrogen is favored [72–77]. Note that kinetic control is vital not only in initial creation of hydrogen, but also in then preserving hydrogen long enough for it to be of use. It is owing to *kinetic* control that the Sun and all other main-sequence stars fuse hydrogen only to helium, not to iron, and are restrained to doing so slowly enough to give them usefully-long lifetimes. Main-sequence fusion of hydrogen to iron is thermodynamically favored, but kinetically its rate of occurrence is for all practical purposes zero. Thus kinetic control wins, limiting main-sequence fusion to helium and at a slow enough rate to give stars usefully-long lifetimes [72–77]. Indeed it is owing to *kinetic* control that not only hydrogen, but also all other elements except iron, do not instantaneously decay to iron. Kinetic control may also argue against positive-cosmological-constant Λ being *completely* transformed into equilibrium blackbody radiation (without iron). A single hydrogen atom can be created at rest with respect to the comoving frame [7]. By contrast, to conserve momentum, at least two photons must be created simultaneously, which may impose a bottleneck that diminishes the rate of such a process kinetically. Hence *<sup>f</sup>* <sup>∼</sup> <sup>10</sup>−<sup>5</sup> of the Planck-power input in the positive-but-much-less-than-maximal-entropy form of hydrogen may at least *prima facie* seem plausible. Again it doesn't seem to hurt to at least mention the numerical concurrence between F ≈ <sup>10</sup>−<sup>5</sup> [70,71] and *<sup>f</sup>* <sup>∼</sup> <sup>10</sup>−5, even if any connection is unlikely.

Note that a zero value for the initial entropy for would also obtain if Planck-power input were *ex nihilo* [21–25] or at the expense of a negative-energy C field (despite its never having been observed and its other difficulties [34,35]) or other negative-energy field rather than at the expense of negative gravitational potential energy: the entropy of (zero positive energy + zero negative energy = zero total energy) would still *perforce* be zero. Thus our considerations of this Sect. 4, including that of dominance of kinetic over thermodynamic control [72–77], would still be applicable.

Our L-region and O-region clearly manifest evolutionary behavior, for example increasing metallicity [52–55] and a decreasing rate of star formation [52–55]. But our Planck-power hypothesis seems to suggest that its evolutionary behavior could gradually merge towards steady-state behavior. Early in the history of our L-region and O-region, star formation occurred at a much faster rate than now, and stars were on the average much more massive and hence *very* much faster-burning [hydrogen-burning rate <sup>∼</sup> (mass of star)3]. Thus hydrogen was consumed faster than a conversion of *<sup>f</sup>* <sup>∼</sup> <sup>10</sup>−<sup>5</sup> of Planck-power input could replace it: Stars were burning capital in addition to (Planck-power) income indeed more capital than income. But with decreasing rate of star formation and decreasing average stellar mass, perhaps a steady-state balance between hydrogen consumption and its replacement via *<sup>f</sup>* <sup>∼</sup> <sup>10</sup>−<sup>5</sup> of Planck-power input could be approached, with stars living solely on (Planck-power) income. Merging of evolutionary towards steady-state behavior could already be beginning or could have even begun in the very recent past with as yet no or at most very limited observational evidence that might be suggestive of it. If such merging exists then both metallicity and star formation rate could stabilize in the future. Perhaps they even could already now be stabilizing or have even already begun stabilizing in the very recent past with as yet no or at most very limited observational evidence that might be suggestive of such stabilization in particular, or of such merging in general. This stabilization, if it exists, would require only a small fraction *<sup>f</sup>* <sup>∼</sup> <sup>10</sup>−<sup>5</sup> of Planck-power as hydrogen to maintain the current status quo in our L-region and O-region. This would allow star formation to continue forever not merely at the peripheries of island Universes, but *even in their central regions* [58]. We note that there is observational evidence that might at least be suggestive of "unexplained" hydrogen [78], which perhaps might qualify as such very limited suggestive observational evidence of merging towards steady-state behavior [78].

(event horizon [2,3]) *<sup>R</sup>* (*τ*) always fixed at *<sup>R</sup>*<sup>0</sup> <sup>=</sup> *<sup>c</sup>*/*H*0. But we wish for a fraction *<sup>f</sup>* <sup>∼</sup> <sup>10</sup>−<sup>5</sup> of Planck-power input hopefully, somehow, via an as-yet-unknown mechanism, being transformed into *hydrogen*. *Hydrogen*, so that stars can have fuel. But why hydrogen? Why not a thermodynamically dead form such as (iron + equilibrium blackbody radiation)? Because kinetically, it would be much more difficult for positive-cosmological-constant Λ to be transformed into a complex atom such as iron than into the simplest one — hydrogen. Thus while *thermodynamic* control would favor iron, if *kinetic* control wins then hydrogen is favored [72–77]. Note that kinetic control is vital not only in initial creation of hydrogen, but also in then preserving hydrogen long enough for it to be of use. It is owing to *kinetic* control that the Sun and all other main-sequence stars fuse hydrogen only to helium, not to iron, and are restrained to doing so slowly enough to give them usefully-long lifetimes. Main-sequence fusion of hydrogen to iron is thermodynamically favored, but kinetically its rate of occurrence is for all practical purposes zero. Thus kinetic control wins, limiting main-sequence fusion to helium and at a slow enough rate to give stars usefully-long lifetimes [72–77]. Indeed it is owing to *kinetic* control that not only hydrogen, but also all other elements except iron, do not instantaneously decay to iron. Kinetic control may also argue against positive-cosmological-constant Λ being *completely* transformed into equilibrium blackbody radiation (without iron). A single hydrogen atom can be created at rest with respect to the comoving frame [7]. By contrast, to conserve momentum, at least two photons must be created simultaneously, which may impose a bottleneck that diminishes the rate of such a process kinetically. Hence *<sup>f</sup>* <sup>∼</sup> <sup>10</sup>−<sup>5</sup> of the Planck-power input in the positive-but-much-less-than-maximal-entropy form of hydrogen may at least *prima facie* seem plausible. Again it doesn't seem to hurt to at least mention the numerical concurrence between F ≈ <sup>10</sup>−<sup>5</sup> [70,71] and *<sup>f</sup>* <sup>∼</sup> <sup>10</sup>−5, even if any connection is unlikely. Note that a zero value for the initial entropy for would also obtain if Planck-power input were *ex nihilo* [21–25] or at the expense of a negative-energy C field (despite its never having been observed and its other difficulties [34,35]) or other negative-energy field rather than at the expense of negative gravitational potential energy: the entropy of (zero positive energy + zero negative energy = zero total energy) would still *perforce* be zero. Thus our considerations of this Sect. 4, including that of dominance of kinetic over thermodynamic control [72–77],

Our L-region and O-region clearly manifest evolutionary behavior, for example increasing metallicity [52–55] and a decreasing rate of star formation [52–55]. But our Planck-power hypothesis seems to suggest that its evolutionary behavior could gradually merge towards steady-state behavior. Early in the history of our L-region and O-region, star formation occurred at a much faster rate than now, and stars were on the average much more massive and hence *very* much faster-burning [hydrogen-burning rate <sup>∼</sup> (mass of star)3]. Thus hydrogen was consumed faster than a conversion of *<sup>f</sup>* <sup>∼</sup> <sup>10</sup>−<sup>5</sup> of Planck-power input could replace it: Stars were burning capital in addition to (Planck-power) income indeed more capital than income. But with decreasing rate of star formation and decreasing average stellar mass, perhaps a steady-state balance between hydrogen consumption and its replacement via *<sup>f</sup>* <sup>∼</sup> <sup>10</sup>−<sup>5</sup> of Planck-power input could be approached, with stars living solely on (Planck-power) income. Merging of evolutionary towards steady-state behavior could already be beginning or could have even begun in the very recent past with as yet no or at most very limited observational evidence that might be suggestive of it. If such merging exists then both metallicity and star formation rate could stabilize in the future.

would still be applicable.

234 Recent Advances in Thermo and Fluid Dynamics

It should perhaps be re-emphasized that even Planck-power input as hydrogen entails *some* entropy increase and therefore is thermodynamically irreversible, consistently with the Second Law of Thermodynamics while still thwarting the heat death. The heat death is thus thwarted via *dilution* of entropy as an island Universe [1] expands indefinitely, which is consistent with the Second Law [59–63] — not via *destruction* of entropy, which is not: Planck-power input as hydrogen represents input at *positive but far less than maximum* entropy. Thus Planck-power input (if it exists) defeats the heat death predicted by the Second Law of Thermodynamics [59–63] even though it does not defeat the Second Law itself.

Thus with only *<sup>f</sup>* <sup>∼</sup> <sup>10</sup>−<sup>5</sup> of the Planck-power input as hydrogen, the heat death predicted by the Second Law of Thermodynamics [59–63] of our L-region of our island Universe [1], and likewise of any L-region of any island Universe [1], is thwarted *forever*. The heat death is thwarted *forever* not only at the periphery but *even at the center* [58] of our and every other island Universe [1]. The heat death is thwarted consistently with, not in violation of, the Second Law of Thermodynamics [59–63]. Hubble flow export of entropy (along with mass-energy) out of our L-region of our island Universe [1], and likewise out of any L-region of any island Universe [1], as its expansion creates more volume *forever*, is compensated *forever* by creation of thermodynamically fresh but still positive-entropy mass-energy — most importantly, hopefully, the fraction *<sup>f</sup>* <sup>∼</sup> <sup>10</sup>−<sup>5</sup> thereof as hydrogen — via Planck-power input.

Steady-state balance between Planck-power input and Hubble-flow expansion of space can allow both the entropy density and the nongravitational mass-energy density in our L-region of our island Universe [1], and likewise in any L-region of any island Universe [1], to remain constant, even as the total entropy and nongravitational mass-energy of the entire island Universe increase indefinitely. As mass-energy creation at the rate of the Planck power and at the expense of negative gravitational energy is matched by mass-energy dilution via an island Universe's expanding space, so is entropy production matched by entropy dilution. Thus the negative-energy gravitational field of an island Universe is an inexhaustible fuel (positive mass-energy and negative-entropy = negentropy = less-than-maximum-entropy) source. Gravity is a bank that provides an infinite line of credit and never requires repayment [79]. Planck-power input draws on this infinite line of credit [79], which never runs out — indeed which can*not* run out. [Of course, if (positive) nongravitational mass-energy density remains constant, then so must (negative) gravitational energy density, if the balance of zero total energy is to be maintained. Thus Planck-power input if it exists is really equally of positive nongravitational mass-energy and negative gravitational energy simultaneously.]

Additional questions bearing on the Second Law of Thermodynamics will be discussed in Sects. 5–7. In this chapter, whether concerning Planck-power input or otherwise, we limit ourselves to considerations of thwarting the heat death within the restrictions of the Second Law. Nonetheless we note that the universal validity of the Second Law of Thermodynamics has been seriously questioned [80–84], albeit with the understanding that even if not universally valid at the very least it has a very wide range of validity [80–84].

There are two difficulties that should at least be briefly mentioned and, even if only briefly and only incompletely, also addressed. (i) In order for negative gravitational energy to balance positive mass-energy of a hydrogen atom (or of any other entity), a hydrogen atom (or other entity) newly created via Planck-power input would have to interact gravitationally *infinitely fast or instantaneously* [85,86] — and hence universally simultaneously [85,86] — with our *entire* L-region of the Universe within our cosmological event horizon [3]. But if a *signal* of mass-energy and/or information is not *transmitted*, no violation of relativity is required [85,86]. Perhaps this may be possible if, as suggested in the third paragraph of this Sect. 4, Planck-power input occurs *initially* as positive-cosmological-constant <sup>Λ</sup> [65–67], with *<sup>f</sup>* <sup>∼</sup> <sup>10</sup>−<sup>5</sup> thereof, then hopefully, somehow, via an as-yet-unknown mechanism, being transformed into hydrogen. Perhaps the gravitational interaction of positive-cosmological-constant Λ [65–67], and thence of hydrogen atoms (and/or other entities) newly created therefrom via Planck-power input can be instantaneously "*rubber-stamped*" onto our entire L-region at once, rather than being *transmitted* as a "*signal*" from one place to another within our L-region. (ii) *Even if* an interaction, or any other process such as "rubber-stamping," can be infinitely fast or instantaneous — and hence universally simultaneous — it can be so in only *one* reference frame [86]. A superluminal phenomenon, even be it only the motion of a geometric point that possesses no mass-energy and carries no information (for example the intersection point of scissors blades) [86], can be infinitely fast and hence instantaneous — universally simultaneous — in only one reference frame [86] (as a subluminal phenomenon can be infinitely slow — at rest — in only one reference frame [86]).<sup>10</sup> But there is a natural choice for this frame: The comoving frame [7], in which the cosmic background radiation and Hubble flow are isotropic [7], even if not an absolute rest frame, is at least a preferred rest frame [87], indeed *the* preferred rest frame [87], of our L-region of the Universe. If any one reference frame can claim to be preferred, it is the comoving frame [7,87]. Since by the cosmological principle [51] there is nothing special about our L-region of the Universe, the same likewise obtains in any other L-region thereof. The existence of this universal preferred frame [7,87] implies the existence of a preferred, perhaps even absolute, cosmic time *τ* [4,5,87]. A clock in the comoving frame measures cosmic time *τ* [4,5,87] — the *longest* possible elapsed time from the Big Bang until now (and also the *longest* possible elapsed time from the Big Bang to the Big Crunch in an oscillating cosmology [16,88–94]) — clocks in all other frames measure *shorter* elapsed times [4,5,88–94].11 A clock in the comoving frame also measures the *longest* possible elapsed time ∆*τ* corresponding to a given decrease in the temperature of the cosmic

<sup>10</sup> (Re: Entry [86], Ref. [2]) Special Relativity permits arbitrarily fast superluminal phenomena that transmit no mass-energy or information, as well as mutual velocities up to 2*c*: see pp. 56 and 70 of Ref. [2]. Section 2.10 of Ref. [2] states that the speed *U* of transmission of information must not exceed *c* if violation of causality is to be prevented in Special Relativity. But Eqs. (2.21) and (2.22) in Sect. 2.10 of Ref. [2] at least suggest the possibility that Special Relativity may be consistent with a somewhat less conservative limit, namely *<sup>U</sup>* <sup>≤</sup> *<sup>c</sup>*2/*v*, where *<sup>v</sup>* is the relative velocity between the transmitter and receiver. Of course to guarantee causality Nature must then have a method to checkmate any attempt by the transmitter and/or receiver to "cheat" by increasing *v* while a signal is en route.

<sup>11</sup> (Re: Entry [88], Ref. [2]) The following is a near-quote from p. 402 of Ref. [2]: "Though unlikely to represent the actual Universe (according to present data) the oscillating-Universe model is interesting in itself."

background radiation (or to a given increase in this temperature during the contracting phase in an oscillating cosmology). This *longest* possible elapsed time *is* cosmic time [4,5,87]. A clock moving at velocity *v relative to the comoving frame* [7,87] measures times *shorter* by a ratio of <sup>1</sup> <sup>−</sup> *<sup>v</sup>*2/*c*<sup>2</sup>1/2 [7,87]. Thus the existence of this universal preferred frame and hence of cosmic time [4,5] weakens [87] the concept of relativity of simultaneity [85] as obtains within "the featureless vacuum of Special Relativity" [4,5,85–87]: Events, *even if spatially separated*, can be considered *absolutely* simultaneous if they occur *when* — with "when" having an *absolute* meaning — the cosmic background radiation as observed in the comoving frame has the same temperature, this temperature currently decreasing monotonically with increasing cosmic time *τ* since the Big Bang [4,5,87].12 (Simultaneity of *non*-spatially-separated events is absolute even in Special Relativity [85].) Also, the contribution to the total nongravitational mass of our L-region of the Universe of a body of rest-mass [95] *m* is equal to *m* only if it is at rest in the comoving frame; if it moves at velocity *v* relative to the comoving frame [7,87] then its contribution is *m* <sup>1</sup> <sup>−</sup> *<sup>v</sup>*2/*c*<sup>2</sup><sup>−</sup>1/2 [7,87]. For a zero-rest-mass particle the contribution is *m* = *E*/*c*<sup>2</sup> where *E* is its energy as measured in the comoving frame (for example *m* = *E*/*c*<sup>2</sup> = *hν*/*c*<sup>2</sup> for a photon of frequency *ν* as measured in the comoving frame). Thus the total nongravitational mass-energy *M*<sup>0</sup> of our L-region as per Eq. (1) is that measured *with respect to the comoving frame*.

the Second Law. Nonetheless we note that the universal validity of the Second Law of Thermodynamics has been seriously questioned [80–84], albeit with the understanding that even if not universally valid at the very least it has a very wide range of validity [80–84]. There are two difficulties that should at least be briefly mentioned and, even if only briefly and only incompletely, also addressed. (i) In order for negative gravitational energy to balance positive mass-energy of a hydrogen atom (or of any other entity), a hydrogen atom (or other entity) newly created via Planck-power input would have to interact gravitationally *infinitely fast or instantaneously* [85,86] — and hence universally simultaneously [85,86] — with our *entire* L-region of the Universe within our cosmological event horizon [3]. But if a *signal* of mass-energy and/or information is not *transmitted*, no violation of relativity is required [85,86]. Perhaps this may be possible if, as suggested in the third paragraph of this Sect. 4, Planck-power input occurs *initially* as positive-cosmological-constant <sup>Λ</sup> [65–67], with *<sup>f</sup>* <sup>∼</sup> <sup>10</sup>−<sup>5</sup> thereof, then hopefully, somehow, via an as-yet-unknown mechanism, being transformed into hydrogen. Perhaps the gravitational interaction of positive-cosmological-constant Λ [65–67], and thence of hydrogen atoms (and/or other entities) newly created therefrom via Planck-power input can be instantaneously "*rubber-stamped*" onto our entire L-region at once, rather than being *transmitted* as a "*signal*" from one place to another within our L-region. (ii) *Even if* an interaction, or any other process such as "rubber-stamping," can be infinitely fast or instantaneous — and hence universally simultaneous — it can be so in only *one* reference frame [86]. A superluminal phenomenon, even be it only the motion of a geometric point that possesses no mass-energy and carries no information (for example the intersection point of scissors blades) [86], can be infinitely fast and hence instantaneous — universally simultaneous — in only one reference frame [86] (as a subluminal phenomenon can be infinitely slow — at rest — in only one reference frame [86]).<sup>10</sup> But there is a natural choice for this frame: The comoving frame [7], in which the cosmic background radiation and Hubble flow are isotropic [7], even if not an absolute rest frame, is at least a preferred rest frame [87], indeed *the* preferred rest frame [87], of our L-region of the Universe. If any one reference frame can claim to be preferred, it is the comoving frame [7,87]. Since by the cosmological principle [51] there is nothing special about our L-region of the Universe, the same likewise obtains in any other L-region thereof. The existence of this universal preferred frame [7,87] implies the existence of a preferred, perhaps even absolute, cosmic time *τ* [4,5,87]. A clock in the comoving frame measures cosmic time *τ* [4,5,87] — the *longest* possible elapsed time from the Big Bang until now (and also the *longest* possible elapsed time from the Big Bang to the Big Crunch in an oscillating cosmology [16,88–94]) — clocks in all other frames measure *shorter* elapsed times [4,5,88–94].11 A clock in the comoving frame also measures the *longest* possible elapsed time ∆*τ* corresponding to a given decrease in the temperature of the cosmic

236 Recent Advances in Thermo and Fluid Dynamics

<sup>10</sup> (Re: Entry [86], Ref. [2]) Special Relativity permits arbitrarily fast superluminal phenomena that transmit no mass-energy or information, as well as mutual velocities up to 2*c*: see pp. 56 and 70 of Ref. [2]. Section 2.10 of Ref. [2] states that the speed *U* of transmission of information must not exceed *c* if violation of causality is to be prevented in Special Relativity. But Eqs. (2.21) and (2.22) in Sect. 2.10 of Ref. [2] at least suggest the possibility that Special Relativity may be consistent with a somewhat less conservative limit, namely *<sup>U</sup>* <sup>≤</sup> *<sup>c</sup>*2/*v*, where *<sup>v</sup>* is the relative velocity between the transmitter and receiver. Of course to guarantee causality Nature must then have a method to checkmate any attempt by the transmitter and/or receiver to "cheat" by increasing *v* while a signal is en

<sup>11</sup> (Re: Entry [88], Ref. [2]) The following is a near-quote from p. 402 of Ref. [2]: "Though unlikely to represent the

actual Universe (according to present data) the oscillating-Universe model is interesting in itself."

route.

It should be noted that these two difficulties (i) and (ii) discussed in the immediately preceding paragraph [85,86] also plague Universes created via the Everett interpretation of quantum mechanics [96–98], provided that creation of Everett Universes [96–98] is required to obey mass-energy conservation (no creation of mass-energy *ex nihilo* [21–25]). The creation of Everett Universes [96–98] with no higher entropy (or entropy density if they are infinite) than that of their precursor Universe could perhaps obtain for reasons similar to Planck-power input into our L-region being at positive but less-than-maximum entropy as per our considerations in this Sect. 4 — perhaps most importantly kinetic control winning over thermodynamic control [72–77].

#### **5. The Planck power: One-time and two-time low-entropy boundary conditions, and minimal Boltzmann brains**

As discussed in Sects. 3 and 4 (recall especially the third paragraph of Sect. 4), the simplest model of Planck-power input entails a fixed positive cosmological constant Λ. Also, from the viewpoint of General Relativity [65–67], a fixed cosmological constant Λ is the simplest choice for Λ [65–67]. Yet we should also consider other possibilities [66].

False-vacuum high-energy-density-scalar-field regions — the inflaton field — of the Multiverse separating island Universes [1] inflate much faster than they decay to non-inflating true-vacuum regions. Hence while inflation had a beginning once begun it is eternal [99]. Within island Universes high-cosmological-"constant" regions play essentially the same role that inflationary regions play between island Universes: they double in size much faster than their half-life against decay, so each island Universe expands forever, albeit more slowly than inflationary regions separating island Universes [1,100]. Yet the

<sup>12</sup> (Re: Entry [87]) The phrase "the featureless vacuum of Special Relativity" is a quote from a very thoughtful and insightful letter from Dr. Wolfgang Rindler, most probably in the 1990s, in reply to a question that I raised concerning relativity of simultaneity.

cosmological "constant" is not high everywhere in an island Universe [1]; in L-regions and O-regions such as ours regions it is sedate. As decay of the inflaton field gives birth to island Universes, within each island Universe decay of high-cosmological-constant-field regions gives birth to new sedate L-regions and O-regions such as ours. In these sedate L-regions and O-regions, the cosmological "constant" may eventually decay to negative values, resulting in a Big Crunch — and perhaps oscillatory behavior, even as entire island Universes expand forever and the spaces between them expand forever even faster. For simplicity, as noted in the first paragraph of this Sect. 5, we thus far in this chapter (except for brief parenthetical remarks in the second-to-last paragraph of Sect. 4) considered our L-region and O-region to be ever-expanding [more often than not assuming constant *H* (*τ*) = *H*<sup>0</sup> for consistency with a fixed positive cosmological constant Λ and for maximum simplicity]. We now offer a few brief speculations concerning the role of the Planck power if the Universe, or at least our L-region and O-region, is oscillating with two-time low-entropy boundary conditions at the Big Bang and at the Big Crunch [16,88–94,101–105]. It is important to note that there exist oscillating cosmological models, including those with thermodynamic rejuvenation, both in conjunction with and apart from the concept of an inflationary Multiverse [16,88–94,101–105]. Some of these models [16,88,89,101–104] were developed well before inflationary cosmology, when our *observable* Universe or O-region was construed to be the *entire* Universe or at least a major fraction thereof. Within inflationary cosmology, it has been theorized based on quantum considerations that the probability that an oscillating L-region and O-region will have a given lifetime *τ*osc from Big Bang to Big Crunch decreases towards zero with increasing *τ*osc such that *τ*osc = ∞ — a nonoscillating L-region and O-region — is impossible [88,90], even as entire island Universes expand forever and the spaces between them expand forever even faster. Based on this theoretical analysis [88–90] the dark energy *must* eventually switch sign and become attractive instead of repulsive [88–90]: Hence according to this theoretical analysis [88–90] not only the current acceleration of our L-region's and O-region's expansion but even the expansion itself *must* be a passing fad — our L-region and O-region *must* be oscillatory [88–90].13 Shortly we will discuss Dr. Roger Penrose's central point [61,62] concerning entropy in the context of both ever-expanding and oscillatory behavior.

If Planck-power input is positive when our L-region expands, could it be negative if and when it contracts? Could this reduce or at least help to reduce the (nongravitational) mass-energy, and hence also entropy, during contraction, possibly to zero, by the time of the Big Crunch? If so, could a singularity at the Big Crunch thereby be evaded, thus ensuring a new thermodynamically fresh Big Bang to begin a new cycle? Moreover, since the Planck power (whether or not divided by *c*2) does *not* contain ¯*h*, but only *G* and *c*, would or at least might this evading of a Big Crunch singularity be a *classical* process *in*dependent of

<sup>13</sup> (Re: Entries [89]–[94], Refs. [1], [12], and [90]–[93]) The analysis showing that our L-region and O-region must be oscillatory is discussed qualitatively in the passages from Ref. [1] cited in Entry [89], with more technical discussions provided in Ref. [90]. At the 27th Texas Symposium on Relativistic Astrophysics, held at the Fairmont Hotel in Dallas, Texas, December 8–13, 2013, I asked Dr. Michael Turner about the theory discussed in Entry [89] and Ref. [90], according to which the Universe, or at least our L-region and O-region, *must* be oscillatory, also mentioning Entry [89] and Ref. [90]. Dr. Turner is familiar with the passages from Ref. [1] cited in Entry [89] and with Ref. [90], but nevertheless seemed to favor an ever-expanding Universe. An alternative model of an oscillating Universe is discussed in the work in Ref. [63] cited in Entry [92]. In the alternative model of an oscillating Universe investigated in this work, even a Big Rip is shown to be consistent with and indeed part of an oscillating Universe's life cycle. The work in Ref. [63] cited in Entry [93] considers related issues. Also, we should mention that an oscillatory Universe is closer to Einstein's conception of cosmology than a nonoscillatory one. A closed oscillating Universe with Λ = 0, similar to that considered by Dr. Albert Einstein in the early 1930s, is discussed in the material from Ref. [12] cited in Entry [94].

quantum effects, if not absolutely then at least via opposing quantum effects canceling out, as ¯*h* cancels out in the division *P*Planck = *E*Planck/*t*Planck [12–15]? Note again that perhaps similar canceling out obtains with respect to the Planck speed *l*Planck/*t*Planck = *c*: *c* is the fundamental speed in the classical (nonquantum) theories of Special and General Relativity.

cosmological "constant" is not high everywhere in an island Universe [1]; in L-regions and O-regions such as ours regions it is sedate. As decay of the inflaton field gives birth to island Universes, within each island Universe decay of high-cosmological-constant-field regions gives birth to new sedate L-regions and O-regions such as ours. In these sedate L-regions and O-regions, the cosmological "constant" may eventually decay to negative values, resulting in a Big Crunch — and perhaps oscillatory behavior, even as entire island Universes expand forever and the spaces between them expand forever even faster. For simplicity, as noted in the first paragraph of this Sect. 5, we thus far in this chapter (except for brief parenthetical remarks in the second-to-last paragraph of Sect. 4) considered our L-region and O-region to be ever-expanding [more often than not assuming constant *H* (*τ*) = *H*<sup>0</sup> for consistency with a fixed positive cosmological constant Λ and for maximum simplicity]. We now offer a few brief speculations concerning the role of the Planck power if the Universe, or at least our L-region and O-region, is oscillating with two-time low-entropy boundary conditions at the Big Bang and at the Big Crunch [16,88–94,101–105]. It is important to note that there exist oscillating cosmological models, including those with thermodynamic rejuvenation, both in conjunction with and apart from the concept of an inflationary Multiverse [16,88–94,101–105]. Some of these models [16,88,89,101–104] were developed well before inflationary cosmology, when our *observable* Universe or O-region was construed to be the *entire* Universe or at least a major fraction thereof. Within inflationary cosmology, it has been theorized based on quantum considerations that the probability that an oscillating L-region and O-region will have a given lifetime *τ*osc from Big Bang to Big Crunch decreases towards zero with increasing *τ*osc such that *τ*osc = ∞ — a nonoscillating L-region and O-region — is impossible [88,90], even as entire island Universes expand forever and the spaces between them expand forever even faster. Based on this theoretical analysis [88–90] the dark energy *must* eventually switch sign and become attractive instead of repulsive [88–90]: Hence according to this theoretical analysis [88–90] not only the current acceleration of our L-region's and O-region's expansion but even the expansion itself *must* be a passing fad — our L-region and O-region *must* be oscillatory [88–90].13 Shortly we will discuss Dr. Roger Penrose's central point [61,62]

238 Recent Advances in Thermo and Fluid Dynamics

concerning entropy in the context of both ever-expanding and oscillatory behavior.

If Planck-power input is positive when our L-region expands, could it be negative if and when it contracts? Could this reduce or at least help to reduce the (nongravitational) mass-energy, and hence also entropy, during contraction, possibly to zero, by the time of the Big Crunch? If so, could a singularity at the Big Crunch thereby be evaded, thus ensuring a new thermodynamically fresh Big Bang to begin a new cycle? Moreover, since the Planck power (whether or not divided by *c*2) does *not* contain ¯*h*, but only *G* and *c*, would or at least might this evading of a Big Crunch singularity be a *classical* process *in*dependent of

<sup>13</sup> (Re: Entries [89]–[94], Refs. [1], [12], and [90]–[93]) The analysis showing that our L-region and O-region must be oscillatory is discussed qualitatively in the passages from Ref. [1] cited in Entry [89], with more technical discussions provided in Ref. [90]. At the 27th Texas Symposium on Relativistic Astrophysics, held at the Fairmont Hotel in Dallas, Texas, December 8–13, 2013, I asked Dr. Michael Turner about the theory discussed in Entry [89] and Ref. [90], according to which the Universe, or at least our L-region and O-region, *must* be oscillatory, also mentioning Entry [89] and Ref. [90]. Dr. Turner is familiar with the passages from Ref. [1] cited in Entry [89] and with Ref. [90], but nevertheless seemed to favor an ever-expanding Universe. An alternative model of an oscillating Universe is discussed in the work in Ref. [63] cited in Entry [92]. In the alternative model of an oscillating Universe investigated in this work, even a Big Rip is shown to be consistent with and indeed part of an oscillating Universe's life cycle. The work in Ref. [63] cited in Entry [93] considers related issues. Also, we should mention that an oscillatory Universe is closer to Einstein's conception of cosmology than a nonoscillatory one. A closed oscillating Universe with Λ = 0, similar to that considered by Dr. Albert Einstein in the early 1930s, is discussed in the material from Ref. [12] cited

in Entry [94].

But *negative* Planck-power input requires entropy *reduction*. Hence it seems to require *two-time* low-entropy boundary conditions [101–105] at the Big Bang *and* at the Big Crunch — although two-time, or one-time, low-entropy boundary conditions can also obtain in a "traditional" oscillating Universe without any (positive or negative) Planck-power input [16,88,89,101–105] and without any (repulsive or attractive) dark energy or cosmological constant [16,88,89,101–105]. (In "traditional" oscillating cosmologies, one-time low-entropy boundary conditions imply increasing entropy from cycle to cycle, with each succeeding cycle being longer and reaching a larger maximum size [106,107]. In nonoscillating, ever-expanding, cosmologies, only one-time low-entropy boundary conditions can occur.) Two-time low-entropy boundary conditions require that not only the Big Bang but also the Big Crunch must be special [61,62,101–105]. But even *one-time* low-entropy boundary conditions at the Big Bang that are required for our L-region and O-region to exist as it is currently observed are equally special [61,62]. We will not address the question of whether or not the decrease in entropy during the contracting phase of an oscillating universal cycle imposed by two-time low-entropy boundary conditions [101–105] should be construed as contravening the Second Law of Thermodynamics. It could perhaps be argued that, *within* the restrictions of the Second Law, given two-time low-entropy boundary conditions [101–105] there is no *net* decrease in entropy for an entire cycle, or that two-time low-entropy boundary conditions [101–105] impose such a tight constraint on an oscillating Universe's journey through phase space that there is *no change* in entropy from the initial and final low value during a cycle. In accordance with the third-to-last paragraph of Sect. 4, in ideas developed in this chapter per se (as opposed to brief descriptions of ideas developed in cited references) we limit ourselves to considerations of thwarting the heat death within the restrictions of the Second Law of Thermodynamics. Nonetheless we again note that the universal validity of the second law has been seriously questioned [80–84], albeit with the understanding that even if not universally valid at the very least it has a very wide range of validity [80–84].

The reduction of the (nongravitational) mass-energy of a contracting Universe to zero or at least close to zero at the Big-Crunch/Big-Bang = Big Bounce event might thus be a way, although not necessarily the only way [101–105], to ensure zero entropy — the entropy of nothing is *perforce* zero — or at least low entropy at the Big Bounce. It should be noted that a zero- or at least low-entropy state at the Big Bounce is *imposed* in models with *two-time* low-entropy boundary conditions [101–105]. Thus the cosmic time [4,5] interval from the Big Bang to the Big Crunch can be incomparably shorter than and is totally unrelated to the Poincaré recurrence time [108].14

But whether low-entropy or equivalently high-negentropy boundary conditions are one-time, or two-time in oscillating cosmologies [61,62,101–105], Dr. Roger Penrose's central

<sup>14</sup> (Re: Entry [108], Ref. [5]) Contrary to what is stated on p. 192 of Ref. [5], in ever-expanding cosmological models Poincaré fluctuations on the scale of galactic — or smaller, indeed, even minimal-Boltzmann-brain — dimensions in spite of the dissipation due to expansion would *not* be expected, because the energy of starlight and ultimately *all* energy would be irrevocably lost from each and every galaxy into infinitely-expanding space and (without compensating input via a Planck-power or other mechanism, which is not considered on p. 192 of Ref. [5]) never replaced.

point [61,62] concerning entropy survives unscathed. This point had been brought out previously [108–110], but Dr. Roger Penrose's more modern analysis [61,62] takes into consideration inflation, which was not generally recognized prior to the late 1970s [108–110]. (See Sect. 6 concerning the connection with inflation.) This point begins with but does not end with recognizing that the L-region and O-region of our Universe are not merely special. They are *much more* special than they have to be — their negentropy is *much* greater than is required for conscious observers to exist. *By far* the *minimum* negentropy consistent with conscious observation would be that required for the *minimal* existence of a *single minimally*-conscious observer — *one and only one minimal Boltzmann brain* [111–118] with no body or sense organs, and with zero information including zero sensory input even if fictitious [112] and zero memory even if fictitious [113], save only the *minimal* information that one exists and is conscious and even this *minimal* information only for most *minimal* fleeting split-second of conscious existence consistent with recognition that one exists and is conscious, in an otherwise *maximum*-entropy and therefore dead L-region and O-region of our Universe — no other observers, no Sun or other stars, no Earth or other planets, no Darwinian evolution, no nothing (at any rate no nothing worthwhile). Input of *any* sensory information even if fictitious [112], and/or *any* memory even if fictitious [113], is incompatible with the *minimalness* of a Boltzmann brain required by Boltzmann's exponential relation between negentropy *σ* ≡ *S*max − *S* and its associated probability Prob (*σ*) = exp (−*σ*/*k*). [Note: *Negentropy σ* ≡ *S*max − *S* should not be confused with the *entropy change* ∆*S* associated with a given reaction or process introduced in the paragraph containing Eq. (8).] Even fictitious sensory input [112] or fictitious memory [113], as in a dream or in a simulated Universe, requires larger *σ* than none at all and hence is exponentially forbidden. Thus Boltzmann's exponential relation Prob (*σ*) = exp(−*σ*/*k*) allows not *any* Boltzmann brain but only a *minimal* Boltzmann brain — and only *one* of them. Based *solely* on Boltzmann's exponential relation Prob (*σ*) = exp(−*σ*/*k*) a lone *minimal* Boltzmann brain is not merely by far but *exponentially by far* the most probable type of observer to be and *exponentially by far* the most probable type of L-region and O-region of our Universe — or of any Universe in the Multiverse — to find oneself in: One should then expect not even fictitious sensory input [112], not even fictitious memory [113], but only the most fleeting split-second of conscious existence consistent with recognition that one is conscious.

But a basis *solely* on Boltzmann's relation Prob (*σ*) = exp(−*σ*/*k*) is incorrect, or at the very least incomplete. Boltzmann's relation Prob (*σ*) = exp(−*σ*/*k*) is valid *only* assuming thermodynamic equilibrium — that the ensemble of L-regions and O-regions corresponds to that at thermodynamic equilibrium. Probably the most powerful argument against this being the case is the *vast* disparity between our L-region and O-region that we *actually* observe and what one *would* observe as per the immediately preceding paragraph based *solely* on Boltzmann's relation Prob (*σ*) = exp(−*σ*/*k*). This disparity, the minimal-Boltzmann-brain disparity, by a factor of *<sup>O</sup>* <sup>∼</sup> 1010123 [61,62], utterly dwarfs the disparity by a factor in the range of *<sup>O</sup>* <sup>∼</sup> <sup>10</sup><sup>120</sup> [119] to *<sup>O</sup>* <sup>∼</sup> 10123 [120] between the observed and predicted values of the cosmological constant [119,120] — indeed it may utterly dwarf *all other disparities combined* [121]. These other disparities [121] relate mainly to the fundamental and effective laws of physics and physical constants requisite for the existence even of a minimal Boltzmann brain. Yet, even apart from viewpoints [122] that not all of them [121] may be significant, they are utterly dwarfed by the minimal-Boltzmann-brain disparity that obtains *even given* these requisite fundamental and effective laws of physics and physical constants.15 In contrast to minimal Boltzmann brains, we are sometimes dubbed "ordinary observers" [115–117] — but based *solely* on Boltzmann's relation Prob (*σ*) = exp(−*σ*/*k*) dubbing us even as *extra*ordinary observers would be a vast understatement. Indeed the same reasoning can be extended to *extra*ordinary observers. For, based *solely* on Boltzmann's Prob (*σ*) = exp(−*σ*/*k*), *exponentially by far* the most probable *extra*ordinary observer (say, a human with a typical life span) is a *minimal extra*ordinary observer, and only *one* of these per L-region or O-region. While *σ* required for a lone *minimal extra*ordinary observer greatly exceeds that required for a lone minimal Boltzmann brain, it is *still* utterly dwarfed by the actual *σ* of our L-region and O-region: The disparity of Prob (*σ*) = exp(−*σ*/*k*) between that corresponding to a lone *minimal extra*ordinary observer and that corresponding to our observed L-region and O-region is *still* by a factor of *<sup>O</sup>* <sup>∼</sup> 1010123 [61,62]. We are privileged to be not merely *minimal extra*ordinary observers but *super-extra*ordinary observers — more correctly *hyper-extra*ordinary observers — with an entire Universe to explore and enjoy [61,62].

point [61,62] concerning entropy survives unscathed. This point had been brought out previously [108–110], but Dr. Roger Penrose's more modern analysis [61,62] takes into consideration inflation, which was not generally recognized prior to the late 1970s [108–110]. (See Sect. 6 concerning the connection with inflation.) This point begins with but does not end with recognizing that the L-region and O-region of our Universe are not merely special. They are *much more* special than they have to be — their negentropy is *much* greater than is required for conscious observers to exist. *By far* the *minimum* negentropy consistent with conscious observation would be that required for the *minimal* existence of a *single minimally*-conscious observer — *one and only one minimal Boltzmann brain* [111–118] with no body or sense organs, and with zero information including zero sensory input even if fictitious [112] and zero memory even if fictitious [113], save only the *minimal* information that one exists and is conscious and even this *minimal* information only for most *minimal* fleeting split-second of conscious existence consistent with recognition that one exists and is conscious, in an otherwise *maximum*-entropy and therefore dead L-region and O-region of our Universe — no other observers, no Sun or other stars, no Earth or other planets, no Darwinian evolution, no nothing (at any rate no nothing worthwhile). Input of *any* sensory information even if fictitious [112], and/or *any* memory even if fictitious [113], is incompatible with the *minimalness* of a Boltzmann brain required by Boltzmann's exponential relation between negentropy *σ* ≡ *S*max − *S* and its associated probability Prob (*σ*) = exp (−*σ*/*k*). [Note: *Negentropy σ* ≡ *S*max − *S* should not be confused with the *entropy change* ∆*S* associated with a given reaction or process introduced in the paragraph containing Eq. (8).] Even fictitious sensory input [112] or fictitious memory [113], as in a dream or in a simulated Universe, requires larger *σ* than none at all and hence is exponentially forbidden. Thus Boltzmann's exponential relation Prob (*σ*) = exp(−*σ*/*k*) allows not *any* Boltzmann brain but only a *minimal* Boltzmann brain — and only *one* of them. Based *solely* on Boltzmann's exponential relation Prob (*σ*) = exp(−*σ*/*k*) a lone *minimal* Boltzmann brain is not merely by far but *exponentially by far* the most probable type of observer to be and *exponentially by far* the most probable type of L-region and O-region of our Universe — or of any Universe in the Multiverse — to find oneself in: One should then expect not even fictitious sensory input [112], not even fictitious memory [113], but only the most fleeting split-second of

conscious existence consistent with recognition that one is conscious.

disparity, by a factor of *<sup>O</sup>* <sup>∼</sup> 1010123

240 Recent Advances in Thermo and Fluid Dynamics

But a basis *solely* on Boltzmann's relation Prob (*σ*) = exp(−*σ*/*k*) is incorrect, or at the very least incomplete. Boltzmann's relation Prob (*σ*) = exp(−*σ*/*k*) is valid *only* assuming thermodynamic equilibrium — that the ensemble of L-regions and O-regions corresponds to that at thermodynamic equilibrium. Probably the most powerful argument against this being the case is the *vast* disparity between our L-region and O-region that we *actually* observe and what one *would* observe as per the immediately preceding paragraph based *solely* on Boltzmann's relation Prob (*σ*) = exp(−*σ*/*k*). This disparity, the minimal-Boltzmann-brain

the range of *<sup>O</sup>* <sup>∼</sup> <sup>10</sup><sup>120</sup> [119] to *<sup>O</sup>* <sup>∼</sup> 10123 [120] between the observed and predicted values of the cosmological constant [119,120] — indeed it may utterly dwarf *all other disparities combined* [121]. These other disparities [121] relate mainly to the fundamental and effective laws of physics and physical constants requisite for the existence even of a minimal Boltzmann brain. Yet, even apart from viewpoints [122] that not all of them [121] may be significant, they are utterly dwarfed by the minimal-Boltzmann-brain disparity that obtains *even given* these requisite fundamental and effective laws of physics and physical

[61,62], utterly dwarfs the disparity by a factor in

There are many arguments against Boltzmann-brain hypotheses [111–118]. Indeed, if there exist (a) *imposed* one-time low-entropy boundary conditions, (b) *imposed* two-time low-entropy boundary conditions [16,88–94,101–105] in an oscillating L-region and O-region, or (c) Planck-power (or other [21–25,33–35]) *imposed* low-entropy mass-energy input such as hydrogen in a nonoscillating one [78], then such *imposition* would *preclude* thermodynamic equilibrium. Indeed, given (b) or (c), thermodynamic equilibrium would not only be precluded but be precluded *forever*. Given (b) or (c), there would be no need to assume a decaying or finite-lived Universe [117] to help explain consistency with our observations. But even given (a) the heat death *σ* = 0 need *not* be the most probable *current* state of the L-region or O-region of our Universe and hence a *minimal* Boltzmann brain [111–118] need *not* be the most probable *current* observer therein, because at the *current* cosmic time decay to maximum entropy has not yet occurred. Since by the cosmological principle [51] our L-region and O-region are nothing special, this must likewise be true with respect to any L-region or O-region in our island Universe — and likewise with respect to those in any other island Universe in the Multiverse. Moreover, it has even been argued that low-entropy boundary conditions are *not* required to avoid minimal Boltzmann brains being exponentially by far the most probable type of observer, or even the most probable type of observer at all [116]. Also, it has been argued that special, i.e., low-entropy, conditions are not required at Big Bangs or Big Bounces [123,124]. [Clustering of matter at *t* = 0, which might typically be expected to increase entropy in the presence of gravity [125,126], does not do so because in this model [123,124] it is prevented owing to positive kinetic energy equaling negative gravitational energy in magnitude, so that the total energy (which in a Newtonian model excludes mass-energy) equals zero. But on pp. 3–4 of Ref. [124], friction, which generates entropy, is invoked during the time evolution of the system. Frictional damping, by degrading part of the macroscopic kinetic energy of any given pair of objects into microscopic kinetic energy (heat), facilitates their settling into a bound Keplerian-orbit state. But because friction thus generates entropy, this may correspond to a hidden, overlooked, pre-friction low-entropy assumption concerning the initial *t* = 0 state of this model [123,124] in either of its two directions of time [123,124]. But a Kepler pair can be formed without friction, for example via a three-body collision wherein a third body removes enough macroscopic

<sup>15</sup> (Re: Entry [122], Ref. [112]) Reference [112] provides discussions of a spectrum of numerous viewpoints concerning Multiverses and related topics, Dr. Steven Weinberg's viewpoint among this spectrum of viewpoints.

kinetic energy from the other two (without degrading any into heat) that they can settle into a bound Keplerian-orbit state.]

Low-entropy Planck-power (or other [21–25,33–35]) input such as hydrogen in nonoscillating cosmologies, or two-time low-entropy boundary conditions in oscillating ones [61,62,101–105], would enable our Universe — and likewise any Universe in the Multiverse — to *forever* thwart the heat death predicted by the Second Law of Thermodynamics. It should be noted that there also are other ways that the heat death can be thwarted: see, for example, Ref. [127]. Hopefully, one way or another, the heat death *is* thwarted in the *real* Universe, whether within an inflationary Multiverse [89–94,105] or otherwise [88,89,101–105,127].

Perhaps we should also note that the fraction *<sup>f</sup>* <sup>∼</sup> <sup>10</sup>−<sup>5</sup> of Planck-power input as hydrogen mentioned in Sect. 4 would maintain our L-region and O-region *much farther* from thermodynamic equilibrium than is required for existence of one and only one minimal-Boltzmann-brain. Thus *if* Planck-power input exists *then f* <sup>∼</sup> <sup>10</sup>−<sup>5</sup> rather than *f* = 0 can*not* be explained owing to our L-region being lucky: Boltzmann's exponential relation Prob (*σ*) = exp(−*σ*/*k*) on the one hand, and *σ* being a monotonically increasing function of *f* on the other, rules out any values of *σ* and *f* larger than the absolute minima that allow the existence of one and only one minimal-Boltzmann-brain obtaining by dumb luck. Thus *if* Planck-power input exists *then* perhaps there is an underlying principle or law of physics *requiring f* <sup>∼</sup> <sup>10</sup>−<sup>5</sup> not only in our L-region but in accordance with the cosmological principle [51] in every L-region of our, and also every other, island Universe [1] in the Multiverse [52–58].

#### **6. Dr. Roger Penrose's concerns: Both sides of the inflation issue**

We still must consider Dr. Roger Penrose's difficulty with inflation per se, the evidence for inflation not yet being totally beyond doubt [36–50]. Dr. Penrose has shown that, as per Boltzmann's relation between entropy and probability Prob (*σ*) = exp(−*σ*/*k*), the probability Prob 1, per "attempt," of creation of a Universe *as far from thermodynamic equilibrium as ours* without inflation, while extremely small, is nevertheless enormously larger than the probability Prob 2 with inflation. That is Prob 2 ≪ Prob 1 ≪ 1. At the 27th Texas Symposium on Relativistic Astrophysics [8], I asked Dr. Penrose the following question (I have streamlined the wording for this chapter): No matter how much smaller Prob 2 is than Prob 1 (so long as Prob 2, however miniscule even compared to the already miniscule Prob 1, is finitely greater than zero), inflation has to initiate *only once* — after initiating *once* it will then overwhelm all noninflationary regions. Dr. Penrose provided a concise and insightful reply [128], and also suggested that I re-read the relevant sections of his book, "The Road to Reality [15,61,62]" I did so. Dr. Penrose's key argument seems to be centered on squaring inflation with the Second Law of Thermodynamics. Dr. Penrose's central point, already briefly discussed in Sect. 5, begins with but does not end with recognizing that our L-region and O-region are *much* more thermodynamically atypical — with *much* lower entropy — than is required for us to exist even as *hyper-extra*ordinary observers, as opposed to only one of us as a minimal extraordinary observer, let alone only one of us as a minimal Boltzmann brain. Our L-region and O-region are thermodynamically extremely atypical not merely with respect to all possible L-regions and O-regions. They are thermodynamically extremely atypical even with respect to the extremely tiny subset of already thermodynamically extremely atypical L-regions and O-regions that allow us to exist as *hyper-extra*ordinary observers, as opposed to only one of us as a minimal extraordinary observer, let alone only one of us as a minimal Boltzmann brain. But now the link to inflation per se: As thermodynamically untypical as our L-region and O-region are today, they become as per Boltzmann's Prob (*σ*) = exp(−*σ*/*k*) *exponentially ever more* thermodynamically untypical as one considers them backwards in time [61,62]. Thus the disparity *today* by a factor of *<sup>O</sup>* <sup>∼</sup> 1010123 between the minimal-Boltzmann-brain or even minimal-extraordinary-observer hypothesis and observation becomes *exponentially* ever more severe as one considers our L-region and O-region backwards in time [61,62]. Thus the connection with inflation: Since inflation smooths out temperature differences and other nonuniformities, the very existence of temperature differences and other nonuniformities prior to inflation implies lower entropy than without such nonuniformities and hence renders the thermodynamic problem of origins worse not better [61,62]. In fact *exponentially* worse as per Boltzmann's *exponential* diminution Prob (*σ*) = exp(−*σ*/*k*) of probability with increasing negentropy ∆*S* [61,62]. As thermodynamically atypical and hence *exponentially* improbable as our Big Bang was, it must have been thermodynamically *more* atypical and hence *exponentially more* improbable if it was inflation-mediated than if it was not. This is the basic reason for Dr. Penrose's extremely strong inequality Prob 2 ≪ Prob 1. (We should, however, cite the remark that prior to inflation there may have been little mass-energy to thermalize [129].) Nevertheless my question still persists: In infinite time, or even in a sufficiently long finite time, even the most improbable event (so long as its probability, however miniscule, is finitely greater than zero) not merely can occur but *must* occur. It has been noted "that whatever physics permitted one Big Bang to occur might well permit many repetitions [130]." But suppose that Universe creations can occur via both noninflationary and inflationary physics. Even if because Prob 2 ≪ Prob 1 there first occurred an enormous but finite number *N*<sup>1</sup> of noninflationary Big Bangs yielding Universes *as far from thermodynamic equilibrium as ours*, so long as Prob 2, however miniscule even compared to the already miniscule Prob 1, is finitely greater than zero, after a sufficiently enormous but finite number *N*<sup>1</sup> of such noninflationary Universe creations inflation *must* initiate. And it need initiate *only once* to kick-start the inflationary Multiverse. Thereafter the inflationary Multiverse rapidly attains overwhelming dominance over the noninflationary one — with the number *N*<sup>2</sup> of inflation-mediated Big Bangs yielding Universes *as far from thermodynamic equilibrium as ours* henceforth overwhelming the number *N*<sup>1</sup> of noninflationary ones by an ever-increasing margin. To reiterate, no matter how much smaller Prob 2 is than Prob 1 (so long as Prob 2, however miniscule even compared to the already miniscule Prob 1, is finitely greater than zero), in infinite time, or even in a sufficiently long finite time, inflation *must* eventually initiate *once*, kick-starting the inflationary Multiverse, which henceforth becomes ever-increasingly overwhelmingly dominant over the noninflationary one. But even if inflation is eternal, it did have a beginning [99], and hence so did the inflationary Multiverse [99].

kinetic energy from the other two (without degrading any into heat) that they can settle into

Low-entropy Planck-power (or other [21–25,33–35]) input such as hydrogen in nonoscillating cosmologies, or two-time low-entropy boundary conditions in oscillating ones [61,62,101–105], would enable our Universe — and likewise any Universe in the Multiverse — to *forever* thwart the heat death predicted by the Second Law of Thermodynamics. It should be noted that there also are other ways that the heat death can be thwarted: see, for example, Ref. [127]. Hopefully, one way or another, the heat death *is* thwarted in the *real* Universe, whether within an inflationary Multiverse [89–94,105] or

Perhaps we should also note that the fraction *<sup>f</sup>* <sup>∼</sup> <sup>10</sup>−<sup>5</sup> of Planck-power input as hydrogen mentioned in Sect. 4 would maintain our L-region and O-region *much farther* from thermodynamic equilibrium than is required for existence of one and only one minimal-Boltzmann-brain. Thus *if* Planck-power input exists *then f* <sup>∼</sup> <sup>10</sup>−<sup>5</sup> rather than *f* = 0 can*not* be explained owing to our L-region being lucky: Boltzmann's exponential relation Prob (*σ*) = exp(−*σ*/*k*) on the one hand, and *σ* being a monotonically increasing function of *f* on the other, rules out any values of *σ* and *f* larger than the absolute minima that allow the existence of one and only one minimal-Boltzmann-brain obtaining by dumb luck. Thus *if* Planck-power input exists *then* perhaps there is an underlying principle or law of physics *requiring f* <sup>∼</sup> <sup>10</sup>−<sup>5</sup> not only in our L-region but in accordance with the cosmological principle [51] in every L-region of our, and also every other, island Universe [1]

**6. Dr. Roger Penrose's concerns: Both sides of the inflation issue**

We still must consider Dr. Roger Penrose's difficulty with inflation per se, the evidence for inflation not yet being totally beyond doubt [36–50]. Dr. Penrose has shown that, as per Boltzmann's relation between entropy and probability Prob (*σ*) = exp(−*σ*/*k*), the probability Prob 1, per "attempt," of creation of a Universe *as far from thermodynamic equilibrium as ours* without inflation, while extremely small, is nevertheless enormously larger than the probability Prob 2 with inflation. That is Prob 2 ≪ Prob 1 ≪ 1. At the 27th Texas Symposium on Relativistic Astrophysics [8], I asked Dr. Penrose the following question (I have streamlined the wording for this chapter): No matter how much smaller Prob 2 is than Prob 1 (so long as Prob 2, however miniscule even compared to the already miniscule Prob 1, is finitely greater than zero), inflation has to initiate *only once* — after initiating *once* it will then overwhelm all noninflationary regions. Dr. Penrose provided a concise and insightful reply [128], and also suggested that I re-read the relevant sections of his book, "The Road to Reality [15,61,62]" I did so. Dr. Penrose's key argument seems to be centered on squaring inflation with the Second Law of Thermodynamics. Dr. Penrose's central point, already briefly discussed in Sect. 5, begins with but does not end with recognizing that our L-region and O-region are *much* more thermodynamically atypical — with *much* lower entropy — than is required for us to exist even as *hyper-extra*ordinary observers, as opposed to only one of us as a minimal extraordinary observer, let alone only one of us as a minimal Boltzmann brain. Our L-region and O-region are thermodynamically extremely atypical not merely with respect to all possible L-regions and O-regions. They

a bound Keplerian-orbit state.]

242 Recent Advances in Thermo and Fluid Dynamics

otherwise [88,89,101–105,127].

in the Multiverse [52–58].

While in this Sect. 6 the focus is on thermodynamic issues concerning inflation, we note that Dr. Penrose also considers nonthermodynamic issues, specifically the flatness problem [131].

#### **7. Kinetic control versus both heat death and Boltzmann brains?**

A tentative solution to the thermodynamic problem of origins, namely dominance of kinetic over thermodynamic control [72–77] has already been proposed, as a reasonable guess, for the special cases of Planck-power input throughout Sect. 4 and Everett-Universe creation in the last paragraph of Sect. 4. We would now like to consider this issue somewhat more generally.

A generalized form of this *prima facie* perhaps reasonable guess might include: (a) Creation in general, by whatever method, both initial via Big Bang with or without inflation, etc. [26–31], via Everett [96–98], and sustained via Planck-power (or other [33–35]) input of equal nonzero quantities of both positive mass-energy and negative gravitational (or other negative [33–35]) energy starting from (zero positive energy + zero negative energy = zero total energy) entails an initial entropy of zero — the entropy of (zero positive energy + zero negative energy = zero total energy) is *perforce* zero. (b) Creation in general, by whatever method, both initial via Big Bang with or without inflation, etc. [26–31], via Everett [96–98], and sustained via Planck-power (or other [33–35]) input of equal nonzero quantities of both positive mass-energy and negative gravitational (or other negative [33–35]) energy starting from (zero positive energy + zero negative energy = zero total energy) is a nonequilibrium process. These processes do not allow enough time for complete thermalization of the input from the initial value of zero entropy of (zero positive energy + zero negative energy = zero total energy) to the maximum possible positive entropy of (nonzero positive energy + nonzero negative energy = zero total energy). Thus even though, *thermodynamically*, *exponentially* the most probable creation, initial or sustained, by any method, would yield a maximum-entropy Universe with *exponentially* the most probable observer a *minimal* Boltzmann brain, *kinetically* the reaction

zero positive energy + zero negative energy = zero total energy −→ nonzero positive energy + nonzero negative energy = zero total energy (8 (restated))

occurs too quickly to allow thermodynamic equilibrium = maximum entropy to be attained. Thus creation, initial or sustained, by whatever method, yields (nonzero positive energy + nonzero negative energy = zero total energy) at positive but far lass than maximum entropy, consistently with the Second Law of Thermodynamics but not with the heat death. Thus the basis of our proposed tentative solution to the thermodynamic problem of both initial and sustained-input origins: the reaction (rx) of Eq. (8) is *kinetically* rather than *thermodynamically* controlled [72–77]. This kinetic control does not defeat thermodynamics (specifically the Second Law of Thermodynamics) but it does defeat the heat death. Thus if the reaction of Eq. (8) is kinetically rather than thermodynamically controlled then the heat death is thwarted, but within the restrictions of the Second Law of Thermodynamics. This kinetic as opposed to thermodynamic control could similarly obtain at the initial creation in accordance with Eq. (8) of an oscillating Universe with two-time low-entropy boundary conditions at the Big Bang and at the Big Crunch [16,61,62,88–94,101–105], and in the case of creation *ex nihilo* [21–25].

**7. Kinetic control versus both heat death and Boltzmann brains?**

zero positive energy + zero negative energy = zero total energy

−→ nonzero positive energy + nonzero negative energy = zero total energy

occurs too quickly to allow thermodynamic equilibrium = maximum entropy to be attained. Thus creation, initial or sustained, by whatever method, yields (nonzero positive energy + nonzero negative energy = zero total energy) at positive but far lass than maximum entropy, consistently with the Second Law of Thermodynamics but not with the heat death. Thus the basis of our proposed tentative solution to the thermodynamic problem of both initial and sustained-input origins: the reaction (rx) of Eq. (8) is *kinetically* rather than *thermodynamically* controlled [72–77]. This kinetic control does not defeat thermodynamics (specifically the Second Law of Thermodynamics) but it does defeat the heat death. Thus if the reaction of Eq. (8) is kinetically rather than thermodynamically controlled then the heat death is thwarted, but within the restrictions of the Second Law of Thermodynamics. This kinetic as opposed to thermodynamic control could similarly obtain at the initial creation in accordance with Eq. (8) of an oscillating Universe with two-time low-entropy boundary conditions at

(8 (restated))

generally.

244 Recent Advances in Thermo and Fluid Dynamics

the reaction

A tentative solution to the thermodynamic problem of origins, namely dominance of kinetic over thermodynamic control [72–77] has already been proposed, as a reasonable guess, for the special cases of Planck-power input throughout Sect. 4 and Everett-Universe creation in the last paragraph of Sect. 4. We would now like to consider this issue somewhat more

A generalized form of this *prima facie* perhaps reasonable guess might include: (a) Creation in general, by whatever method, both initial via Big Bang with or without inflation, etc. [26–31], via Everett [96–98], and sustained via Planck-power (or other [33–35]) input of equal nonzero quantities of both positive mass-energy and negative gravitational (or other negative [33–35]) energy starting from (zero positive energy + zero negative energy = zero total energy) entails an initial entropy of zero — the entropy of (zero positive energy + zero negative energy = zero total energy) is *perforce* zero. (b) Creation in general, by whatever method, both initial via Big Bang with or without inflation, etc. [26–31], via Everett [96–98], and sustained via Planck-power (or other [33–35]) input of equal nonzero quantities of both positive mass-energy and negative gravitational (or other negative [33–35]) energy starting from (zero positive energy + zero negative energy = zero total energy) is a nonequilibrium process. These processes do not allow enough time for complete thermalization of the input from the initial value of zero entropy of (zero positive energy + zero negative energy = zero total energy) to the maximum possible positive entropy of (nonzero positive energy + nonzero negative energy = zero total energy). Thus even though, *thermodynamically*, *exponentially* the most probable creation, initial or sustained, by any method, would yield a maximum-entropy Universe with *exponentially* the most probable observer a *minimal* Boltzmann brain, *kinetically* But as we discussed in the third paragraph of Sect. 4, perhaps the *simplest* model of Planck-power input is *initially* in the form of the *simplest* possible type of dark energy, corresponding to *positive constant* Λ — a *positive* cosmological *constant* [65–67]. The *simplest* possible type of dark energy, corresponding to *positive constant* Λ — a *positive* cosmological *constant* [65–67] — is perhaps the type of dark energy that is most easily reconcilable with Planck-power input, in particular with *positive constant* Planck-power input. As we have mentioned, it is also simplest with respect to General Relativity [65–67], and it also implies, or at least is consistent with, *constant H* (*τ*) = *H*<sup>0</sup> at all cosmic times *τ*, and hence a fixed size of our L-region, with its boundary (event horizon [2,3]) *R* (*τ*) always fixed at *R*<sup>0</sup> = *c*/*H*0.

Let ∆*S*rx be the increase in entropy associated with the reaction (rx) of Eq. (8), with respect to our L-region. If 0 <sup>∆</sup>*S*rx *<sup>S</sup>*max <sup>∼</sup> 10123*k*, then, on the one hand, the strong inequality 0 ∆*S*rx ensures an equilibrium constant *K*eq = exp(∆*S*rx/*k*) sufficiently large that the reverse reaction is forbidden for all practical purposes, thus stabilizing creation [72–77]. Thus the strong inequality 0 ∆*S*rx justifies the placement of only a forward arrow (no reverse arrow) at the beginning of the second line of Eq. (8) [72–77]. On the other hand, the strong inequality <sup>∆</sup>*S*rx *<sup>S</sup>*max <sup>∼</sup> 10123*<sup>k</sup>* ensures against the doom and gloom that one would dread based *solely* on Boltzmann's relation Prob (∆*S*) = exp(−∆*S*/*k*). Note for example that even if ∆*S*rx = 10120*k* and hence for the reaction (rx) of Eq. (8) *K*eq = *e*<sup>10120</sup> , the entropy of our L-region is still only *<sup>O</sup>* <sup>∼</sup> <sup>10</sup>−<sup>3</sup> of that corresponding to thermodynamic equilibrium and hence still *<sup>σ</sup>* <sup>∼</sup> <sup>10</sup>123*k*. [References [73–77] express the equilibrium constant as *<sup>K</sup>*eq <sup>=</sup> exp(−∆*G*rx/*kT*), where ∆*G*rx is the Gibbs free energy change associated with a reaction in the special case of a system maintained at constant temperature *T* and constant ambient pressure. (To be precise, the ambient pressure must be maintained strictly constant during a reaction, but the temperature of the reactive system can vary in intermediate states so long as at the very least the initial and final states are at the same temperature, for this definition of ∆*G*rx to be valid [132–135].<sup>16</sup> In this special case, <sup>|</sup>∆*G*rx<sup>|</sup> is the maximum work that a reaction can yield if ∆*G*rx *<* 0 and the minimum work required to enable it if ∆*G*rx *>* 0. But in this special case ∆*G*rx = −*T*∆*S*rx where ∆*S*rx is the *total* entropy change of the (system + surroundings). Hence *K*eq = exp(−∆*G*rx/*kT*) is the corresponding special case of *K*eq = exp(∆*S*rx/*k*). In this chapter ∆*S* and ∆*S*rx are always taken to be *total* entropy changes of the entire Universe or at least of our L-region thereof.]

<sup>16</sup> (Re: Entry [132], Ref. [132]) One point: On p. 479 of Ref. [132], it is stated that in an adiabatic process all of the energy lost by a system can be converted to work, but that in a nonadiabatic process less than all of the energy lost by a system can be converted to work. But if the entropy of a system undergoing a nonadiabatic process *increases*, then *more* than all of the energy lost by this system can be converted to work, because energy extracted from the surroundings can then also contribute to the work output. In some such cases positive work output can be obtained at the expense of the surroundings even if the change in a system's energy is *zero*, indeed even if a system *gains* energy. Examples: (a) Isothermal expansion of an ideal gas is a thermodynamically spontaneous process, yielding work even though the energy change of the ideal gas is *zero*. (b) Evaporation of water into an unsaturated atmosphere (relative humidity less than 100%) is a thermodynamically spontaneous process, yielding work even though it costs heat, i.e., yielding work even though liquid water *gains* energy in becoming water vapor: see Refs. [133–135] concerning this point.

Thus the doom and gloom that one would dread based *solely* on Boltzmann's relation Prob (*σ*) = exp(−*σ*/*k*) does *not* obtain, and furthermore will *never* obtain if there exists *imposed* two-time low-entropy boundary conditions in an oscillating cosmology [16,61,62,88–94,101–105], or Planck-power (or other [21–25,33–35]) *imposed* sustained low-entropy mass-energy input such as hydrogen in a nonoscillating one [78]. Thus creation — initial via Big Bang with or without inflation, etc. [26–31], via Everett [96–98], and sustained via Planck-power (or other [21–25,33–35]) input — being kinetically rather than thermodynamically controlled [72–77] seems to be at least a reasonable tentative explanation of why we are privileged to be not merely *minimal extra*ordinary observers but *super-extra*ordinary observers — more correctly *hyper-extra*ordinary observers — with an entire Universe to explore and enjoy [61,62]. By the cosmological principle [51] we may hope that this is true everywhere in the Multiverse.

As a brief aside, we note that many chemical reactions are similarly kinetically rather than thermodynamically controlled [72–77], in like manner as Eq. (8). While only chemical reactions are discussed in Refs. [72–77], the same principle likewise applies with respect to *all* kinetically rather than thermodynamically controlled processes, for example kinetically rather than thermodynamically controlled physical and nuclear reactions. As we discussed in Sect. 4 if nuclear reactions were thermodynamically rather than kinetically controlled then there would be nothing but (iron + equilibrium blackbody radiation) — an iron-dead Universe.

#### **8. A brief review concerning the Multiverse, and some alternative viewpoints**

Four Levels of the Multiverse have been recognized [136–141]: Level I, the infinite number of L-regions and O-regions within an island Universe, with identical fundamental and effective laws of physics but with generally different histories (given the infinite number of L-regions and O-regions per island Universe, identical histories must occur in sufficiently widely separated ones); Level II, an infinite number of island Universes with identical fundamental but different effective laws of physics; Level III, Dr. Hugh Everett's many worlds [96–98]; and Level IV, wherein — within limits [136–142] — different fundamental laws of physics are allowed [136–142].17

Dr. Max Tegmark [138,139] writes that Level III is at least in some sense may be equivalent to Levels I+II: Level I incorporates different quantum branches within one single given Hubble volume of an infinity of such volumes contained in an island Universe. Level II incorporates different quantum branches within an entire island Universe. Level III incorporates different Level I and Level II Universes within one single given quantum branch. But it seems that Levels I+II, or at the very least Level I, must exist *first*, because Levels I+II, or at the very least Level I, seems *prerequisite* for the existence of entities capable of executing Dr. Hugh Everett's program [96–98].

<sup>17</sup> (Re: Entry [137], Ref. [112]) Reference [112] provides discussions of a spectrum of numerous viewpoints concerning Multiverses and related topics, Dr. Max Tegmark's viewpoint among this spectrum of viewpoints.

We should note that *if* conscious observers, also referred to as self-aware substructures (SASs) [143–145], are not merely self-aware but also have free will, *then* they have at least *some* degree of *choice* concerning creation of Level III Universes: They then have at least *some* freedom to *choose* whether or not to make a given observation or measurement, which observations and measurements to make, and when to make them. Even if the Everett interpretation [96–98] of quantum mechanics is incorrect [146] and Level III Universes exist only in potentiality until one and only one of them is actualized [146], say via wave-function collapse [147], then an SAS with free will still has this degree of *choice*. Even if the probabilities of the possible outcomes of any given observation or measurement cannot be altered, the set of possible outcomes on offer to Nature depends on which observations and measurements are chosen by an SAS with free will, and when they are on offer depends on when an SAS with free will chooses to observe or measure. Thus irrespective of the character of Level III Universes, *if* free will exists *then* there is this *qualitative* difference between unchosen observations and measurements made by Nature herself, say via decoherence [148,149], and chosen ones made by an SAS with free will. Moreover, "decoherence" is perhaps too strong a term; "delocalization of coherence" seems more correct. Since quantum-mechanical information in general cannot be destroyed, quantum-mechanical coherence in particular is never really destroyed, merely delocalized. As with any delocalization process there is an accompanying increase in entropy. But within a system of finite volume this increase in entropy is limited to a finite maximum value, implying recoherence, or more correctly relocalization of coherence, after a Poincaré recurrence time [108,150,151]. Of course, typical Poincaré recurrence times [108,150,151] of all but very small systems are inconceivably long, but in a very small system at least partial recoherence, or more correctly relocalization of coherence, may occur in a reasonable time. We should note that even before the term "decoherence" had been coined, some aspects of decoherence, or more correctly delocalization of coherence, had been partially anticipated [152,153]. For general reviews concerning the quantum-mechanical measurement problem see, for example, Refs. [149] and [152–155].18

Thus the doom and gloom that one would dread based *solely* on Boltzmann's relation Prob (*σ*) = exp(−*σ*/*k*) does *not* obtain, and furthermore will *never* obtain if there exists *imposed* two-time low-entropy boundary conditions in an oscillating cosmology [16,61,62,88–94,101–105], or Planck-power (or other [21–25,33–35]) *imposed* sustained low-entropy mass-energy input such as hydrogen in a nonoscillating one [78]. Thus creation — initial via Big Bang with or without inflation, etc. [26–31], via Everett [96–98], and sustained via Planck-power (or other [21–25,33–35]) input — being kinetically rather than thermodynamically controlled [72–77] seems to be at least a reasonable tentative explanation of why we are privileged to be not merely *minimal extra*ordinary observers but *super-extra*ordinary observers — more correctly *hyper-extra*ordinary observers — with an entire Universe to explore and enjoy [61,62]. By the cosmological principle [51] we may

As a brief aside, we note that many chemical reactions are similarly kinetically rather than thermodynamically controlled [72–77], in like manner as Eq. (8). While only chemical reactions are discussed in Refs. [72–77], the same principle likewise applies with respect to *all* kinetically rather than thermodynamically controlled processes, for example kinetically rather than thermodynamically controlled physical and nuclear reactions. As we discussed in Sect. 4 if nuclear reactions were thermodynamically rather than kinetically controlled then there would be nothing but (iron + equilibrium blackbody radiation) — an iron-dead

**8. A brief review concerning the Multiverse, and some alternative**

Four Levels of the Multiverse have been recognized [136–141]: Level I, the infinite number of L-regions and O-regions within an island Universe, with identical fundamental and effective laws of physics but with generally different histories (given the infinite number of L-regions and O-regions per island Universe, identical histories must occur in sufficiently widely separated ones); Level II, an infinite number of island Universes with identical fundamental but different effective laws of physics; Level III, Dr. Hugh Everett's many worlds [96–98]; and Level IV, wherein — within limits [136–142] — different fundamental laws of physics

Dr. Max Tegmark [138,139] writes that Level III is at least in some sense may be equivalent to Levels I+II: Level I incorporates different quantum branches within one single given Hubble volume of an infinity of such volumes contained in an island Universe. Level II incorporates different quantum branches within an entire island Universe. Level III incorporates different Level I and Level II Universes within one single given quantum branch. But it seems that Levels I+II, or at the very least Level I, must exist *first*, because Levels I+II, or at the very least Level I, seems *prerequisite* for the existence of entities capable of executing Dr. Hugh Everett's

<sup>17</sup> (Re: Entry [137], Ref. [112]) Reference [112] provides discussions of a spectrum of numerous viewpoints concerning

Multiverses and related topics, Dr. Max Tegmark's viewpoint among this spectrum of viewpoints.

hope that this is true everywhere in the Multiverse.

246 Recent Advances in Thermo and Fluid Dynamics

Universe.

**viewpoints**

are allowed [136–142].17

program [96–98].

Perhaps the concepts considered in this chapter may be at least to some degree applicable to the maximal proposed version of the Multiverse, the Level IV Multiverse [136–145], wherein all well-defined mathematical structures [140–145] — but *not* all arbitrary figments or fantasies of one's imagination [140–142] — would be realized as physically-existing Universes [140–145]. But as Dr. Alex Vilenkin points out, not all mathematical structures, indeed not even all *allowable* mathematical structures given the restrictions stated by Dr. Max Tegmark [140–142], are equal: some are more beautiful and hence more equal than others [156]. Alex Vilenkin writes: "Beautiful mathematics combines simplicity with depth [156]." (But also that "simplicity" and "depth" are almost as difficult to define as "beauty [156].") But Dr. Alex Vilenkin also writes: "Mathematical beauty may be useful as a guide, but it is hard to imagine that it would suffice to select a unique theory out of

<sup>18</sup> In Chap. 23 of Ref. [152], Dr. David Bohm expresses the viewpoint that classical mechanics should be considered in its own right and as prerequisite for quantum mechanics, rather than as a limiting case of quantum mechanics. This is opposed to the more generally accepted viewpoint that classical mechanics should be considered as a limiting case of quantum mechanics. Moreover, even Dr. David Bohm expresses the latter viewpoint in his own recognition of the Universe as ultimately quantum-mechanical, in Chap. 8 (especially Sects. 8.22–8.23) and Chap. 22 (especially Sects. 22.2–22.3) of Ref. [152]. But, in any case, this is apart from Dr. David Bohm's partial anticipation of certain aspects of decoherence, or more correctly delocalization of coherence, in Sect. 6.12 and Chap. 22 (especially Sects. 22.11–22.12) of Ref. [152].

the infinite number of possibilities [157]." These points are also considered by Dr. Roger Penrose [158]. Yet even so mathematical beauty should have at least *some* selective power. A case in point: Newton's laws have both simplicity and depth, and hence are beautiful. But Einstein's laws have both *greater* simplicity and *greater* depth, and hence are *more* beautiful. The laws of motion have the same form in all reference frames in General Relativity but not in Newton's theory (for example, Newton's theory requires extra terms for centrifugal and Coriolis forces in rotating reference frames), thus General Relativity has greater simplicity; additionally, Newton's theory is a *limiting case* of Einstein's but not vice versa, thus General Relativity also has greater depth. Hence might a Universe wherein Newton's laws are the *fundamental* laws, not merely a limiting case of relativity and quantum mechanics, be denied physical existence in a Level IV Multiverse — because even though it is a beautiful mathematical structure, it is not the *maximally*-beautiful one that *maximally* entails both simplicity and depth? While (even neglecting quantum mechanics) we can*not* be sure if General Relativity *is* the maximally-beautiful mathematical structure, we *can* be sure that Newtonian theory, while beautiful, is *not* maximally beautiful. Moreover, while the Multiverse is eternal, it nonetheless, at least below Level IV [136], did have a beginning [99]. The laws of quantum mechanics — *our* laws of quantum mechanics governed the initial tunneling event that created not merely our Universe but the Multiverse, at least through Level II [99,136]. Thus these laws, on whatever tablets they are written, must have existed before, and must exist independently of, the Multiverse at least through Level II [99,136] — not merely of our island Universe [99]. Concerning Level III, it seems that Levels I+II, or at the very least Level I, must exist *first*, because Levels I+II, or at the very least Level I, seems *prerequisite* for the existence of entities capable of executing Dr. Hugh Everett's program [96–98]. But might the prerequisites for a beginning and for the pre-existence of *our* laws of quantum mechanics be general, operative even at Level IV [136]? But if so then might Level IV — but not Levels I, II, and III — be more restricted than has been suggested [136]? For then might *our* laws of quantum mechanics be part of the *one* maximally-beautiful mathematical structure that *maximally* entails both simplicity and depth — *our* fundamental (not merely effective) laws of physics [136] — after all? Then perhaps this *one* maximally beautiful mathematical structure, this *maximal* possible entailment of both simplicity and depth, is the only one realized via physically-existing Universes. But if this is the case then the question arises: Why does this *one* maximally beautiful mathematical structure permit life [159] (at the very least, carbon-based life as we know it on Earth)?

We must admit that in this chapter we have not even scratched the surface, as per this paragraph and the two immediately following ones. There are many alternative viewpoints concerning the Multiverse and related issues. We should at least mention a few of them that we have not mentioned until now. According to at least one of these viewpoints, inflation is eternal into the past as well as into the future, and hence has no beginning as well as no end [160–162]. But perhaps this is compatible with inflation having a beginning if regions of inflation in the forward and backward time directions are disjoint and incapable of any interaction with each other [163]. Then perhaps observers in both types of regions would consider their home region to be evolving forward, not backward, in time. According to other viewpoints, inflation not only has a beginning but also has an end — eternal inflation is impossible [164,165]. According to one of these viewpoints, the end of inflation is imposed by the increasingly fractal nature of spacetime [164,165]. We also note that Dr. Roger Penrose considered another difficulty associated with possible fractal nature of spacetime: inflation does not solve the smoothness and flatness problems if the structure of spacetime is fractal, or worse than fractal [166]. According to another of these viewpoints, the end of inflation is imposed by the Big Snap, according to which expansion of space will eventually dilute the number of degrees of freedom per any unit volume, and specifically per Hubble volume, to less than one, although the Universe will probably be in trouble well before the number of degrees of freedom per Hubble volume is reduced to one [167,168]. But perhaps new degrees of freedom can be created to compensate [167,168]. Perhaps Planck-power input, if it exists, can, because it replenishes mass-energy, also replenish degrees of freedom — thereby precluding the Big Snap. In Dr. Max Tegmark's rubber-band analogy, this corresponds to new molecules of rubber being created as the rubber band stretches, thereby keeping the density of rubber constant [167,168]. But, with or without a Big Snap [167,168], if inflation does have an end for any reason whatsoever, then my question to Dr. Roger Penrose in Sect. 6 is answered negatively.

There are also many proposed solutions to the entropy problem (why there is so very much more than one minimal Boltzmann brain in our L-region and O-region), some of which we have already discussed and/or cited in Sects. 5–8, other than Planck-power input. But there are still other proposed solutions to the entropy problem. One other proposed solution that we have not yet cited entails quantum fluctuations ensuring that every baby Universe starts out with an unstable large cosmological-constant, which corresponds to low total entropy because it is thermodynamically favorable for the consequent high-energy false vacuum to decay spontaneously [169,170]. Yet another proposed solution that we have not yet cited entails observer-assisted low entropy [168].

There are also many alternative viewpoints concerning fine-tuning and life in the Universe. It has been noted that since physical parameters such as constants of nature, strengths of forces, masses of elementary particles, etc., all have real-number values, and the range of real numbers is infinite, then if the probability of occurrence of a given real-number value of a given parameter is uniform, or at least non-convergent, then there is only an infinitessimal probability of this value being within any finite range [171]. But if thee are an infinite number of L-regions and O-regions, this infinity may be as large or even larger. We should also note that while some scientists are favorable towards the idea of fine tuning [172], others are sceptical to the point of not requiring a Multiverse to explain it away, but stating that it is an invalid concept even if our O-region constituted the entire Universe [173–175]. Even this sceptical viewpoint admits that only a very small range of parameter space is consistent with carbon-based life as we know it on Earth [], but assumes that a much larger range of parameter space is consistent with life in general [174]. But life, at least chemically-based life, probably must be based on carbon, because no other element even comes close to matching carbon's ability to form highly complex, information-rich molecules. Even carbon's closest competitor, silicon, falls woefully short. Also, nucleosynthesis in stars forms carbon more easily that silicon [176], so carbon is more abundant [176].

#### **Acknowledgments**

the infinite number of possibilities [157]." These points are also considered by Dr. Roger Penrose [158]. Yet even so mathematical beauty should have at least *some* selective power. A case in point: Newton's laws have both simplicity and depth, and hence are beautiful. But Einstein's laws have both *greater* simplicity and *greater* depth, and hence are *more* beautiful. The laws of motion have the same form in all reference frames in General Relativity but not in Newton's theory (for example, Newton's theory requires extra terms for centrifugal and Coriolis forces in rotating reference frames), thus General Relativity has greater simplicity; additionally, Newton's theory is a *limiting case* of Einstein's but not vice versa, thus General Relativity also has greater depth. Hence might a Universe wherein Newton's laws are the *fundamental* laws, not merely a limiting case of relativity and quantum mechanics, be denied physical existence in a Level IV Multiverse — because even though it is a beautiful mathematical structure, it is not the *maximally*-beautiful one that *maximally* entails both simplicity and depth? While (even neglecting quantum mechanics) we can*not* be sure if General Relativity *is* the maximally-beautiful mathematical structure, we *can* be sure that Newtonian theory, while beautiful, is *not* maximally beautiful. Moreover, while the Multiverse is eternal, it nonetheless, at least below Level IV [136], did have a beginning [99]. The laws of quantum mechanics — *our* laws of quantum mechanics governed the initial tunneling event that created not merely our Universe but the Multiverse, at least through Level II [99,136]. Thus these laws, on whatever tablets they are written, must have existed before, and must exist independently of, the Multiverse at least through Level II [99,136] — not merely of our island Universe [99]. Concerning Level III, it seems that Levels I+II, or at the very least Level I, must exist *first*, because Levels I+II, or at the very least Level I, seems *prerequisite* for the existence of entities capable of executing Dr. Hugh Everett's program [96–98]. But might the prerequisites for a beginning and for the pre-existence of *our* laws of quantum mechanics be general, operative even at Level IV [136]? But if so then might Level IV — but not Levels I, II, and III — be more restricted than has been suggested [136]? For then might *our* laws of quantum mechanics be part of the *one* maximally-beautiful mathematical structure that *maximally* entails both simplicity and depth — *our* fundamental (not merely effective) laws of physics [136] — after all? Then perhaps this *one* maximally beautiful mathematical structure, this *maximal* possible entailment of both simplicity and depth, is the only one realized via physically-existing Universes. But if this is the case then the question arises: Why does this *one* maximally beautiful mathematical structure permit life [159] (at the very least, carbon-based life as we know it on Earth)?

248 Recent Advances in Thermo and Fluid Dynamics

We must admit that in this chapter we have not even scratched the surface, as per this paragraph and the two immediately following ones. There are many alternative viewpoints concerning the Multiverse and related issues. We should at least mention a few of them that we have not mentioned until now. According to at least one of these viewpoints, inflation is eternal into the past as well as into the future, and hence has no beginning as well as no end [160–162]. But perhaps this is compatible with inflation having a beginning if regions of inflation in the forward and backward time directions are disjoint and incapable of any interaction with each other [163]. Then perhaps observers in both types of regions would consider their home region to be evolving forward, not backward, in time. According to other viewpoints, inflation not only has a beginning but also has an end — eternal inflation is impossible [164,165]. According to one of these viewpoints, the end of inflation is imposed by the increasingly fractal nature of spacetime [164,165]. We also note that Dr. Roger Penrose considered another difficulty associated with possible fractal nature of spacetime: inflation does not solve the smoothness and flatness problems if the structure of spacetime is fractal,

I am especially grateful to Dr. Wolfgang Rindler, Dr. Donald H. Kobe, Dr. Bruce N. Miller, and Dr. Roger Penrose for very helpful, thoughtful, and insightful discussions, communications, and advice concerning relativity, cosmology, and thermodynamics. I am also grateful to Dr. Roger Penrose for valuable insights and clarifications concerning the relation between thermodynamics on the one hand, and inflation and cosmology on the other, and to Dr. Michael Turner for valuable insights and clarifications concerning oscillating versus nonoscillating universes, both at the 27th Texas Symposium on Relativistic Astrophysics, held at the Fairmont Hotel in Dallas, Texas on December 8–13, 2013 (website: nsm.utdallas.edu/texas2013/). I am also grateful to Dr. Wolfgang Rindler, Dr. Bruce N. Miller, and Dr. Roger Penrose for helpful general discussions concerning physics and astrophysics, both at the 27th Texas Symposium on Relativistic Astrophysics and otherwise, and to Dr. Donald H. Kobe for such discussions on various occasions. I also thank Dr. Marlan O. Scully and Dr. Donald H. Kobe for helpful insights concerning decoherence. I am thankful to Dr. Paolo Grigolini for very helpful and thoughtful considerations concerning both earlier and the most recent versions of this manuscript in Special Problems courses. Also, I thank Dr. S. Mort Zimmerman for engaging in general scientific discussions over many years, and both Dan Zimmerman and Dr. Kurt W. Hess for brief yet helpful discussions concerning this chapter and for engaging in general scientific discussions at times. I also thank Dr. Iva Simcic, Publishing Process Manager, for much very helpful advice in preparing this chapter and for much extra time to prepare it, and Technical Support at MacKichan Software for their very helpful advice concerning Scientific WorkPlace 5.5.

#### **Author details**

Jack Denur

Address all correspondence to: jackdenur@my.unt.edu

Electric & Gas Technology, Inc., Rowlett, Texas, USA

#### **References**

[1] Vilenkin A. Many Worlds in One: The Search for Other Universes. New York: Hall and Wang; 2007. DOI: 10.1063/1.2743129. See Chaps. 10 and 11, pp. 203–205, and the associated Notes including references cited therein.

[2] Rindler W. Relativity: Special, General, and Cosmological. 2nd ed. New York: Oxford University Press; 2006, pp. 367, 374, and 384.

[3] Reference [2], Sects. 16.1G and 17.1–17.4 (especially Sect. 17.3), also Exercise 16.3 on p. 369.

[4] Reference [2], Sects. 16.2–16.4 (especially pp. 359–360 and 367).

[5] Davies PCW. The Physics of Time Asymmetry. 2nd ed. Berkeley: University of California Press; 1977, Sect. 1.4.

[6] Reference [2], pp. 376–377.

[7] Reference [2], Exercise 7.22 on pp. 160–161, Sects. 16.1E–16.1G, p. 362, p. 366, Sect. 17.2, and Exercise 17.4 on p. 388.

[8] 27th Texas Symposium on Relativistic Astrophysics, held at the Fairmont Hotel in Dallas, Texas, December 8–13, 2013. [Internet]. 2013. Available from: nsm.utdallas.edu/texas2013/ [Accessed 2015-12-05]

[10] Lang KR. Essential Astrophysics. Berlin: Springer-Verlag; 2013, Sect. 14.3 and references cited therein.

[11] Planck Collaboration. *Planck* 2015 results. XIII. Cosmological parameters. arXiv:1502.01589v2 [astro-ph.CO] 6 Feb 2015: 67 pages. [Internet]. 2015. Available from: xxx.lanl.gov/abs/1502.01589v2 or arxiv.org/abs/1502.01589v2 [Accessed: 2015-12-05]. See Sects. 5.4 and 5.6.

[12] Misner CW, Thorne KS, Wheeler JA. Gravitation. New York: Oxford; 1973, pp. 10–13 and 1180, and Sects. 43.4, 44.3, and 44.6 (especially pp. 1215–1217).

[13] Planck M. Über irreversible Strahlangdvorgänge. Sitzungsber. Deut. Akad. Wiss Berlin Kl. Math–Phys. Tech. 1899; 440–480.

[14] Bradley W. Carroll BW, Ostlie DA. An Introduction to Modern Astrophysics. 2nd ed. San Francisco: Pearson Addison Wesley; 2007, pp. 1233–1235.

[15] Penrose R. The Road to Reality: A Complete Guide to the Laws of the Universe. New York: Alfred A. Knopf; 2005, Sects. 27.3, 27.10, and 31.1. See also references cited in Sect. 31.1.

[16] Reference [2], Sects. 16.1G–16.1K.

[17] Reference [2], Sects. 16.4, 18.3, and 18.4.

[18] Reference [10], Sects. 14.6 and 15.6.2.

[19] Reference [11], Sect. 6.2.4.

the relation between thermodynamics on the one hand, and inflation and cosmology on the other, and to Dr. Michael Turner for valuable insights and clarifications concerning oscillating versus nonoscillating universes, both at the 27th Texas Symposium on Relativistic Astrophysics, held at the Fairmont Hotel in Dallas, Texas on December 8–13, 2013 (website: nsm.utdallas.edu/texas2013/). I am also grateful to Dr. Wolfgang Rindler, Dr. Bruce N. Miller, and Dr. Roger Penrose for helpful general discussions concerning physics and astrophysics, both at the 27th Texas Symposium on Relativistic Astrophysics and otherwise, and to Dr. Donald H. Kobe for such discussions on various occasions. I also thank Dr. Marlan O. Scully and Dr. Donald H. Kobe for helpful insights concerning decoherence. I am thankful to Dr. Paolo Grigolini for very helpful and thoughtful considerations concerning both earlier and the most recent versions of this manuscript in Special Problems courses. Also, I thank Dr. S. Mort Zimmerman for engaging in general scientific discussions over many years, and both Dan Zimmerman and Dr. Kurt W. Hess for brief yet helpful discussions concerning this chapter and for engaging in general scientific discussions at times. I also thank Dr. Iva Simcic, Publishing Process Manager, for much very helpful advice in preparing this chapter and for much extra time to prepare it, and Technical Support at MacKichan Software for their

[1] Vilenkin A. Many Worlds in One: The Search for Other Universes. New York: Hall and Wang; 2007. DOI: 10.1063/1.2743129. See Chaps. 10 and 11, pp. 203–205, and the associated

[2] Rindler W. Relativity: Special, General, and Cosmological. 2nd ed. New York: Oxford

[3] Reference [2], Sects. 16.1G and 17.1–17.4 (especially Sect. 17.3), also Exercise 16.3 on p. 369.

[5] Davies PCW. The Physics of Time Asymmetry. 2nd ed. Berkeley: University of California

[7] Reference [2], Exercise 7.22 on pp. 160–161, Sects. 16.1E–16.1G, p. 362, p. 366, Sect. 17.2,

[8] 27th Texas Symposium on Relativistic Astrophysics, held at the Fairmont Hotel in Dallas, Texas, December 8–13, 2013. [Internet]. 2013. Available from: nsm.utdallas.edu/texas2013/

very helpful advice concerning Scientific WorkPlace 5.5.

Address all correspondence to: jackdenur@my.unt.edu

Electric & Gas Technology, Inc., Rowlett, Texas, USA

Notes including references cited therein.

University Press; 2006, pp. 367, 374, and 384.

[4] Reference [2], Sects. 16.2–16.4 (especially pp. 359–360 and 367).

**Author details**

250 Recent Advances in Thermo and Fluid Dynamics

Jack Denur

**References**

Press; 1977, Sect. 1.4.

[Accessed 2015-12-05]

[6] Reference [2], pp. 376–377.

and Exercise 17.4 on p. 388.

[20] Reference [14], Chap. 29.

[21] Reference [1], pp. 27–28, 37, and 170–171.

[22] Reference [2], pp. 359, 368–369, 387–388, and 411.

[23] Reference [14], pp. 1163–1167.

[24] Harrison ER. Mining Energy in an Expanding Universe. Astrophys. Jour. 1995; 446: 63–66. DOI: 10.1086/175767

[25] Sheehan DP, Kriss VG: Energy Emission by Quantum Systems in an Expanding FRW Metric. arXiv:astro-ph/0411299v1. 11 Nov 2004. 12 pages. [Internet]. 2004. Available from: xxx.lanl.gov/abs/astro-ph/0411299 or arxiv.org/abs/astro-ph/0411299 [Accessed: 2015-12-05]

[26] Tryon EP. Is the Universe a Vacuum Fluctuation? Nature 1973; **246**: 396–397. DOI: 10.1038/246396a0

[27] Reference [1], pp. 11–12 and 183–186.

[28] Filippenko A, Pasachoff JM. A Universe From Nothing. 2 pages. [Internet]. 2001. Available from: https:www.astrosociety.org/publications/a-universe-from-nothing [Accessed: 2015-12-05]

[29] Potter F, Jargodski C. Mad About Modern Physics. Hoboken, NJ; 2005, Question 224, "The Total Energy," on p. 115, and Answer 224, "The Total Energy," along with cited references, on pp. 275–276.

[30] Berman MS. On the Zero-Energy Universe. Int. J. Theor. Phys. 2009; **48**: 3278–3286. DOI: 10.1007/s10773-009-0125-8

[31] Berman MS, Trevisan LA. On Creation of Universe Out of Nothing. Int. J. Modern Phys. B. 2010; **19**: 1309–1313. DOI: 10.1142/S021827181007342

[32] Reference [15], Sects. 19.7–19.8 (especially Sect. 19.8 and most especially pp. 468–469). See also references cited therein; also Notes for Sects. 19.7–19.8 on pp. 469–470. Of these references, see especially the references cited in Note 19.17 on p. 470.

[33] Reference [5], Sect. 7.2 and references cited therein.

[34] Reference [5], the last sentence on p. 187 and the reference cited therein.

[35] Davies PCW. Is the Universe Transparent or Opaque? J. Phys. A: Math. Gen. 1972; **5**: 1722–1737. DOI: 10.1088/0305-4470/5/12/012. See especially Sect. 8.

[36] Reference [1], pp. 48–52 and 67–69; also Notes for Chaps. 5 and 6 on pp. 211–212, especially Notes 2 and 3 for Chap. 5 on p. 211.

[37] Reference [2], pp. 379–380 and Sects. 18.5–18.6.

[38] Reference [14], Sect. 30.1.

[39] Reference [15], Chap. 28, including Notes for Chap. 28 on pp. 778–781.

[40] Reference [10], pp. 523–524.

[41] Crockett C. Primordial gravitational waves found: Researchers see traces of cosmic expansion just after Big Bang. Science News. April 5, 2014; **185 (7)**.

[42] Krauss LM. Beacon from the Big Bang. Sci. Amer. October, 2014; **311 (4)**: 58–67.

[43] Flauger R, Hill JC, Spergel DN. Toward an understanding of foreground emission in the BICEP2 region. Journal of Cosmology and Astroparticle Physics. 2014; **08**: 039. DOI: 10.1088/1475-7516/2014/08/039

[44] Crockett C. Gravitational wave discovery gives way to dust. Science News. October 18, 2014; **186 (8)**: 7.

[45] Crockett C. Dust obscures possible gravitational wave discovery. Science News. December 27, 2014; **186 (13)**: 16.

[46] Grant A. Gravitational wave claim bites dust. Science News. February 21, 2015; **187 (4)**: 13.

[47] Grant A. The past according to Planck: Cosmologists got a lot right. Science News. March 21, 2015; **187 (6)**: 7.

[48] Milne PA, Foley RJ, Brown PJ, Narayan G. The Changing Fractions of Type 1A Supernova NUV-Optical Subclasses with Redshift. Astrophys. Jour. 2015; **803:20**: 15 pages. DOI: 10.1088/0004-637X/803/120

[49] Reference [15], Sects. 28.4–28.7. See also references cited therein; also Notes for Sects. 28.4–28.7 on pp. 779–781; also pp. 1037–1038.

[50] Witze A. Inflation on trial. Science News. July 28, 2012; **182 (2)**: 20–21.

[51] Reference [2], Sects. 1.11 and 16.1–16.2.

[52] Reference [14], Sect. 13.3 and pp. 886–887, 957–958, 986–987, 1011–1013, 1018–1020, and 1028.

[53] Reference [14], pp. 1011–1013, 1018–1020, and 1028.

[54] Reference [10], Sect. 15.4.2 on pp. 542–545 and references cited therein.

[55] Reference [1], Chap. 10. Also see Notes for Chap. 10. on p. 213 and the reference cited in Note 1.

[56] Reference [14], Chaps. 7–16.

[30] Berman MS. On the Zero-Energy Universe. Int. J. Theor. Phys. 2009; **48**: 3278–3286. DOI:

[31] Berman MS, Trevisan LA. On Creation of Universe Out of Nothing. Int. J. Modern Phys.

[32] Reference [15], Sects. 19.7–19.8 (especially Sect. 19.8 and most especially pp. 468–469). See also references cited therein; also Notes for Sects. 19.7–19.8 on pp. 469–470. Of these

[35] Davies PCW. Is the Universe Transparent or Opaque? J. Phys. A: Math. Gen. 1972; **5**:

[36] Reference [1], pp. 48–52 and 67–69; also Notes for Chaps. 5 and 6 on pp. 211–212,

[41] Crockett C. Primordial gravitational waves found: Researchers see traces of cosmic

[43] Flauger R, Hill JC, Spergel DN. Toward an understanding of foreground emission in the BICEP2 region. Journal of Cosmology and Astroparticle Physics. 2014; **08**: 039. DOI:

[44] Crockett C. Gravitational wave discovery gives way to dust. Science News. October 18,

[45] Crockett C. Dust obscures possible gravitational wave discovery. Science News.

[46] Grant A. Gravitational wave claim bites dust. Science News. February 21, 2015; **187 (4)**:

[47] Grant A. The past according to Planck: Cosmologists got a lot right. Science News.

[48] Milne PA, Foley RJ, Brown PJ, Narayan G. The Changing Fractions of Type 1A Supernova NUV-Optical Subclasses with Redshift. Astrophys. Jour. 2015; **803:20**: 15 pages. DOI:

[49] Reference [15], Sects. 28.4–28.7. See also references cited therein; also Notes for

[50] Witze A. Inflation on trial. Science News. July 28, 2012; **182 (2)**: 20–21.

[42] Krauss LM. Beacon from the Big Bang. Sci. Amer. October, 2014; **311 (4)**: 58–67.

10.1007/s10773-009-0125-8

252 Recent Advances in Thermo and Fluid Dynamics

B. 2010; **19**: 1309–1313. DOI: 10.1142/S021827181007342

[33] Reference [5], Sect. 7.2 and references cited therein.

especially Notes 2 and 3 for Chap. 5 on p. 211.

[38] Reference [14], Sect. 30.1.

[40] Reference [10], pp. 523–524.

10.1088/1475-7516/2014/08/039

December 27, 2014; **186 (13)**: 16.

March 21, 2015; **187 (6)**: 7.

10.1088/0004-637X/803/120

Sects. 28.4–28.7 on pp. 779–781; also pp. 1037–1038.

2014; **186 (8)**: 7.

13.

[37] Reference [2], pp. 379–380 and Sects. 18.5–18.6.

references, see especially the references cited in Note 19.17 on p. 470.

1722–1737. DOI: 10.1088/0305-4470/5/12/012. See especially Sect. 8.

[34] Reference [5], the last sentence on p. 187 and the reference cited therein.

[39] Reference [15], Chap. 28, including Notes for Chap. 28 on pp. 778–781.

expansion just after Big Bang. Science News. April 5, 2014; **185 (7)**.

[57] Reference [10], Chaps. 2 and 8–13.

[58] Reference [1], Chap. 10, especially pp. 93–94. Also see Notes for Chap. 10. on p. 213 and the reference cited in Note 1.

[59] Reference [1], pp. 25–28 and 171–175, also Note 4 for Chap. 3 on p. 210.

[60] Reference [5]. See especially Chaps. 1, 2, 4, and 7.

[61] Reference [15], Sects. 19.7–19.8, Chaps. 27 and 28, and references cited therein; also Notes for Sects. 19.7–19.8 and Chaps. 27 and 28 on pp. 469–470, 732–734, and 778–781, respectively; also pp. 1037–1038. See especially Sects. 27.13 and 28.1–28.5, and the associated Notes.

[62] Reference [2], p. 380.

[63] Mersini-Houghton L, Vaas R, Editors. The Arrows of Time: A Debate in Cosmology. Berlin: Springer-Verlag; 2012.

[64] See, for example, Ref. [2], Sect. 16.1D.

[65] Reference [2], Sects. 10.5, 14.2–14.3 and 18.2E. See also the reference cited in Sects. 14.2–14.3.

[66] Reference [2], Sect. 18.2E.

[67] Lovelock D. The Four-dimensionality of Space and the Einstein Tensor. Jour. Math. Phys. 1972; **13**: 874–876. DOI: 10.1063/1.1666069

[68] Reference [11], Sect. 6.3.

[69] Planck Collaboration. *Planck* 2015 results. XIV. Dark energy and modified gravity. arXiv:1502.01590v1 [astro-ph.CO] 5 Feb 2015: 64 pages. [Internet]. 2015. Available from: xxx.lanl.gov/abs/1502.01590 or arxiv.org/abs/1502.01590 [Accessed: 2015-12-05]

[70] Reference [10], Sect. 15.2. See especially Subsects. 15.2.3–15.2.4, and most especially Table 15.1.

[71] Planck Collaboration. *Planck* 2015 results. I. Overview of products and scientific results. arXiv:1502.01582v2 [astro-ph.CO] 10 Aug 2015: 40 pages. [Internet]. 2015. Available from: xxx.lanl.gov/abs/1502.01582v2 or arxiv.org/abs/1502.01582v2 [Accessed: 2015-12-05]. See Fig. 9 and its caption on p. 19.

[72] Noller CR. Chemistry of Organic Compounds, 2nd ed. Philadelphia: W. B. Saunders; 1957, pp. 41–42 and 165–170.

[73] Noller CR. Chemistry of Organic Compounds, 3rd ed. Philadelphia: W. B. Saunders; 1965, Chap. 3.

[74] Noller CR. Textbook of Organic Chemistry, 3rd ed. Philadelphia: W. B. Saunders; 1966, Chap. 3.

[75] Morrison RT, Boyd RN. Organic Chemistry, 6th ed. Englewood Cliffs, N. J.: W. B. Prentice Hall; 1992.

[76] Mahan BM. University Chemistry, Reading, Mass.: Addison Wesley; 1965, Chaps. 8–9.

[77] Mahan BM, Myers RJ. University Chemistry, 4th ed. Menlo Park, Calif.: Bejamin/Cummings; 1980, Chaps. 8–9.

[78] Redd NT. Rivers of Hydrogen Gas May Fuel Spiral Galaxies [Internet]. 2014. Available from: http://www.space.com/24780-spiral-galaxies-hydrogen-gas-river.html (February 21, 2014, 05:25 PM ET) [Accessed 2015-12-05]

[79] Dan Zimmerman, private communications, 2014. When I mentioned this analogy to Dan Zimmerman, he encouraged me to include it in this chapter.

[80] Sheehan, DP, editor. Quantum Limits to the Second Law, AIP Conference Proceedings Volume 643. Melville, N. Y.: American Institute of Physics; 2002.

[81] Nukulov AV, Sheehan DP, editors. Special Issue: Quantum Limits to the Second Law of Thermodynamics. Entropy. March 2004: Vol. 6, Issue 1.

[82] Cápek V, Sheehan DP. Challenges to the Second Law of Thermodynamics: Theory and ˇ Experiment. Dordrecht, The Netherlands: Springer; 2005.

[83] Sheehan, DP, editor. Special Issue: The Second Law of Thermodynamics: Foundations and Status. Found. Phys. December 2007: Vol. 37, Issue 12.

[84] Sheehan, DP, editor. Second Law of Thermodynamics: Status and Challenges, AIP Conference Proceedings Volume 1411. Melville, N. Y.: American Institute of Physics; 2011.

[85] See, for example, Ref. [2], Sects. 2.4–2.10 and 3.5, especially Sects. 2.4–2.6 and pp. 56–57. Relativity of simultaneity is most directly discussed in Sects. 2.4–2.6.

[86] Reference [2], Sects. 2.10 and 3.6.

[87] Dr. Wolfgang Rindler, private communications, 1980s–2010s, including at the 27th Texas Symposium on Relativistic Astrophysics, held at the Fairmont Hotel in Dallas, Texas, December 8–13, 2013. [Internet]. 2013. Available from: nsm.utdallas.edu/texas2013/ [Accessed 2015-12-05]

[88] Reference [2], pp. 398 and 402–403.

[89] Reference [1]. See pp. 120–121 and Note 4 for Chap. 16 on p. 219 for pre-inflationary ideas concerning oscillating cosmologies, and Chap. 18 (especially pp. 197–198), Note 3 for Chap. 18 on p. 221, pp. 203–205, and Note 9 for Chap. 19 on p. 222 for oscillating cosmologies considered in light of inflation.

[90] Vilenkin A, Garriga J. Testable anthropic predictions for dark energy. Phys. Rev. D 2003; **67**: 043503–1-11.

[91] Dr. Michael Turner. private communications, at the 27th Texas Symposium on Relativistic Astrophysics, held at the Fairmont Hotel in Dallas, Texas, December 8–13, 2013. [Internet]. 2013. Available from: nsm.utdallas.edu/texas2013/ [Accessed 2015-12-05]

[92] An alternative model of an oscillating Universe is discussed in Freese K, Brown MG, Kinney WH. The Phantom Bounce: A New Proposal for an Oscillating Cosmology. In Ref. [63], pp. 149–156.

[93] Zeh HD. Open Questions Regarding the Arrow of Time. In Ref. [63], pp. 205–217.

[94] Reference [12], Sect. 27.10 (especially Box 27.4 on p. 738).

[95] Reference [2], Sects. 6.2–6.3.

[72] Noller CR. Chemistry of Organic Compounds, 2nd ed. Philadelphia: W. B. Saunders;

[73] Noller CR. Chemistry of Organic Compounds, 3rd ed. Philadelphia: W. B. Saunders;

[74] Noller CR. Textbook of Organic Chemistry, 3rd ed. Philadelphia: W. B. Saunders; 1966,

[75] Morrison RT, Boyd RN. Organic Chemistry, 6th ed. Englewood Cliffs, N. J.: W. B. Prentice

[76] Mahan BM. University Chemistry, Reading, Mass.: Addison Wesley; 1965, Chaps. 8–9. [77] Mahan BM, Myers RJ. University Chemistry, 4th ed. Menlo Park, Calif.:

[78] Redd NT. Rivers of Hydrogen Gas May Fuel Spiral Galaxies [Internet]. 2014. Available from: http://www.space.com/24780-spiral-galaxies-hydrogen-gas-river.html (February 21,

[79] Dan Zimmerman, private communications, 2014. When I mentioned this analogy to Dan

[80] Sheehan, DP, editor. Quantum Limits to the Second Law, AIP Conference Proceedings

[81] Nukulov AV, Sheehan DP, editors. Special Issue: Quantum Limits to the Second Law of

[82] Cápek V, Sheehan DP. Challenges to the Second Law of Thermodynamics: Theory and ˇ

[83] Sheehan, DP, editor. Special Issue: The Second Law of Thermodynamics: Foundations

[84] Sheehan, DP, editor. Second Law of Thermodynamics: Status and Challenges, AIP Conference Proceedings Volume 1411. Melville, N. Y.: American Institute of Physics; 2011. [85] See, for example, Ref. [2], Sects. 2.4–2.10 and 3.5, especially Sects. 2.4–2.6 and pp. 56–57.

[87] Dr. Wolfgang Rindler, private communications, 1980s–2010s, including at the 27th Texas Symposium on Relativistic Astrophysics, held at the Fairmont Hotel in Dallas, Texas, December 8–13, 2013. [Internet]. 2013. Available from: nsm.utdallas.edu/texas2013/

[89] Reference [1]. See pp. 120–121 and Note 4 for Chap. 16 on p. 219 for pre-inflationary ideas concerning oscillating cosmologies, and Chap. 18 (especially pp. 197–198), Note 3 for Chap. 18 on p. 221, pp. 203–205, and Note 9 for Chap. 19 on p. 222 for oscillating cosmologies

1957, pp. 41–42 and 165–170.

254 Recent Advances in Thermo and Fluid Dynamics

Bejamin/Cummings; 1980, Chaps. 8–9.

2014, 05:25 PM ET) [Accessed 2015-12-05]

[86] Reference [2], Sects. 2.10 and 3.6.

[88] Reference [2], pp. 398 and 402–403.

considered in light of inflation.

[Accessed 2015-12-05]

Zimmerman, he encouraged me to include it in this chapter.

Thermodynamics. Entropy. March 2004: Vol. 6, Issue 1.

Experiment. Dordrecht, The Netherlands: Springer; 2005.

and Status. Found. Phys. December 2007: Vol. 37, Issue 12.

Relativity of simultaneity is most directly discussed in Sects. 2.4–2.6.

Volume 643. Melville, N. Y.: American Institute of Physics; 2002.

1965, Chap. 3.

Chap. 3.

Hall; 1992.

[96] Reference [1], pp. 114–116 and 187–188.

[97] Tegmark M. Our Mathematical Universe. New York: Vintage Books; 2015, p. 179, Chap. 8, pp. 284–286 and 314–315.

[98] Tegmark M. The Mathematical Universe. Found Phys. 2008; 38: 101–150. DOI: 10.1007/s10701-007-9186-9. See Sect. 5.2.

[99] Reference [1], Chaps. 5–8 (especially Chaps. 6 and 8), Chaps. 16 and 17, and pp. 203–205. Also see Notes for Chaps. 5, 6, 8, 16, and 17 and for pp. 203–205 including references cited therein on pp. 211–212 and 219–222. See especially Chaps. 16 and 17, and most especially pp. 180–181 and 204–205.

[100] Reference [1], Chaps. 12 and 13, and Notes for Chaps. 12 and 13 including references cited therein on pp. 214–216.

[101] Reference [5], Chap. 7, especially Sect. 7.4, and most especially the first complete paragraph on p. 193 and the references cited therein, especially Refs. [102–104] of this chapter immediately following.

[102] Cocke J. Statistical Time Symmetry and Two-Time Boundary Conditions in Physics and Cosmology. Phys. Rev. 1967; **160**: 1165–1170. DOI: 10.1103/PhysRev.160.1165

[103] Schmidt H. Model of an oscillating cosmos which rejuvenates during contraction. J. Math. Phys. 1966; **7**: 494–509. DOI: 10.1063/1.1704949

[104] Davies PCW. Closed Time as an Explanation of the Black Body Background Radiation. Nature Physical Science. 1972; **240**: 3–5. DOI: 10.1103/physci240003a0

[105] Hartle J, Hertog T. Arrows of Time in the Bouncing Universes of the no-boundary quantum state. Phys. Rev. D. 2012; **85**: 13 pages. DOI: 10.1038/PhysRevD.85.103524

[106] Reference [5], Sect. 7.3.

[107] Tolman RC. Relativity, Thermodynamics, and Cosmology. Oxford, England: Oxford University Press; 1934. Unaltered and unabridged republication: New York: Dover; 1987, Chaps. IX–X, especially Sects. 130–131, and 169–175 (most especially Sects. 131 and 175).

[108] The Poincaré recurrence time is discussed in Ref. [5], Chap. 3, Sect. 5.2, pp. 131, 144, 164, 173–175, and 192–193, and Sect. 7.4. See also references cited therein.

[109] Reference [5], p. 103.

[110] Reference [2], p. 380 and Sects. 18.5–18.6.

[111] Reference [63]. Boltzmann-brain hypotheses and closely related topics are reviewed on pp. 3, 12–22, 29, 98–102, 193–195, and 205–211. See also relevant references cited therein.

[112] Carr B, Editor. Universe or Multiverse? Cambridge, U. K.: Cambridge University Press; 2007, p. 67.

[113] Reference [97], pp. 305–308 and 313–314.

[114] Reference [98], Sect. 4.2.4.

[115] Bousso R. Vacuum Structure and the Arrow of Time. Phys. Rev. D. 2012; **86**: 16 pages. DOI: 10.1103.PhysRevD.86.123509

[116] Garriga J, Vilenkin A. Watchers of the Multiverse. Journal of Cosmology and Astroparticle Physics. 2013; **05**: 037. DOI: 10.1088/1475-7516/2013/05/037

[117] Page DN. Is the Universe Decaying at an Astronomical Rate? Phys. Lett. B. 2008; **669**: 197–200. DOI: 10.1016/j.physletb.2008.039

[118] Susskind L. Fractal-Flows and Time's Arrow. arXiv:1203.6440v2 [hep-th] 7 Apr 2012. 24 pages. [Internet]. 2012. Available from: xxx.lanl.gov/abs/1203.6440v2 or arxiv.org/abs/1203.6440v2 [Accessed: 2015-12-05]

[119] Reference [1], Chap. 12, and pp. 135–136 and 164.

[120] Reference [97], pp. 140–141, 353–354, and 362–363.

[121] Reference [97], pp. 132–150 and 311–312.

[122] Reference [112]. See especially Chaps. 1, 2, 3, 5, 22, 23, and 25; most especially Chap. 2: Weinberg, S. Living in the Multiverse.

[123] Grant A. Time's Arrow: Maybe Gravity Shapes the Universe Into Two Opposing Futures. Science News. July 25, 2015; **188 (2)**: 15–18.

[124] Barbour J, Koslowski T, Flavio M. Identification of a Gravitational Arrow of Time. Phys. Rev. Lett. 2014; **113**: 118101 (5 pages).

[125] Reference [5], Sect. 4.6.

[126] Vaas R. Time After Time — The Big Bang Cosmology and the Arrows of Time. In Ref. [63], pp. 5–42. See especially pp. 8–9, including Figure 1 on p. 9.

[127] Reference [1], pp. 171–172 and Note 5 for Chap. 16 on p. 219; also, Refs. [24], [25], and [59–63] of this chapter.

[128] Dr. Roger Penrose. private communications, at the 27th Texas Symposium on Relativistic Astrophysics, held at the Fairmont Hotel in Dallas, Texas, December 8–13, 2013. [Internet]. 2013. Available from: nsm.utdallas.edu/texas2013/ [Accessed 2015-12-05]

[129] Reference [2], the last paragraph on p. 413.

[130] Reference [2], p. 416.

[108] The Poincaré recurrence time is discussed in Ref. [5], Chap. 3, Sect. 5.2, pp. 131, 144,

[111] Reference [63]. Boltzmann-brain hypotheses and closely related topics are reviewed on pp. 3, 12–22, 29, 98–102, 193–195, and 205–211. See also relevant references cited therein.

[112] Carr B, Editor. Universe or Multiverse? Cambridge, U. K.: Cambridge University Press;

[115] Bousso R. Vacuum Structure and the Arrow of Time. Phys. Rev. D. 2012; **86**: 16 pages.

[116] Garriga J, Vilenkin A. Watchers of the Multiverse. Journal of Cosmology and

[117] Page DN. Is the Universe Decaying at an Astronomical Rate? Phys. Lett. B. 2008; **669**:

[118] Susskind L. Fractal-Flows and Time's Arrow. arXiv:1203.6440v2 [hep-th] 7 Apr 2012. 24 pages. [Internet]. 2012. Available from: xxx.lanl.gov/abs/1203.6440v2 or

[122] Reference [112]. See especially Chaps. 1, 2, 3, 5, 22, 23, and 25; most especially Chap. 2:

[123] Grant A. Time's Arrow: Maybe Gravity Shapes the Universe Into Two Opposing

[124] Barbour J, Koslowski T, Flavio M. Identification of a Gravitational Arrow of Time. Phys.

[126] Vaas R. Time After Time — The Big Bang Cosmology and the Arrows of Time. In

[127] Reference [1], pp. 171–172 and Note 5 for Chap. 16 on p. 219; also, Refs. [24], [25], and

[128] Dr. Roger Penrose. private communications, at the 27th Texas Symposium on Relativistic Astrophysics, held at the Fairmont Hotel in Dallas, Texas, December 8–13, 2013.

[Internet]. 2013. Available from: nsm.utdallas.edu/texas2013/ [Accessed 2015-12-05]

Astroparticle Physics. 2013; **05**: 037. DOI: 10.1088/1475-7516/2013/05/037

164, 173–175, and 192–193, and Sect. 7.4. See also references cited therein.

[109] Reference [5], p. 103.

256 Recent Advances in Thermo and Fluid Dynamics

[114] Reference [98], Sect. 4.2.4.

DOI: 10.1103.PhysRevD.86.123509

197–200. DOI: 10.1016/j.physletb.2008.039

arxiv.org/abs/1203.6440v2 [Accessed: 2015-12-05]

[121] Reference [97], pp. 132–150 and 311–312.

Futures. Science News. July 25, 2015; **188 (2)**: 15–18.

Ref. [63], pp. 5–42. See especially pp. 8–9, including Figure 1 on p. 9.

Weinberg, S. Living in the Multiverse.

Rev. Lett. 2014; **113**: 118101 (5 pages).

[125] Reference [5], Sect. 4.6.

[59–63] of this chapter.

[119] Reference [1], Chap. 12, and pp. 135–136 and 164. [120] Reference [97], pp. 140–141, 353–354, and 362–363.

2007, p. 67.

[110] Reference [2], p. 380 and Sects. 18.5–18.6.

[113] Reference [97], pp. 305–308 and 313–314.

[131] Reference [15], p. 756.

[132] Berry RS, Rice SA, Ross J. Physical Chemistry. 2nd ed. New York: Oxford University Press; 2000, pp. 476–479.

[133] Bachhuber C. Energy from the evaporation of water. Am. J. Phys. 1983; **51**: 259–264.

[134] Güémez J, Valiente B, Fiolhais C, Fiolhais M. Experiments with the drinking bird, Am J. Phys. 2003; **71**: 1257–1264.

[135] Temming M. Water, Water Everywhere. Sci. Am. 2015; **313 (3)**: 26.

[136] Reference [97], pp. 134–140, Chap. 12, and pp. 358–370. Concise summaries of Multiverses are provided in Table 6.1 on p. 139, Figure 12.2 on p. 322, and Figure 13.1 on p. 358.

[137] Tegmark M. The Multiverse Hierarchy (Chap. 7). In Ref. [112]. Carr B, Editor. Universe or Multiverse? Cambridge, U. K.: Cambridge University Press; 2007.

[138] Reference [97], pp. 220–226.

[139] Tegmark M. Sect. 7.4. In Ref. [112].

[140] Reference [97]. See especially Chaps. 6, 8, and 10–12, also pp. 358–370 (most especially Chaps. 10 and 12).

[141] Tegmark M. The Multiverse Hierarchy (Chap. 7). In Ref. [112]. See especially Sects. 7.4.4–7.6.

[142] Reference [98], Sect. 5.4.

[143] Reference [97], Chaps. 8 and 11, especially pp. 291–299.

[144] Tegmark M. pp. 116–120. In Ref. [112].

[145] Reference [98], Sects. 2.3, 4.2.4, and 5.3.

[146] Lindley D. Where Does the Weirdness Go?: Why Quantum Mechanics is Strange, But Not as Strange as You Think. New York: Basic Books; 1996, pp. 107–111 and the 2nd Note on p. 233.

[147] Reference [97], pp. 175–183 and Chap. 8.

[148] Reference [97], Chap. 8.

[149] Reference [146], Act III, also Notes for Act III on pp. 234–240.

[150] Reference [5], pp. 173–175.

[151] Reference [146], pp. 177–203, especially pp. 199–203; also the last 3 Notes on p. 237 (the 3rd of these Notes continuing on p. 238), and the 2 Notes on p. 240.

[152] Bohm D. Quantum Theory. Englewood Cliffs, N. J.: Prentice-Hall; 1951. Unaltered and unabridged republication: New York: Dover; 1989, Sect. 6.12 and Chap. 22 (especially Sects. 22.11–22.12).

[153] Reference [5], Sect. 6.3.

[154] Reference [15], Chaps. 29 and 30.

[155] Reference [97], Chaps. 7 and 8, especially Chap. 8.

[156] Reference [1], pp. 202–203 and Notes 6–9 for Chap. 19 on p. 222. See also the reference cited in Note 6.

[157] Reference [1], pp. 201–202 and Notes 2–6 for Chap. 19 on p. 222. See also references cited in Notes 3, 4, and 6.

[158] Reference [15], Sect. 34.9.

[159] Reference [2], Sect. 18.6.

[160] Aguirre A, Gratton S. Steady-state eternal inflation. Phys. Rev. D. 2002; **65**: 083507, 6 pages. DOI: 10.1103/PhysRevD.65.083507

[161] Aguirre A, Gratton S. Inflation without a beginning: A null boundary proposal. Phys. Rev. D. 2003; **67**: 083515, 16 pages. DOI: 10.1103/PhysRevD.67.083515

[162] Aguirre A. Eternal Inflation, past and future. arXiv:0712.0571v1 [hep-th] 4 Dec 2007: 38 pages. [Internet]. 2007. Available from: xxx.lanl.gov/abs/0712.0571 or arxiv.org/abs/0712.0571 [Accessed: 2015-12-14]

[163] Vilenkin A. Arrows of time and the beginning of the universe. Phys. Rev. D. 2013; **88**: 043516, 10 pages. DOI: 10.1103/PhysRevD.88.043516

[164] Mersini-Houghton L, Perry MJ. The end of eternal inflation. Class. Quantum Grav. 2014; **31**: 165005, 11 pages. DOI: 10.1088/0264-9381/31/16/165005

[165] Mersini-Houghton L, Perry MJ. Localization on the landscape and eternal inflation. Class. Quantum Grav. 2014; **31**: 215008, 17 pages. DOI: 10.1088/0264-9381/31/21/215008

[166] Reference [15], Sects. 28.4–28.5. See especially pp. 746–747 and 756–757.

[167] Reference [97], pp. 366–370.

[168] Tegmark M. How unitary cosmology generalizes thermodynamics and solves the inflationary entropy problem. Phys. Rev. D. 2012; **85**: 123517, 19 pages. DOI: 10.1103/PhysRevD.85.123517

[169] Carroll SM, Chen J. Spontaneous Inflation and the Origin of the Arrow of Time. arXiv:0410270v1 [hep-th] 27 Oct 2004: 36 pages. [Internet]. 2004. Available from: xxx.lanl.gov/abs/0410270 or arxiv.org/abs/0410270 [Accessed: 2015-12-15]

[170] Carroll SM. From Eternity to Here: The Quest for the Ultimate Theory of Time (Kindle Edition, Version 6). New York: Penguin; 2010. See especially Chaps. 15–16 and most especially Locations 6576–6846; also, Notes 291–295 at Locations 8218–8233.

[171] Davies P. Universes galore: where will it all end? (Chap. 28). In Ref. [112]. Carr B, Editor. Universe or Multiverse? Cambridge, U. K.: Cambridge University Press; 2007. See especially Sect. 28.3.2.

[172] Barnes LA. The Fine-Tuning of the Universe for Intelligent Life. Publications of the Astronomical Society of Australia. 2012; **29**: 529–564. DOI: 10.1071/AS12015

[153] Reference [5], Sect. 6.3.

258 Recent Advances in Thermo and Fluid Dynamics

cited in Notes 3, 4, and 6.

[158] Reference [15], Sect. 34.9. [159] Reference [2], Sect. 18.6.

[167] Reference [97], pp. 366–370.

10.1103/PhysRevD.85.123517

especially Sect. 28.3.2.

pages. DOI: 10.1103/PhysRevD.65.083507

arxiv.org/abs/0712.0571 [Accessed: 2015-12-14]

043516, 10 pages. DOI: 10.1103/PhysRevD.88.043516

cited in Note 6.

[154] Reference [15], Chaps. 29 and 30.

[155] Reference [97], Chaps. 7 and 8, especially Chap. 8.

[156] Reference [1], pp. 202–203 and Notes 6–9 for Chap. 19 on p. 222. See also the reference

[157] Reference [1], pp. 201–202 and Notes 2–6 for Chap. 19 on p. 222. See also references

[160] Aguirre A, Gratton S. Steady-state eternal inflation. Phys. Rev. D. 2002; **65**: 083507, 6

[161] Aguirre A, Gratton S. Inflation without a beginning: A null boundary proposal. Phys.

[162] Aguirre A. Eternal Inflation, past and future. arXiv:0712.0571v1 [hep-th] 4 Dec 2007: 38 pages. [Internet]. 2007. Available from: xxx.lanl.gov/abs/0712.0571 or

[163] Vilenkin A. Arrows of time and the beginning of the universe. Phys. Rev. D. 2013; **88**:

[164] Mersini-Houghton L, Perry MJ. The end of eternal inflation. Class. Quantum Grav.

[165] Mersini-Houghton L, Perry MJ. Localization on the landscape and eternal inflation. Class. Quantum Grav. 2014; **31**: 215008, 17 pages. DOI: 10.1088/0264-9381/31/21/215008

[168] Tegmark M. How unitary cosmology generalizes thermodynamics and solves the inflationary entropy problem. Phys. Rev. D. 2012; **85**: 123517, 19 pages. DOI:

[169] Carroll SM, Chen J. Spontaneous Inflation and the Origin of the Arrow of Time. arXiv:0410270v1 [hep-th] 27 Oct 2004: 36 pages. [Internet]. 2004. Available from:

[170] Carroll SM. From Eternity to Here: The Quest for the Ultimate Theory of Time (Kindle Edition, Version 6). New York: Penguin; 2010. See especially Chaps. 15–16 and most

[171] Davies P. Universes galore: where will it all end? (Chap. 28). In Ref. [112]. Carr B, Editor. Universe or Multiverse? Cambridge, U. K.: Cambridge University Press; 2007. See

Rev. D. 2003; **67**: 083515, 16 pages. DOI: 10.1103/PhysRevD.67.083515

2014; **31**: 165005, 11 pages. DOI: 10.1088/0264-9381/31/16/165005

[166] Reference [15], Sects. 28.4–28.5. See especially pp. 746–747 and 756–757.

xxx.lanl.gov/abs/0410270 or arxiv.org/abs/0410270 [Accessed: 2015-12-15]

especially Locations 6576–6846; also, Notes 291–295 at Locations 8218–8233.

[173] Stenger VJ. The Fallacy of Fine-Tuning: Why the Universe Is Not Designed for Us (Kindle Edition). Amherst, N. Y.: Prometheus Books; 2011.

[174] Stenger VJ. God and the Multiverse. Amherst, N. Y.: Prometheus Books; 2014. See especially Chap. 16.

[175] Stenger VJ. Defending The Fallacy of Fine-Tuning. arXiv:1202.4359v1 [physics.pop-ph] 28 Jan 2012: 12 pages. [Internet]. 2012. Available from: xxx.lanl.gov/abs/1202.4359 or arxiv.org/abs/1202.4359 [Accessed: 2015-12-14]

[176] See the Wikipedia article entitled "Abundance of the chemical elements." [Internet]. 2015. Available from: www.wikipedia.org [Accessed: 2015-12-16]

#### **Chapter 10**

## **Absolute Zero and Even Colder?**

Jack Denur

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/61641

#### **Abstract**

We consider first the absolute zero of temperature and then negative Kelvin temperatures. The unattainability formulation of the Third Law of Thermodynamics is briefly reviewed. It puts limitations on the quest for absolute zero, and in its strongest mode forbids the attainment of absolute zero by any method whatsoever. But typically it is stated principally with respect to thermal-entropy-reduction refrigeration (TSRR). TSRR entails reduction of a refrigerated system's thermal entropy, i.e., its localization in momentum space. The possibility or impossibility of overcoming these limitations via TSRR is considered, with respect to both standard and absorption TSRR. (In standard TSRR, refrigeration is achieved at the expense of work input; in absorption TSRR, at the expense of high-temperature heat input.) We then consider the possibility or impossibility of the attainability of absolute zero temperature via configurational-entropy-reduction refrigeration (CSRR). CSRR entails reduction of a refrigerated system's configurational entropy, i.e., its localization in position space, via positional isolation of entities that happen to be in their ground states. Of course, the Second Law of Thermodynamics requires any decrease in entropy of a refrigerated system to be paid for by a compensating greater (in the limit of perfection, equal) increase in eLtropy. Or, in other words, the Second law of Thermodynamics requires any localization in the total momentum-plus-position phase space of a refrigerated system to be paid for by a compensating greater (in the limit of perfection, equal) delocalization in the total momentum-plus-position phase space of the refrigerated system and/or of its surroundings. We also briefly consider energy-reduction refrigeration (ERR), which entails extraction of energy but not entropy from a refrigerated system, and quantum-control refrigeration (QCR). (S not E denotes entropy in TSRR and CSRR, and E denotes energy in ERR, because S is the standard symbol for entropy, and E for energy.) With respect to both TSRR and CSRR, we consider not only the issue of attainability of absolute zero, but also the separate issues, even if absolute zero can be attained, of maintaining it, and of verifying that it has been attained. Purely dynamic – as opposed to thermodynamic – limitations on the quest for absolute

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

zero under classical versus quantum mechanics are compared and contrasted. Then hot true and cold effective negative Kelvin temperatures are considered. A few fine points concerning the Third Law of Thermodynamics are briefly mentioned in the Appendix.

**Keywords**: absolute zero, unattainability formulation of the Third Law of Thermodynamics, quantization, energy-time uncertainty principle, negative Kelvin temperatures.

#### **1. Introduction**

We consider first the absolute zero of temperature and then negative Kelvin temperatures.

The unattainability formulation of the Third Law of Thermodynamics is briefly reviewed in Sect. 2.1. It puts limitations of the quest for absolute zero, and in its *strongest mode* forbids the attainment of absolute zero by *any method* whatsoever. But typically it is stated principally with respect to thermal-entropy-reduction refrigeration (TSRR). TSRR entails reduction of a refrigerated system's thermal entropy, i.e., its localization in the momentum part of phase space (in momentum space for short). The possibility or impossibility of overcoming these limitations via TSRR is considered, in Sects. 2.2. and 2.3. with respect to standard TSRR, and in Sect. 2.4. with respect to absorption TSRR. (In standard TSRR, refrigeration is achieved at the expense of work input; in absorption TSRR, at the expense of high-temperature heat input.)

In Sect. 3, we consider the possibility or impossibility of the attainability of absolute zero temperature via configurational-entropy-reduction refrigeration (CSRR). CSRR entails reduction of a refrigerated system's configurational entropy, i.e., its localization in the position part of phase space (in position space for short), via positional isolation of entities that happen to be in their ground states. In TSRR, whether standard or absorption, a refrigerated system's thermal energy, as well as its thermal entropy, is reduced. By contrast, in CSRR only its configurational entropy is reduced: since the entities to be positionally isolated are already in their ground states, their thermal energy can*not* be reduced. In Sect. 3, we consider CSRR via positional isolation, by means of weighing or Stern-Gerlach apparatus, of entities that happen to be in their ground states, with reference to a specific one of the quantum-control-refrigeration (QCR) methods investigated in Ref. [1], but we will not employ this or any other QCR method per se.

Refrigeration of a system is also possible via extraction only of energy, but not of entropy, from this system. We dub this type of refrigeration as energy-reduction refrigeration (ERR). In order for energy to be extracted from a system without entropy being extracted from it, the energy must be extracted solely as work and not at all as heat. Simple examples include a one-time perfect (isentropic, reversible) adiabatic expansion of a gas, with energy but not entropy extracted from the gas solely via its doing work on its surroundings during expansion, and the one-time expansion of the photon gas comprising cosmic background radiation in an ever-expanding Universe (with no steady-state-theory-type "replacement"). In Sect. 2.1. and especially in Sect. 2.5. we will very briefly discuss one-time-expansion ERR, and in Sect. 3. we will very briefly discuss but not employ another type of ERR that is part of a QCR method. (S not E denotes entropy in TSRR and CSRR, and E denotes energy in ERR, because *S* is the standard symbol for entropy, and *E* for energy.)

zero under classical versus quantum mechanics are compared and contrasted. Then hot true and cold effective negative Kelvin temperatures are considered. A few fine points concerning the Third Law of Thermodynamics are briefly mentioned in the

**Keywords**: absolute zero, unattainability formulation of the Third Law of Thermodynamics, quantization, energy-time uncertainty principle, negative Kelvin

We consider first the absolute zero of temperature and then negative Kelvin temperatures. The unattainability formulation of the Third Law of Thermodynamics is briefly reviewed in Sect. 2.1. It puts limitations of the quest for absolute zero, and in its *strongest mode* forbids the attainment of absolute zero by *any method* whatsoever. But typically it is stated principally with respect to thermal-entropy-reduction refrigeration (TSRR). TSRR entails reduction of a refrigerated system's thermal entropy, i.e., its localization in the momentum part of phase space (in momentum space for short). The possibility or impossibility of overcoming these limitations via TSRR is considered, in Sects. 2.2. and 2.3. with respect to standard TSRR, and in Sect. 2.4. with respect to absorption TSRR. (In standard TSRR, refrigeration is achieved at the expense of work input; in absorption TSRR, at the expense of high-temperature heat

In Sect. 3, we consider the possibility or impossibility of the attainability of absolute zero temperature via configurational-entropy-reduction refrigeration (CSRR). CSRR entails reduction of a refrigerated system's configurational entropy, i.e., its localization in the position part of phase space (in position space for short), via positional isolation of entities that happen to be in their ground states. In TSRR, whether standard or absorption, a refrigerated system's thermal energy, as well as its thermal entropy, is reduced. By contrast, in CSRR only its configurational entropy is reduced: since the entities to be positionally isolated are already in their ground states, their thermal energy can*not* be reduced. In Sect. 3, we consider CSRR via positional isolation, by means of weighing or Stern-Gerlach apparatus, of entities that happen to be in their ground states, with reference to a specific one of the quantum-control-refrigeration (QCR) methods investigated in Ref. [1], but we will

Refrigeration of a system is also possible via extraction only of energy, but not of entropy, from this system. We dub this type of refrigeration as energy-reduction refrigeration (ERR). In order for energy to be extracted from a system without entropy being extracted from it, the energy must be extracted solely as work and not at all as heat. Simple examples include a one-time perfect (isentropic, reversible) adiabatic expansion of a gas, with energy but not entropy extracted from the gas solely via its doing work on its surroundings during expansion, and the one-time expansion of the photon gas comprising cosmic background

not employ this or any other QCR method per se.

Appendix.

262 Recent Advances in Thermo and Fluid Dynamics

temperatures.

**1. Introduction**

input.)

Of course, the Second Law of Thermodynamics requires any decrease in entropy of a refrigerated system to be paid for by a compensating greater (in the limit of perfection or reversibility, equal) increase in entropy. Or, in other words, the Second Law of Thermodynamics requires any localization in the total momentum-plus-position phase space of a refrigerated system to be paid for by a compensating greater (in the limit of perfection or reversibility, equal) delocalization in the total momentum-plus-position phase space of the refrigerated system and/or of its surroundings. If *all* of the entropy increase and associated waste heat owing to imperfection or irreversibility can be dumped into the surroundings rather than into a refrigerated system, then the refrigerated system will still be cooled to as low a temperature *as if* refrigeration were perfect (reversible), albeit at higher thermodynamic cost. Although perfect (isentropic, reversible) ERR entails zero *net* entropy change of a refrigerated system, within this zero ERR *does* entail a decrease in this system's thermal or momentum-space entropy (localization in momentum space) and an increase in its configurational or position-space entropy (delocalization in position space). If ERR is imperfect (irreversible) then the increase exceeds the decrease and hence the net entropy change is positive. But again if *all* of the entropy increase and associated waste heat owing to imperfection or irreversibility can be dumped into the surroundings rather than into refrigerated system, then the refrigerated system will still be cooled to as low a temperature *as if* ERR were perfect (reversible), albeit at higher thermodynamic cost.

All other things being equal, only if *all* waste heat owing to irreversibility is dumped outside of the system being refrigerated can imperfect (irreversible) refrigeration by *any* method (standard or absorption TSRR, CSRR, ERR, QCR, etc.) attain a temperature as low as that attainable via perfect (reversible) refrigeration, and then only at higher thermodynamic cost than via perfect (reversible) refrigeration. Otherwise, all other things being equal, imperfect (irreversible) refrigeration cannot attain as low a temperature as that attainable via perfect (reversible) refrigeration, even at higher thermodynamic cost than via perfect (reversible) refrigeration.

The Second Law of Thermodynamics forbids a negative change in total entropy, which would correspond to better-than-perfect refrigeration by *any* method (standard, absorption, or other) TSRR, CSRR, ERR, QCR, etc. (In Sect. 3.6, we will give a brief hypothetical consideration of better-than-perfect refrigeration.)

With respect to both TSRR and CSRR, we consider not only the issue of *attainability* of absolute zero, but also the separate issues, *even if* absolute zero can be *attained*, of *maintaining* it, and of *verifying* that it has been attained. The issues of attaining and maintaining absolute zero are considered in both Sects. 2. and 3. The issues of verifiability and *purely* dynamic as opposed to *thermo*dynamic — limitations on the quest for absolute zero are considered in Sect. 3, because they seem to be more transparently understandable with respect to CSRR, but in Sect. 3. we then relate them also with respect to TSRR. *Purely* dynamic — as opposed to *thermo*dynamic — limitations on the quest for absolute zero under classical versus quantum mechanics are compared in Sect. 3. Our considerations in Sects. 2. and 3. are general in nature, rather than of specific technical aspects of any particular refrigeration apparatus.

In Sect. 4, we briefly review hot true negative Kelvin temperatures, and then consider cold effective negative Kelvin temperatures. Brief concluding remarks are provided in Sect. 5. A few fine points concerning the third law of thermodynamics are briefly mentioned in the Appendix.

#### **2. The quest for absolute zero via TSRR**

#### **2.1. Limits imposed by the Second Law and by the unattainability formulation of the Third Law on TSRR**

According to the unattainability formulation of the Third Law of Thermodynamics, the absolute zero of temperature, 0 K, is unattainable in a finite number of finite operations [2–5].<sup>1</sup> But these operations are usually assumed to be TSRR operations [2–5], and most usually *standard* TSRR operations [2–5]. [It might be argued that a one-time infinite operation, for example a one-time infinite adiabatic expansion of a gas, or of the photon gas comprising cosmic background radiation in an ever-expanding Universe (with no steady-state-theory-type "replacement"), *can* via ERR attain 0 K. But (except perhaps for an ever-expanding Universe) a one-time infinite operation is as physically impossible and physically unrealizable as an infinite number of finite operations. Hence we will not employ one–time-expansion ERR in this chapter. (In this Sect. 2.1. and especially in Sect. 2.5. we will very briefly discuss one–time-expansion ERR. In Sect. 3. we will very briefly discuss but not employ another type of ERR that does not require infinite volume (and that is part of a QCR method) for cooling to 0 K, but which still encounters another difficulty with respect to cooling to 0 K.)]

*Standard* TSRR most typically entails, first, reducing the position-space or configurational entropy of a system to be refrigerated without a compensating increase in its momentum-space or thermal entropy. This first, isothermal, step is *necessary* but *preparatory*, itself *not* yielding a lowering of temperature. An example is isothermal compression of a gas, the heat of compression being expelled to the surroundings, with the surroundings rather than the system to be refrigerated thus suffering the required compensating increase in momentum-space or thermal entropy. The Second Law of Thermodynamics requires that the system's surroundings must suffer a larger (in the limit of perfection, equal) increase in momentum-space or thermal entropy than the decrease in the refrigerated system's position-space or configurational entropy owing to compression of the gas to within a smaller volume. In standard TSRR this is accomplished via heat transfer from the system to be refrigerated to its surroundings [2–5]. Thus momentum-space or thermal entropy is dumped from the system into its surroundings [2–5]. Other examples include isothermal condensation or magnetization, with the heat and thermal entropy thereby released similarly dumped into the surroundings [2–5]. In the second, adiabatic, step, the system to be refrigerated is thermally isolated, so that it can receive no heat from its surroundings. Then, via doing work

<sup>1</sup> (Re: Entries [2] and [3], Refs. [2] and [3]) Contrary to one minor statement on p. 30 of Ref. [3], internal energy *does* have a uniquely-defined zero, in accordance with *E* = *mc*2. Reference [3] does not render Ref. [2] obsolete, because Ref. [2] discusses aspects not discussed in Ref. [3], and vice versa.

on its surroundings and/or internally within itself, the refrigerated system trades an increase in its position-space or configurational entropy for a decrease in its momentum-space or thermal entropy. Examples include adiabatic expansion of a gas, wherein work is done on the surroundings, and adiabatic evaporation or demagnetization, wherein at least some of the work is done internally against attractive forces within the refrigerated system itself. The refrigerated system thus follows an adiabat towards a decrease in its temperature. Note that lowering of temperature occurs in this second step, the first step being necessary but preparatory. The *net* result of *both* steps is a decrease in our refrigerated system's momentum-space or thermal entropy but no change in its position-space or configurational entropy — localization in position space in the first step is undone by delocalization in position space in the second step. Thus the refrigerated system's *net* localization is in momentum space but not in position space: hence the designation TSRR. The second, adiabatic, step of standard TSRR, being the temperature-lowering step, may seem to be the more important one. But it would be impossible without the first, isothermal, preparatory step. Note that TSRR requires *heat* to be extracted from a refrigerated system at *some* point in the refrigeration process, even if not at the temperature-lowering step. In the examples of standard TSRR given in this paragraph, this required extraction of heat occurs in the first, isothermal, preparatory step.

mechanics are compared in Sect. 3. Our considerations in Sects. 2. and 3. are general in nature, rather than of specific technical aspects of any particular refrigeration apparatus. In Sect. 4, we briefly review hot true negative Kelvin temperatures, and then consider cold effective negative Kelvin temperatures. Brief concluding remarks are provided in Sect. 5. A few fine points concerning the third law of thermodynamics are briefly mentioned in the

**2.1. Limits imposed by the Second Law and by the unattainability formulation**

According to the unattainability formulation of the Third Law of Thermodynamics, the absolute zero of temperature, 0 K, is unattainable in a finite number of finite operations [2–5].<sup>1</sup> But these operations are usually assumed to be TSRR operations [2–5], and most usually *standard* TSRR operations [2–5]. [It might be argued that a one-time infinite operation, for example a one-time infinite adiabatic expansion of a gas, or of the photon gas comprising cosmic background radiation in an ever-expanding Universe (with no steady-state-theory-type "replacement"), *can* via ERR attain 0 K. But (except perhaps for an ever-expanding Universe) a one-time infinite operation is as physically impossible and physically unrealizable as an infinite number of finite operations. Hence we will not employ one–time-expansion ERR in this chapter. (In this Sect. 2.1. and especially in Sect. 2.5. we will very briefly discuss one–time-expansion ERR. In Sect. 3. we will very briefly discuss but not employ another type of ERR that does not require infinite volume (and that is part of a QCR method) for cooling to 0 K, but which still encounters another difficulty with respect to

*Standard* TSRR most typically entails, first, reducing the position-space or configurational entropy of a system to be refrigerated without a compensating increase in its momentum-space or thermal entropy. This first, isothermal, step is *necessary* but *preparatory*, itself *not* yielding a lowering of temperature. An example is isothermal compression of a gas, the heat of compression being expelled to the surroundings, with the surroundings rather than the system to be refrigerated thus suffering the required compensating increase in momentum-space or thermal entropy. The Second Law of Thermodynamics requires that the system's surroundings must suffer a larger (in the limit of perfection, equal) increase in momentum-space or thermal entropy than the decrease in the refrigerated system's position-space or configurational entropy owing to compression of the gas to within a smaller volume. In standard TSRR this is accomplished via heat transfer from the system to be refrigerated to its surroundings [2–5]. Thus momentum-space or thermal entropy is dumped from the system into its surroundings [2–5]. Other examples include isothermal condensation or magnetization, with the heat and thermal entropy thereby released similarly dumped into the surroundings [2–5]. In the second, adiabatic, step, the system to be refrigerated is thermally isolated, so that it can receive no heat from its surroundings. Then, via doing work

<sup>1</sup> (Re: Entries [2] and [3], Refs. [2] and [3]) Contrary to one minor statement on p. 30 of Ref. [3], internal energy *does* have a uniquely-defined zero, in accordance with *E* = *mc*2. Reference [3] does not render Ref. [2] obsolete, because

Ref. [2] discusses aspects not discussed in Ref. [3], and vice versa.

Appendix.

cooling to 0 K.)]

**2. The quest for absolute zero via TSRR**

**of the Third Law on TSRR**

264 Recent Advances in Thermo and Fluid Dynamics

Standard TSRR is perfect (reversible) if, in the first step, the decrease in the system's position-space or configurational entropy equals the increase in the surroundings' momentum-space or thermal entropy, and if, in the second step, the decrease in the system's momentum-space or thermal entropy equals the increase in the system's position-space or configurational entropy. Thus standard TSRR is perfect (reversible) if the *total* entropy change is zero for each step considered individually and for both steps combined. TSRR is imperfect (irreversible) if the total entropy change is positive. As per the fourth through sixth paragraphs of Sect. 1. applied specifically to TSRR, all other things being equal, only if *all* waste heat is dumped outside of the system being refrigerated can imperfect (irreversible) TSRR attain a temperature as low as that attainable via perfect (reversible) TSRR, and then only at greater thermodynamic cost than via perfect (reversible) TSRR. Otherwise, all other things being equal, imperfect (irreversible) TSRR cannot attain as low a temperature as that attainable via perfect (reversible) TSRR.

We will consider two types of TSRR. *Standard* TSRR, which we described briefly in the first two paragraphs of this Sect. 2.1, and which we will consider more detail in Sects. 2.2. and 2.3, is executed at the expense of *work* input. In standard TSRR, heat is extracted from a refrigerated system at the expense of *work* input. In Sect. 2.4, we will provide a brief comparison with *absorption* TSRR, which is executed at the expense of *heat* input from a high-temperature reservoir. In absorption TSRR, heat is extracted from a refrigerated system at the expense of *heat* input from a high-temperature reservoir.

To re-emphasize, TSRR, whether standard or absorption, always requires energy to be extracted from a refrigerated system via *heat* at *some* point in the refrigeration process, even if not at the temperature-lowering step. In standard TSRR methods described in the first paragraph of this Sect. 2.1. heat must be extracted from a refrigerated system in the first step to maintain the system isothermal despite compression and/or other localization in position space, even though no heat is extracted from it in the second, adiabatic, temperature-lowering step. Energy may also be extracted from a refrigerated system via work during standard TSRR. As we will see in Sect. 2.4, in absorption TSRR energy is extracted from a refrigerated system continuously via heat, but never via work.

Because entropy changes become ever smaller as 0 K is approached [2–5], and also because the rate of change of entropy with respect to temperature never becomes infinite [3], the temperature decrease attainable with each successive two-step isothermal-adiabatic standard-TSRR cycle becomes ever smaller [2–5]. Two adiabats can never intercept, and in particular no other adiabat can intercept that corresponding to zero-point entropy and hence to 0 K [2–5]: This is probably the *paramount* reason why, according to the unattainability formulation of the Third Law, of Thermodynamics 0 K cannot be reached in a finite number of finite standard-TSRR operations; the reasons cited in the first sentence of this paragraph probably being more of a supplementary nature [2–5]. [It might be argued that a one-time infinite adiabatic expansion of a gas or of the photon gas comprising cosmic background radiation in an ever-expanding Universe (with no steady-state-theory-type "replacement"), can via ERR attain 0 K, and thus via such infinite expansion its adiabat can intercept that corresponding to zero-point entropy and hence to 0 K. But (except perhaps for an ever-expanding Universe), a one-time infinite expansion is as physically impossible and physically unrealizable as an infinite number of finite operations.]

Beyond these considerations [2–5] concerning limitations on the quest for 0 K in the classical regime, there is of course a vast literature concerning the quest for 0 K, as well as concerning optimizing refrigerator operation, in the quantum regime. Extensive and thorough discussions, reviews, and bibliographies are provided in Refs. [6] and [7].<sup>2</sup> We also cite one specific study [8] (of very many). This study [8] investigates optimization of quantum refrigerator operation via maximization of the product of the time rate of heat extraction from a refrigerated system and the thermodynamic efficiency (coefficient of performance) of the refrigerator, maximization of both simultaneously being impossible [8]. The bottom line based on these sources [6–8] seems to be that even in the quantum regime 0 K cannot be attained with finite resources and in finite time [6–8]. Thus, based on these sources [6–8], quantum refrigeration also seems to be under the governance of the unattainability statement of the Third Law, at least in its strongest mode [6–8]. Yet there is an alternative viewpoint [1].

#### **2.2. The Third Law does** *not* **require** *infinite* **work to** *attain TC* = 0 K **via standard TSRR, but forbids performance of the required** *finite* **work**

The work required to cool any finite sample of matter (and/or of energy such as equilibrium blackbody radiation) maintained within a fixed finite volume *V* or at constant pressure *P* from any initial finite fixed, relatively hot ambient temperature *TH* to what is generally considered to be the ultimate cold temperature *TC* = 0 K via *standard* TSRR is *finite* indeed for typical room-temperature *TH* and for typical laboratory-size samples typically *small*. Hence the unattainability formulation of the Third Law of Thermodynamics does *not* forbid *attainment* of 0 K via standard TSRR *by requiring infinite work* for the process. Rather, it forbids *attainment* of 0 K via standard TSRR by forbidding the performance of the required *finite*, typically *small*, amount of work.

<sup>2</sup> (Re: Entry [6], Ref. [6]) The "quest for absolute zero" in the titles of Sects. 2. and 3. of this chapter was borrowed from the titles of Refs. 15 and 36 cited in Ref. [6] of this chapter.

While the coefficient of performance of standard TSRR decreases towards 0 as 0 K is approached, specific heats and heat capacities decrease even more rapidly towards 0 as 0 K is approached. Hence the *finite* — indeed typically *small* — amount of work required to *attain* 0 K via standard TSRR is *not* an issue. Let the cold temperature of a refrigerated system at a given stage of a standard-TSRR process be *TC* (*TH* assumed fixed). Let *dQC* be the maximum differential increment of heat that the Second Law of Thermodynamics allows to be extracted from this refrigerated system at temperature *TC*, with differential increment of heat *dQH* ejected at temperature *TH*, at the expense of a given differential increment of work *dW*. By the First Law of thermodynamics

$$dQ\_H = dQ\_{\overline{\mathcal{C}}} + dW. \tag{1}$$

The best possible standard-TSRR operation and hence the highest possible standard-TSRR coefficient of performance *COP*std allowed by the Second Law of Thermodynamics is in accordance with [9,10]

$$\begin{split} dS\_{\text{total}} &= dS\_{\text{C}} + dS\_{H} = \frac{dQ\_{\text{C}}}{T\_{\text{C}}} - \frac{dQ\_{H}}{T\_{H}} = \frac{dQ\_{\text{C}}}{T\_{\text{C}}} - \frac{dQ\_{\text{C}} + dW}{T\_{H}} = 0 \\ \implies \frac{dW}{T\_{H}} &= dQ\_{\text{C}} \left(\frac{1}{T\_{\text{C}}} - \frac{1}{T\_{H}}\right) = dQ\_{\text{C}} \frac{T\_{H} - T\_{\text{C}}}{T\_{\text{C}} \cdot T\_{H}} \\ \implies dW &= dQ\_{\text{C}} \frac{T\_{H} - T\_{\text{C}}}{T\_{\text{C}}} \\ \implies \text{COP}\_{\text{std}} &= \frac{dQ\_{\text{C}}}{dW} = \frac{T\_{\text{C}}}{T\_{H} - T\_{\text{C}}} \\ \implies \lim\_{T\_{\text{C}} \to 0 \,\text{K}} \text{COP}\_{\text{std}} &= \frac{T\_{\text{C}}}{T\_{H}}. \end{split} \tag{2}$$

In the third step of the first line line of Eq. (2) we applied the First Law of Thermodynamics [Eq. (1)]. In the last two lines of Eq. (2) the coefficient of performance *COP*std for standard TSRR is given in general and then in the limiting case *TC* → 0 K.

Let *QC* be the heat that must be extracted from our refrigerated system to cool it from *TH* to 0 K, *CX* (*TC*) be the heat capacity given condition *X* (*X* = *V* if constant volume, *X* = *P* if constant pressure) of this system at temperature *TC* as its temperature *TC* is lowered from *TH* towards 0 K, and *W* be the minimum work that the Second Law of Thermodynamics requires to cool it from *TH* to 0 K via standard TSRR. Then

$$Q\_{\mathbb{C}} = \int\_{0\,\mathrm{K}}^{T\_H} dQ\_{\mathbb{C}} = \int\_{0\,\mathrm{K}}^{T\_H} \mathbb{C}\_X \left( T\_{\mathbb{C}} \right) dT\_{\mathbb{C}} \tag{3}$$

and

TSRR. As we will see in Sect. 2.4, in absorption TSRR energy is extracted from a refrigerated

Because entropy changes become ever smaller as 0 K is approached [2–5], and also because the rate of change of entropy with respect to temperature never becomes infinite [3], the temperature decrease attainable with each successive two-step isothermal-adiabatic standard-TSRR cycle becomes ever smaller [2–5]. Two adiabats can never intercept, and in particular no other adiabat can intercept that corresponding to zero-point entropy and hence to 0 K [2–5]: This is probably the *paramount* reason why, according to the unattainability formulation of the Third Law, of Thermodynamics 0 K cannot be reached in a finite number of finite standard-TSRR operations; the reasons cited in the first sentence of this paragraph probably being more of a supplementary nature [2–5]. [It might be argued that a one-time infinite adiabatic expansion of a gas or of the photon gas comprising cosmic background radiation in an ever-expanding Universe (with no steady-state-theory-type "replacement"), can via ERR attain 0 K, and thus via such infinite expansion its adiabat can intercept that corresponding to zero-point entropy and hence to 0 K. But (except perhaps for an ever-expanding Universe), a one-time infinite expansion is as physically impossible and

Beyond these considerations [2–5] concerning limitations on the quest for 0 K in the classical regime, there is of course a vast literature concerning the quest for 0 K, as well as concerning optimizing refrigerator operation, in the quantum regime. Extensive and thorough discussions, reviews, and bibliographies are provided in Refs. [6] and [7].<sup>2</sup> We also cite one specific study [8] (of very many). This study [8] investigates optimization of quantum refrigerator operation via maximization of the product of the time rate of heat extraction from a refrigerated system and the thermodynamic efficiency (coefficient of performance) of the refrigerator, maximization of both simultaneously being impossible [8]. The bottom line based on these sources [6–8] seems to be that even in the quantum regime 0 K cannot be attained with finite resources and in finite time [6–8]. Thus, based on these sources [6–8], quantum refrigeration also seems to be under the governance of the unattainability statement of the Third Law, at least in its strongest mode [6–8]. Yet there is an alternative viewpoint [1].

**2.2. The Third Law does** *not* **require** *infinite* **work to** *attain TC* = 0 K **via standard**

The work required to cool any finite sample of matter (and/or of energy such as equilibrium blackbody radiation) maintained within a fixed finite volume *V* or at constant pressure *P* from any initial finite fixed, relatively hot ambient temperature *TH* to what is generally considered to be the ultimate cold temperature *TC* = 0 K via *standard* TSRR is *finite* indeed for typical room-temperature *TH* and for typical laboratory-size samples typically *small*. Hence the unattainability formulation of the Third Law of Thermodynamics does *not* forbid *attainment* of 0 K via standard TSRR *by requiring infinite work* for the process. Rather, it forbids *attainment* of 0 K via standard TSRR by forbidding the performance of the required

<sup>2</sup> (Re: Entry [6], Ref. [6]) The "quest for absolute zero" in the titles of Sects. 2. and 3. of this chapter was borrowed

system continuously via heat, but never via work.

266 Recent Advances in Thermo and Fluid Dynamics

physically unrealizable as an infinite number of finite operations.]

**TSRR, but forbids performance of the required** *finite* **work**

*finite*, typically *small*, amount of work.

from the titles of Refs. 15 and 36 cited in Ref. [6] of this chapter.

$$d\mathcal{W} = \frac{d\mathcal{Q}\_{\mathbb{C}}}{\text{COP}\_{\text{std}}} = \frac{T\_{H} - T\_{\text{C}}}{T\_{\text{C}}} d\mathcal{Q}\_{\text{C}} = \frac{T\_{H} - T\_{\text{C}}}{T\_{\text{C}}} \mathcal{C}\_{X} \left(T\_{\text{C}}\right) dT\_{\text{C}} < \frac{T\_{H}}{T\_{\text{C}}} d\mathcal{Q}\_{\text{C}} = \frac{T\_{H}}{T\_{\text{C}}} \mathcal{C}\_{X} \left(T\_{\text{C}}\right) dT\_{\text{C}}$$

$$\implies W = \int\_{0\text{K}}^{T\_{H}} d\mathcal{W} = \int\_{0\text{K}}^{T\_{H}} \frac{T\_{H} - T\_{\text{C}}}{T\_{\text{C}}} \mathcal{C}\_{X} \left(T\_{\text{C}}\right) dT\_{\text{C}} < T\_{H} \int\_{0\text{K}}^{T\_{H}} \frac{\mathcal{C}\_{X} \left(T\_{\text{C}}\right)}{T\_{\text{C}}} dT\_{\text{C}}.\tag{4}$$

It has been stated [12,13]: "The coefficient of performance becomes progressively smaller as the temperature *TC* decreases relative to *TH*. And if the temperature *TC* approaches zero, the coefficient of performance also approaches zero (assuming *TH* fixed). It therefore requires huge amounts of work to extract even trivially small quantities of heat from a system near *TC* = 0 K." But the quantities of heat that must be extracted from *any* laboratory-size system as *TC* → 0 K are *less* than trivially small; hence the required amounts of work are *less* than huge. If *CX* (*TC*) were constant, independent of *TC*, then by Eq. (4) the minimum work *W* required by the second law for cooling *any* finite system to 0 K, i.e., for reducing *TC* from *TH* to 0 K, would indeed diverge towards ∞ — but even then just barely (logarithmically) — as *TC* → 0 K. But if *CX* (*TC*) decreases *at all* — even however slowly — with decreasing *TC*, then by Eq. (4) the divergence of *W* as *TC* → 0 K is cured. In fact *CX* (*TC*) not merely decreases but decreases *rapidly* [14,15] as *TC* → 0 K; hence the divergence of *W* as *TC* → 0 K is not merely cured but cured by an extremely wide margin. *Any* system constrained within a fixed finite volume, or even within an unfixed but always finite volume for example corresponding to maintenance of constant pressure, *must* obey the two-state model [16] in the limit *TC* → 0 K [16], because (as will be discussed in Sect. 3.4.) quantum mechanics *requires* discrete energy levels and *forbids* an energy continuum in any system constrained within a fixed finite volume, or even within an unfixed but always finite volume for example corresponding to maintenance of constant pressure.<sup>3</sup> And for the two-state model [16] the heat capacity decreases nearly exponentially with decreasing temperature in the limit *TC* → 0 K [16]. We note that if *TH* is not too high it makes little difference whether we put *CX* (*TC*) = *CV* (*TC*) or *CX* (*TC*) = *CP* (*TC*) in Eqs. (3) and (4) [14,15], because always *TC* ≤ *TH*, with *CP* (*TC*) − *CV* (*TC*) → 0 and *CP* (*TC*) /*CV* (*TC*) → 1 (both from above) as *TC* → 0 K [14,15]. Indeed, since solids and liquids are typically only slightly compressible, for solids and liquids *CP* (*TC*) is typically only marginally larger than *CV* (*TC*) even at *TC* well above 0 K, even as far above 0 K as is consistent with the existence of solids and liquids [17,18].

But as was discussed in Sect. 2.1, in accordance with the unattainability formulation of the third law of thermodynamics [2–5], entropy changes become ever smaller as 0 K is approached, rates of change of entropy with respect to temperature never become infinite [3], and two adiabats can never intercept [2–5]. Hence the temperature decrease attainable with each successive two-step isothermal-adiabatic TSRR cycle becomes ever smaller [2–5]. Especially, two adiabats can never intercept, and in particular no other adiabat can intercept that corresponding to zero-point entropy and hence to 0 K [2–5]. Thus the unattainability formulation of the third law forbids the attainment of 0 K via standard TSRR *not* by requiring an *infinite* amount of work to cool a finite sample of matter (and/or energy) within a finite volume to 0 K, but rather by forbidding the *finite* — typically *small* — required amount of work from being performed via any standard-TSRR process [2–5]. While *COP*std as per Eq. (2) is the theoretical maximum and the work *W* as per Eq. (4) is the theoretical minimum allowed by the Second Law of Thermodynamics, for well-designed real-world standard TSRR systems the actual *COP*std is not more than a few times smaller and the actual required work

<sup>3</sup> (Re: Entry [16], Ref. [3]) In the version of the two-state model presented in Sects. 15-3 and 16-2 of Ref. [3], both states are nondegenerate. But the nearly exponential decrease of heat capacity with decreasing temperature as *T* → 0 K still obtains irrespective of any (finite) degeneracy of one or both states. Degeneracy of the ground state does not affect the heat capacity at all; *g*-fold degeneracy of the excited state multiplies *g*-fold the heat capacity as compared to that given a nondegenerate excited state as per Sect. 15-3 of Ref. [3].

is not more than a few times larger these theoretical limits, and this small numerical factor in no way contravenes our result.

#### **2.3. Can the difficulty of infinite** *power* **required to** *maintain TC* = 0 K **be overcome?**

It has been stated [12,13]: "The coefficient of performance becomes progressively smaller as the temperature *TC* decreases relative to *TH*. And if the temperature *TC* approaches zero, the coefficient of performance also approaches zero (assuming *TH* fixed). It therefore requires huge amounts of work to extract even trivially small quantities of heat from a system near *TC* = 0 K." But the quantities of heat that must be extracted from *any* laboratory-size system as *TC* → 0 K are *less* than trivially small; hence the required amounts of work are *less* than huge. If *CX* (*TC*) were constant, independent of *TC*, then by Eq. (4) the minimum work *W* required by the second law for cooling *any* finite system to 0 K, i.e., for reducing *TC* from *TH* to 0 K, would indeed diverge towards ∞ — but even then just barely (logarithmically) — as *TC* → 0 K. But if *CX* (*TC*) decreases *at all* — even however slowly — with decreasing *TC*, then by Eq. (4) the divergence of *W* as *TC* → 0 K is cured. In fact *CX* (*TC*) not merely decreases but decreases *rapidly* [14,15] as *TC* → 0 K; hence the divergence of *W* as *TC* → 0 K is not merely cured but cured by an extremely wide margin. *Any* system constrained within a fixed finite volume, or even within an unfixed but always finite volume for example corresponding to maintenance of constant pressure, *must* obey the two-state model [16] in the limit *TC* → 0 K [16], because (as will be discussed in Sect. 3.4.) quantum mechanics *requires* discrete energy levels and *forbids* an energy continuum in any system constrained within a fixed finite volume, or even within an unfixed but always finite volume for example corresponding to maintenance of constant pressure.<sup>3</sup> And for the two-state model [16] the heat capacity decreases nearly exponentially with decreasing temperature in the limit *TC* → 0 K [16]. We note that if *TH* is not too high it makes little difference whether we put *CX* (*TC*) = *CV* (*TC*) or *CX* (*TC*) = *CP* (*TC*) in Eqs. (3) and (4) [14,15], because always *TC* ≤ *TH*, with *CP* (*TC*) − *CV* (*TC*) → 0 and *CP* (*TC*) /*CV* (*TC*) → 1 (both from above) as *TC* → 0 K [14,15]. Indeed, since solids and liquids are typically only slightly compressible, for solids and liquids *CP* (*TC*) is typically only marginally larger than *CV* (*TC*) even at *TC* well above 0 K, even as far above

268 Recent Advances in Thermo and Fluid Dynamics

0 K as is consistent with the existence of solids and liquids [17,18].

to that given a nondegenerate excited state as per Sect. 15-3 of Ref. [3].

But as was discussed in Sect. 2.1, in accordance with the unattainability formulation of the third law of thermodynamics [2–5], entropy changes become ever smaller as 0 K is approached, rates of change of entropy with respect to temperature never become infinite [3], and two adiabats can never intercept [2–5]. Hence the temperature decrease attainable with each successive two-step isothermal-adiabatic TSRR cycle becomes ever smaller [2–5]. Especially, two adiabats can never intercept, and in particular no other adiabat can intercept that corresponding to zero-point entropy and hence to 0 K [2–5]. Thus the unattainability formulation of the third law forbids the attainment of 0 K via standard TSRR *not* by requiring an *infinite* amount of work to cool a finite sample of matter (and/or energy) within a finite volume to 0 K, but rather by forbidding the *finite* — typically *small* — required amount of work from being performed via any standard-TSRR process [2–5]. While *COP*std as per Eq. (2) is the theoretical maximum and the work *W* as per Eq. (4) is the theoretical minimum allowed by the Second Law of Thermodynamics, for well-designed real-world standard TSRR systems the actual *COP*std is not more than a few times smaller and the actual required work

<sup>3</sup> (Re: Entry [16], Ref. [3]) In the version of the two-state model presented in Sects. 15-3 and 16-2 of Ref. [3], both states are nondegenerate. But the nearly exponential decrease of heat capacity with decreasing temperature as *T* → 0 K still obtains irrespective of any (finite) degeneracy of one or both states. Degeneracy of the ground state does not affect the heat capacity at all; *g*-fold degeneracy of the excited state multiplies *g*-fold the heat capacity as compared The work *W* required to *attain TC* = 0 K as derived in Sect. 2.2. is that required *only* to *extract* all of the *internal* thermal energy *out of* a system to be refrigerated via standard TSRR, thus cooling it to *TC* = 0 K. But this *un*realistically assumes that *strictly zero external* thermal energy flows *into* this system in the meantime. Thus we must consider *not only* the work *W* required to *extract* all of the *internal* thermal energy *out of* a system to be refrigerated, thus cooling it to *TC* = 0 K, i.e., to *attain TC* = 0 K. We must *also* consider the power *P* = *dW* /*dt* and work *W* = *Pdt* required to overcome the flow of *external* thermal energy, i.e., heat flow *dQC*/*dt*, *into* our refrigerated system, requisite to *maintain* it at *TC* = 0 K. (Note: Time *t* should not be confused with temperature *T*.) Heat transfer into our refrigerated system, indeed heat transfer in general, occurs via three processes: conduction, radiation, and convection [19,20]. Heat transfer *dQC*/*dt* into our refrigerated system via conduction is proportional to *TH* <sup>−</sup> *TC* and via radiation to *<sup>T</sup>*<sup>4</sup> *<sup>H</sup>* <sup>−</sup> *<sup>T</sup>*<sup>4</sup> *<sup>C</sup>* [19,20]. While convection is a complex phenomenon, which may be either natural or forced, for simplicity and for argument's sake let us accept the most usual result [21], according to which heat transfer via natural convection is proportional to (*TH* − *TC*) 5/4 [21]. If we must have convection, we prefer natural convection to forced convection, because the former transfers heat less efficiently. Thus heat transfer *dQC*/*dt* via conduction is *a* (*TH* − *TC*), via radiation *b T*4 *<sup>H</sup>* <sup>−</sup> *<sup>T</sup>*<sup>4</sup> *C* , and via natural convection *c* (*TH* − *TC*) 5/4 [19,20]; the prefactors *a*, *b*, and *c* corresponding to conductive, radiative, and natural-convective heat transfer, respectively [19,20]. These three prefactors for a given system to be refrigerated are determined by its geometry (size, shape, surface area, etc.), by the type of insulation, usually at least to some extent by *TH* and/or *TC*, and by any other relevant properties [19,20]. Thus, applying the result for *COP*std from Eq. (2),

$$\begin{split} &\frac{d\mathbf{Q\_C}}{dt} = a\left(T\_H - T\_\mathcal{C}\right) + b\left(T\_H^4 - T\_\mathcal{C}^4\right) + c\left(T\_H - T\_\mathcal{C}\right)^{5/4} \\ &\implies P = \frac{d\mathbf{W'}}{dt} = \frac{d\mathbf{Q\_C}/dt}{CO\_\text{std}} = \frac{T\_H - T\_\mathcal{C}}{T\_\mathcal{C}} \frac{d\mathbf{Q\_C}}{dt} = \frac{T\_H - T\_\mathcal{C}}{T\_\mathcal{C}} \left[a\left(T\_H - T\_\mathcal{C}\right) + b\left(T\_H^4 - T\_\mathcal{C}^4\right) + c\left(T\_H - T\_\mathcal{C}\right)^{5/4}\right] \\ &\implies \lim\_{\mathbf{Y\_C}\to\mathbf{K}} \frac{d\mathbf{Q\_C}}{dt} = aT\_H + b\mathbf{T}\_H^4 + c\mathbf{T}\_H^{5/4} \\ &\implies \lim\_{\mathbf{Y\_C}\to\mathbf{K}} P = \lim\_{\mathbf{Y\_C}\to\mathbf{K}} \frac{d\mathbf{W'}}{dt} = \lim\_{\mathbf{Y\_C}\to\mathbf{K}} \frac{d\mathbf{Q\_C}/dt}{CO\_\text{std}} = \lim\_{\mathbf{Y\_C}\to\mathbf{K}} \frac{T\_H - T\_\mathcal{C}}{T\_\mathcal{C}} \frac{d\mathbf{Q\_C}}{dt} \\ &= \lim\_{\mathbf{Y\_C}\to\mathbf{K}} \frac{T\_H}{T\_\mathcal{C}} \left(aT\_H + b\mathbf{T}\_H^4 + c\mathbf{T}\_H^{5/4}\right) = \lim\_{\mathbf{Y\_C}\to\mathbf{K}} \frac{a\mathbf{T}\_H^2 + b\mathbf{T}\_H^2 + c\mathbf{T}\_H^{9/4}}{T\_\mathcal{C}}. \end{split} \tag{5}$$

This expression diverges towards ∞ as *TC* → 0 K unless *a*, *b*, and *c* decrease with decreasing *TC* at least as rapidly as *TC* itself, with *a* = *b* = *c* = 0 at least at *TC* = 0 K. But given that *a*, *b*, and *c* in general depend on *TH* as well as on *TC* and that *TH* is fixed, such a functional dependency of *a*, *b*, and *c* on *TC* seems unlikely. But perhaps we should not *a priori* rule it out as impossible.

The decrease, indeed the typically *rapid* decrease, of *CX* (*TC*) as *TC* → 0 K [14,15] allows *all* of a finite refrigerated system's *internal* thermal energy to be pumped *out* of it thus cooling it to *TC* = 0 K via standard TSRR with the expenditure of a finite (typically small) amount of work. The vanishing of *CX* (*TC*) as *TC* → 0 K within a refrigerated system more than compensates for the vanishing of *COP*std in Eq. (4) as *TC* → 0 K. But the vanishing of *CX* (*TC*) as *TC* → 0 K within a refrigerated system does *not* help insofar as overcoming the flow of *external* thermal energy, i.e., heat flow, from surroundings at ambient temperature *TH into* a refrigerated system, is concerned. Thus in Eq. (5) the vanishing of *COP*std as *TC* → 0 K is *not* compensated for. Hence *even if TC* = 0 K is *attained* infinite power is required to *maintain* it — *unless a*, *b*, and *c* decrease with decreasing *TC* at least as rapidly as *TC* itself, with *a* = *b* = *c* = 0 at least at *TC* = 0 K. Can this "*unless*" — implying that insulation must become *perfect* [23] as *TC* → 0 K — be realized? Again, given that *a*, *b*, and *c* in general depend on *TH* as well as on *TC* and that *TH* is fixed, such a functional dependency of *a*, *b*, and *c* on *TC*, it seems unlikely, but perhaps we should not *a priori* rule it out as impossible. But *even if* it is impossible, *TC* = 0 K may be not merely *attainable*, but also *maintainable*, *but only for an instant, or at most for a finite number of instants*.

Note that *COP*std = 0 obtains *only* at *exactly* the *point* value *TC* = 0 K [9,10]. This leaves open the possibility that even if *a*, *b*, and *c* do *not* decrease with decreasing *TC* at least as rapidly as *TC* itself and remain finite and positive at *TC* = 0 K, i.e., even if insulation does *not* become perfect as *TC* → 0 K, *TC* = 0 K could be *attained* and then *maintained for an instant*, because *P* need be infinite for only an infinitesimally short time so *W* = *Pdt* could still be finite. But if even given this *im*perfection *TC* = 0 K could thus be attained *even for an instant*, then it could be likewise re-attained for any arbitrarily large (but finite) number N of additional instants, since N *Pdt* would then still be finite. Any finite number of infinitesimally short time intervals still sum to an infinitesimally short time interval. Let the refrigeration process begin at time *t*<sup>0</sup> and be completed at time *t*1. Let *TH* be fixed, and for simplicity and for argument's sake let

$$T\_{\mathbb{C}}\left(t\right) = T\_H \left(1 - \frac{t - t\_0}{t\_1 - t\_0}\right)^{\gamma},\tag{6}$$

where *γ* is a fixed positive real number. Then, for simplicity and for argument's sake, let us consider only the part of the refrigeration process at *TC TH*. Within this part of the refrigeration process, *TC* has little room to decrease towards *TC* = 0 K, so since *TH* is fixed letting *a*, *b*, and *c* be constants independent of *TC* may be a good approximation. Thus we have, applying the last two lines of Eq. (5),

$$\begin{split} \mathcal{W} &= \int\_{t\_0}^{t\_1} P dt = \left( aT\_H^2 + bT\_H^5 + cT\_H^{9/4} \right) \int\_{t\_0}^{t\_1} \frac{dt}{T\_\mathbb{C}\left( t \right)} \\ &= \left( aT\_H^2 + bT\_H^5 + cT\_H^{9/4} \right) \int\_{t\_0}^{t\_1} \frac{dt}{T\_H\left( 1 - \frac{t - t\_0}{t\_1 - t\_0} \right)^\gamma} = \left( aT\_H + bT\_H^4 + cT\_H^{5/4} \right) \int\_{t\_0}^{t\_1} \frac{dt}{\left( 1 - \frac{t - t\_0}{t\_1 - t\_0} \right)^\gamma} \\ &= \left( aT\_H + bT\_H^4 + cT\_H^{5/4} \right) \int\_{t\_0}^{t\_1} \frac{dt}{\left[ \frac{t\_1 - t\_0 - (t - t\_0)}{t\_1 - t\_0} \right]^\gamma} = \left( aT\_H + bT\_H^4 + cT\_H^{5/4} \right) \left( t\_1 - t\_0 \right)^\gamma \int\_{t\_0}^{t\_1} \frac{dt}{\left( t\_1 - t \right)^\gamma} \end{split}$$

$$\implies \text{if } 0 < \gamma < 1 \text{ then } W' = \left( aT\_H + bT\_H^4 + cT\_H^{5/4} \right) \left( t\_1 - t\_0 \right)^{\gamma} \frac{(t\_1 - t\_0)^{1 - \gamma}}{1 - \gamma} = \left( aT\_H + bT\_H^4 + cT\_H^{5/4} \right) \frac{t\_1 - t\_0}{1 - \gamma}$$

$$\implies \begin{aligned} &= \left( aT\_H + bT\_H^4 + cT\_H^{5/4} \right) \frac{t\_1 - t\_0}{1 - \gamma} \\ &\implies W' \text{ is finite if } 0 < \gamma < 1 \end{aligned}$$

$$\implies W\_{\text{total}} = W + W' \text{ is finite if } 0 < \gamma < 1. \tag{7}$$

In the last step of Eq. (7) we applied the finiteness of our result for *W* as per Eq. (4) and the associated discussions. Thus it seems that the difficulty of infinite power and infinite work required to *maintain TC* = 0 K can, at least to this very limited extent, be overcome.

The decrease, indeed the typically *rapid* decrease, of *CX* (*TC*) as *TC* → 0 K [14,15] allows *all* of a finite refrigerated system's *internal* thermal energy to be pumped *out* of it thus cooling it to *TC* = 0 K via standard TSRR with the expenditure of a finite (typically small) amount of work. The vanishing of *CX* (*TC*) as *TC* → 0 K within a refrigerated system more than compensates for the vanishing of *COP*std in Eq. (4) as *TC* → 0 K. But the vanishing of *CX* (*TC*) as *TC* → 0 K within a refrigerated system does *not* help insofar as overcoming the flow of *external* thermal energy, i.e., heat flow, from surroundings at ambient temperature *TH into* a refrigerated system, is concerned. Thus in Eq. (5) the vanishing of *COP*std as *TC* → 0 K is *not* compensated for. Hence *even if TC* = 0 K is *attained* infinite power is required to *maintain* it — *unless a*, *b*, and *c* decrease with decreasing *TC* at least as rapidly as *TC* itself, with *a* = *b* = *c* = 0 at least at *TC* = 0 K. Can this "*unless*" — implying that insulation must become *perfect* [23] as *TC* → 0 K — be realized? Again, given that *a*, *b*, and *c* in general depend on *TH* as well as on *TC* and that *TH* is fixed, such a functional dependency of *a*, *b*, and *c* on *TC*, it seems unlikely, but perhaps we should not *a priori* rule it out as impossible. But *even if* it is impossible, *TC* = 0 K may be not merely *attainable*, but also *maintainable*, *but*

Note that *COP*std = 0 obtains *only* at *exactly* the *point* value *TC* = 0 K [9,10]. This leaves open the possibility that even if *a*, *b*, and *c* do *not* decrease with decreasing *TC* at least as rapidly as *TC* itself and remain finite and positive at *TC* = 0 K, i.e., even if insulation does *not* become perfect as *TC* → 0 K, *TC* = 0 K could be *attained* and then *maintained for an instant*, because *P* need be infinite for only an infinitesimally short time so *W* = *Pdt* could still be finite. But if even given this *im*perfection *TC* = 0 K could thus be attained *even for an instant*, then it could be likewise re-attained for any arbitrarily large (but finite) number N of additional instants, since N *Pdt* would then still be finite. Any finite number of infinitesimally short time intervals still sum to an infinitesimally short time interval. Let the refrigeration process begin at time *t*<sup>0</sup> and be completed at time *t*1. Let *TH* be fixed, and for simplicity and for

*TC* (*t*) = *TH*

*<sup>H</sup>* + *cT*9/4 *H*

> *TH* <sup>1</sup> <sup>−</sup> *<sup>t</sup>*−*t*<sup>0</sup> *t*1−*t*<sup>0</sup>

 *<sup>t</sup>*<sup>1</sup> *t*0

 *<sup>t</sup>*<sup>1</sup> *t*0

 *<sup>t</sup>*<sup>1</sup> *t*0

*dt*

*dt t*1−*t*0−(*t*−*t*0) *t*1−*t*<sup>0</sup>

where *γ* is a fixed positive real number. Then, for simplicity and for argument's sake, let us consider only the part of the refrigeration process at *TC TH*. Within this part of the refrigeration process, *TC* has little room to decrease towards *TC* = 0 K, so since *TH* is fixed letting *a*, *b*, and *c* be constants independent of *TC* may be a good approximation. Thus we

> *dt TC* (*t*)

*<sup>γ</sup>* <sup>=</sup> 

*<sup>γ</sup>* <sup>=</sup>  *aTH* + *bT*<sup>4</sup>

*aTH* + *bT*<sup>4</sup>

*<sup>H</sup>* + *cT*5/4 *H*

*<sup>H</sup>* + *cT*5/4 *H* (*t*<sup>1</sup> − *t*0) *γ <sup>t</sup>*<sup>1</sup> *t*0

 *<sup>t</sup>*<sup>1</sup> *t*0

 <sup>1</sup> <sup>−</sup> *<sup>t</sup>*−*t*<sup>0</sup> *t*1−*t*<sup>0</sup> *γ*

<sup>1</sup> <sup>−</sup> *<sup>t</sup>* <sup>−</sup> *<sup>t</sup>*<sup>0</sup> *t*<sup>1</sup> − *t*<sup>0</sup> *γ*

, (6)

*dt*

*dt* (*t*<sup>1</sup> − *t*) *γ*

*only for an instant, or at most for a finite number of instants*.

270 Recent Advances in Thermo and Fluid Dynamics

argument's sake let

*W* = *<sup>t</sup>*<sup>1</sup> *t*0

= *aT*<sup>2</sup> *<sup>H</sup>* + *bT*<sup>5</sup>

= 

have, applying the last two lines of Eq. (5),

*<sup>H</sup>* + *cT*9/4 *H*

*<sup>H</sup>* + *cT*5/4 *H*

*Pdt* = *aT*<sup>2</sup> *<sup>H</sup>* + *bT*<sup>5</sup>

*aTH* + *bT*<sup>4</sup>

*W* and hence *W*total = *W* + *W* can remain finite if *TC* = 0 K is to be maintained for *finitely longer than* an instant or finite number of instants given given fixed finite *TH >* 0 K only if *a*, *b*, and *c* decrease with decreasing *TC* at least as rapidly as *TC* itself, with *a* = *b* = *c* = 0 at least at *TC* = 0 K. And this in the face of *a*, *b*, and *c* in general depending on *TH* as well as on *TC*, with *TH* being fixed. Thus we require *perfect* [23] — not merely good — insulation at *TC* = 0 K, yet also with fixed finite *TH >* 0 K. And perfect insulation — *a* = *b* = *c* = 0 — is hard to come by. Hard, but perhaps not impossible. Again, for simplicity and for argument's sake, let us consider only the part of the refrigeration process at *TC TH*. By surrounding our refrigerated system including its insulation with a vacuum, and with the insulation being comprised entirely of solids (not fluids: liquids or gases), we can indeed achieve *c* = 0 solids certainly exist at finite *TH >* 0 K. Convection (whether natural or forced) occurs only in fluids (gases and liquids), and is nonexistent in solids or in a vacuum. But even though we have thus achieved *c* = 0, we must still achieve *a* = 0 and *b* = 0. Superinsulators — perfect (not merely good) — insulators with respect to electricity have recently been discovered [23], with the superinsulating state existing at temperatures up to *TSI*,elec,max finitely greater than 0 K [23]. (Reference [23] provides a thorough and excellent review, as well as an extensive bibliography.) So perhaps we should not *a priori* rule out superinsulators with respect to heat, with the superinsulating state existing at 0 K ≤ *TH* ≤ *TSI*,heat,max [23]. Even if superinsulation with respect to heat exists, we do not know if *TSI*,heat,max = *TSI*,elec,max. But all we require is that 0 K *< TH < TSI*,heat,max [23]. If superinsulation with respect to heat exists, then *a* = 0 can obtain at finite *TH >* 0 K. [Superinsulation should not be confused with the typical exponential improvement of ordinary insulation with decreasing temperature. The latter obtains, for example, if conduction of heat and electricity is via electrons thermally promoted from the valence band to the conduction band with the two bands separated by a fixed finite energy gap ∆*E*. Then the probability of such promotion per attempt to jump the gap decreases exponentially with decreasing *TH* in accordance with the Boltzmann factor *e*−∆*E*/*kTH* (*k* is Boltzmann's constant) and hence is very small for low *TH*. But it does not vanish *perfectly* except at *TH* = 0 K and hence ordinary insulation remains *im*perfect at any finite *TH >* 0 K.] If furthermore the vacuum surrounding our refrigerated system with its superinsulating shield with respect to heat is permeated by equilibrium blackbody radiation at fixed finite *TH* below the upper temperature limit *TSI*,heat,max of the superinsulating state with respect to heat, then this radiation will not destroy the superinsulating state. (Indeed a vacuum *must* be permeated by equilibrium blackbody radiation at any temperature finitely greater than 0 K.) If the superinsulator is opaque to this equilibrium blackbody radiation, with the radiation being thermalized in its outer layer to internal energy at *TH < TSI*,heat,max, or scattered or reflected away, then *b* = 0. A nonopaque superinsulator can be shielded by an opaque material, with the radiation being thermalized in the outer layer of this opaque material to internal energy at *TH < TSI*,max, or scattered or reflected away, so that *b* = 0. Of course for *b* = 0 *exactly* the opacity must be not merely good but *perfect*. While this perfection may be impossible to achieve *exactly*, it can be achieved *for all practical purposes*. Typically the fraction of incident radiation not thermalized as internal energy within an opaque material, or not scattered or reflected away, decreases exponentially increasing thickness of an opaque material. An incident photon has a probability of *e*−N of penetrating through a thickness of N or more *e*-folding lengths without being thermalized as internal energy within an opaque material, or being scattered or reflected away. If, say, N 1000, then for all practical purposes we can rest assured that not even 1 photon will get through during the time required for any refrigeration experiment and hence that, even if not exactly then for all practical purposes, *b* = 0.

Thus at least *prima facie* it seems that there seems to be no difficulty in principle in achieving *c* = 0, and even if not perfectly then for all practical purposes also *b* = 0. The main concern is whether or not *a* = 0 is achievable, namely whether or not superinsulation exists with respect to heat as it does with respect to electricity [23]. Probably the best that we can do at this point is to admit that we do not know; that this is an open question [23].

#### **2.4. A brief comparison with absorption TSRR**

The discussions in Sects. 2.2. and 2.3. presuppose *standard* TSRR, which operates as a heat engine in reverse. In heat engine operation heat flows from a hot reservoir via the engine into a cold reservoir; within the limit imposed by the Second Law of Thermodynamics, the engine can convert part of this heat flow into work output. Standard TSRR operates as a heat engine in reverse, with work input driving heat flow from a cold reservoir into a hot one, also within the limit imposed by the Second Law of Thermodynamics.

However, there is one other commonly-employed type of TSRR that we wish to consider *absorption* TSRR [24]. While absorption TSRR is not employed in practice to reach cryogenic temperatures, let alone to approach 0 K, it may be of interest to consider it even if only in principle. Absorption TSRR requires zero work input [24]. Instead, heat *QH* is supplied to the refrigeration apparatus from a hot reservoir at temperature *TH*, heat *QC* is extracted by the refrigeration apparatus from a refrigerated system at cold temperature *TC*, and heat *QI* is ejected from the refrigeration apparatus at intermediate temperature *TI* (*TH > TI > TC*) [24]. By the First Law of Thermodynamics

$$dQ\_I = dQ\_\mathbb{C} + dQ\_H. \tag{8}$$

Let the cold temperature of a refrigerated system at a given stage of an absorption-TSRR process be *TC* (*TI* and *TH* assumed fixed). Let *dQC* be the maximum differential increment of heat that the Second Law of Thermodynamics allows to be extracted from this refrigerated system at temperature *TC* at the expense of a given differential increment of heat input *dQH* at temperature *TH*, with differential increment of heat *dQI* = *dQC* + *dQH* ejected at temperature *TI*. The best possible absorption-TSRR operation and hence the highest possible absorption-TSRR coefficient of performance *COP*abs allowed by the Second Law of Thermodynamics is in accordance with

$$\begin{split}d\mathcal{S}\_{\text{total}} &= dS\_{I} + dS\_{\text{C}} + dS\_{H} = \frac{dQ\_{1}}{T\_{I}} - \frac{dQ\_{C}}{T\_{\text{C}}} - \frac{dQ\_{H}}{T\_{H}} = 0\\ \implies & \frac{dQ\_{C} + dQ\_{H}}{T\_{I}} - \frac{dQ\_{C}}{T\_{\text{C}}} - \frac{dQ\_{H}}{T\_{H}} = \frac{dQ\_{C}}{T\_{I}} + \frac{dQ\_{H}}{T\_{I}} - \frac{dQ\_{C}}{T\_{\text{C}}} - \frac{dQ\_{H}}{T\_{H}} = 0\\ \implies & dQ\_{C} \left(\frac{1}{T\_{\text{C}}} - \frac{1}{T\_{I}}\right) = dQ\_{H} \left(\frac{1}{T\_{I}} - \frac{1}{T\_{H}}\right)\\ \implies & dQ\_{C} \frac{T\_{I} - T\_{\text{C}}}{T\_{I}T\_{\text{C}}} = dQ\_{H} \frac{T\_{H} - T\_{I}}{T\_{I}T\_{H}}\\ \implies & dQ\_{C} \frac{T\_{I} - T\_{\text{C}}}{T\_{\text{C}}} = dQ\_{H} \frac{T\_{H} - T\_{I}}{T\_{H}}\\ \implies & \underline{\nabla P}\_{\text{abs}} = \frac{dQ\_{\text{C}}}{dQ\_{H}} = \frac{T\_{\text{C}} \left(T\_{H} - T\_{I}\right)}{T\_{H} \left(T\_{I} - T\_{\text{C}}\right)}\\ \implies & \lim\_{T\_{\text{C}} \to \text{K}} \underline{\text{CO}}\_{\text{abs}} = \frac{T\_{\text{C}} \left(T\_{H} - T\_{I}\right)}{T\_{I}T\_{H}}.\end{split} \tag{9}$$

In the second line of Eq. (9) we applied the First Law of Thermodynamics [Eq. (8)]. In the last two lines of Eq. (9) the coefficient of performance *COP*abs for absorption TSRR is given in general and then in the limiting case *TC* → 0 K.

Now let us compare *COP*abs with *COP*std. By comparing Eqs. (2) and (9), we obtain

$$\frac{\text{COP}\_{\text{abs}}}{\text{COP}\_{\text{std}}} = \frac{\frac{T\_{\text{C}}(T\_{H} - T\_{I})}{T\_{\text{H}}(T\_{I} - T\_{\text{C}})}}{\frac{T\_{\text{C}}}{T\_{H} - T\_{\text{C}}}} = \frac{(T\_{H} - T\_{I})}{T\_{H}(T\_{I} - T\_{\text{C}})} = \left(1 - \frac{T\_{\text{C}}}{T\_{H}}\right)\frac{T\_{H} - T\_{I}}{T\_{I} - T\_{\text{C}}} = \left(1 - \frac{T\_{I}}{T\_{H}}\right)\frac{T\_{H} - T\_{\text{C}}}{T\_{I} - T\_{\text{C}}}.\tag{10}$$

$$\implies \lim\_{T\_{\text{C}} \to \text{0K}} \frac{\text{COP}\_{\text{abs}}}{\text{COP}\_{\text{std}}} = \frac{T\_{H} - T\_{I}}{T\_{I}} = \frac{T\_{H}}{T\_{I}} - 1.\tag{10}$$

Thus

with the radiation being thermalized in its outer layer to internal energy at *TH < TSI*,heat,max, or scattered or reflected away, then *b* = 0. A nonopaque superinsulator can be shielded by an opaque material, with the radiation being thermalized in the outer layer of this opaque material to internal energy at *TH < TSI*,max, or scattered or reflected away, so that *b* = 0. Of course for *b* = 0 *exactly* the opacity must be not merely good but *perfect*. While this perfection may be impossible to achieve *exactly*, it can be achieved *for all practical purposes*. Typically the fraction of incident radiation not thermalized as internal energy within an opaque material, or not scattered or reflected away, decreases exponentially increasing thickness of an opaque material. An incident photon has a probability of *e*−N of penetrating through a thickness of N or more *e*-folding lengths without being thermalized as internal energy within an opaque material, or being scattered or reflected away. If, say, N 1000, then for all practical purposes we can rest assured that not even 1 photon will get through during the time required for any refrigeration experiment and hence that, even if not exactly then for all practical purposes,

Thus at least *prima facie* it seems that there seems to be no difficulty in principle in achieving *c* = 0, and even if not perfectly then for all practical purposes also *b* = 0. The main concern is whether or not *a* = 0 is achievable, namely whether or not superinsulation exists with respect to heat as it does with respect to electricity [23]. Probably the best that we can do at

The discussions in Sects. 2.2. and 2.3. presuppose *standard* TSRR, which operates as a heat engine in reverse. In heat engine operation heat flows from a hot reservoir via the engine into a cold reservoir; within the limit imposed by the Second Law of Thermodynamics, the engine can convert part of this heat flow into work output. Standard TSRR operates as a heat engine in reverse, with work input driving heat flow from a cold reservoir into a hot one,

However, there is one other commonly-employed type of TSRR that we wish to consider *absorption* TSRR [24]. While absorption TSRR is not employed in practice to reach cryogenic temperatures, let alone to approach 0 K, it may be of interest to consider it even if only in principle. Absorption TSRR requires zero work input [24]. Instead, heat *QH* is supplied to the refrigeration apparatus from a hot reservoir at temperature *TH*, heat *QC* is extracted by the refrigeration apparatus from a refrigerated system at cold temperature *TC*, and heat *QI* is ejected from the refrigeration apparatus at intermediate temperature *TI* (*TH > TI > TC*) [24].

Let the cold temperature of a refrigerated system at a given stage of an absorption-TSRR process be *TC* (*TI* and *TH* assumed fixed). Let *dQC* be the maximum differential increment of heat that the Second Law of Thermodynamics allows to be extracted from this refrigerated system at temperature *TC* at the expense of a given differential increment of heat input *dQH* at temperature *TH*, with differential increment of heat *dQI* = *dQC* + *dQH* ejected at temperature *TI*. The best possible absorption-TSRR operation and hence the highest

*dQI* = *dQC* + *dQH*. (8)

this point is to admit that we do not know; that this is an open question [23].

also within the limit imposed by the Second Law of Thermodynamics.

**2.4. A brief comparison with absorption TSRR**

By the First Law of Thermodynamics

*b* = 0.

272 Recent Advances in Thermo and Fluid Dynamics

$$\begin{aligned} \text{COP}\_{\text{abs}} &> \text{COP}\_{\text{std}} \text{ if } \left(T\_H - T\_I\right) \left(T\_H - T\_{\text{C}}\right) > T\_H \left(T\_I - T\_{\text{C}}\right) \\ \implies T\_H^2 - T\_I T\_H - T\_{\text{C}} T\_H + T\_{\text{C}} T\_I &> T\_I T\_H - T\_{\text{C}} T\_H \\ \implies T\_H^2 - 2T\_I T\_H + T\_{\text{C}} T\_I &> 0 \\ \implies T\_I \left(T\_C - 2T\_H\right) &> -T\_H^2 \\ \implies T\_I \left(2T\_H - T\_{\text{C}}\right) &< T\_H^2 \\ \implies T\_I &< \frac{T\_H^2}{2T\_H - T\_{\text{C}}} \\ \implies T\_I &< \frac{T\_H}{2 - \frac{T\_{\text{C}}}{T\_H}} \\ \implies T\_I &< \frac{T\_H}{2} \text{ if } T\_{\text{C}} \ll T\_H. \end{aligned}$$

Thus it may be of interest to consider absorption TSRR, even if only in principle, because [24]: (a) It is thermodynamically less costly to supply a *given* quantity of energy input as *heat*, which is sufficient for absorption TSRR, than as *work*, which is required for standard TSRR. (b) If Inequality (11) is fulfilled, then absorption TSRR requires a *smaller* quantity of energy input as *heat* than standard TSRR does as *work*. (c) Some absorption-TSRR systems, notably the Munters/von-Platen system [24] and the Einstein/Szilárd system [25] (which however at least in their original forms cannot attain cryogenic temperatures, let alone approach 0 K), have no moving parts, which minimizes waste of negentropy and free energy via friction while maximizing reliability; also, they operate essentially silently, thus wasting essentially no negentropy and free energy as sound. By Inequality (11), the upper limit of *TI* consistent with *COP*abs *> COP*std, i.e., with absorption TSRR requiring less heat input than standard TSRR does work input for given *TH* and *TC*, never falls below *TH*/2 even in the limit *TC* → 0 K. Hence even after *TC* has been reduced sufficiently that the last lines of Eqs. (2), (9), and (10) and Inequality (11) are applicable, if *TI < TH*/2 then the quantities of *QH*, *dQH*/*dt*, and *Q <sup>H</sup>* <sup>=</sup> *dQH dt dt* required for absorption TSRR are *smaller* than those of *W*, *P*, and *W* , respectively, required in accordance with Eqs. (4), (5), and (7), respectively, for standard TSRR — besides being thermodynamically less costly *per given quantity*. Thus it seems that perhaps we should not *a priori* rule out that approaching or even attaining *TC* = 0 K may be easier, at least in principle even if not in practice, via absorption TSRR, or perhaps via some variant or modification thereof, than via standard TSRR.

Nevertheless, for *TI* finitely greater than 0 K (of course *TI > TC*) the advantage of absorption TSRR over standard TSRR is finite. Hence we should restate the first paragraph of Sect. 2.2. with respect to absorption TSRR: The heat input *QH* required to cool any finite sample of matter (and/or of energy such as equilibrium blackbody radiation) maintained within a fixed finite volume *V* or at fixed finite pressure *P* from any initial finite fixed, relatively hot ambient temperature *TH* to *TC* = 0 K via absorption TSRR is *finite* — indeed for typical room-temperature *TH* and for typical laboratory-size samples typically *small*. Hence the unattainability formulation of the Third Law of Thermodynamics does *not* forbid attainment of 0 K via absorption TSRR *by requiring infinite QH* for the process. Rather, it forbids *attainment* of 0 K via absorption TSRR by forbidding the utilization of the required *finite*, typically *small*, *QH*. While *COP*abs as per Eq. (9) is the theoretical maximum allowed by the Second Law of Thermodynamics, for well-designed real-world absorption TSRR systems the actual *COP*abs is not more than a few times smaller this theoretical maximum, and this small numerical factor in no way contravenes our result.

But *even if TC* = 0 K *could* be precisely *attained*, whether via standard, absorption, or other TSRR, the question of *maintaining TC* = 0 K discussed in Sect. 2.3. is still open. *Even if TC* = 0 K *could* be precisely *attained*, whether via standard, absorption, or other TSRR, whether or not it is *maintainable* for finitely longer than the infinitesimally short time allowed in accordance with Eqs. (6) and (7) and the associated discussions is still open. But least in principle even if not in practice *if TC* = 0 K can be *attained*, *then maintaining TC* = 0 K, whether this is possible only for infinitesimally short time or for finite time, may be more easily achievable via absorption TSRR than via standard TSRR.

#### **2.5. Brief remarks concerning one–time-expansion ERR**

One-time-expansion ERR can, at least in principle, be achieved via a sample of gas at ambient pressure at Earth's surface (or the surface of any other planet with an atmosphere) being transported to a vacuum (either to a vacuum chamber on the planet or to the vacuum of space), and there being allowed to expand adiabatically. For a given expansion ratio, maximum cooling is attained via a perfect (reversible) adiabatic expansion, wherein the decrease in thermal (momentum-space) entropy exactly offsets the increase in configurational (position-space) entropy owing to expansion. But even an irreversible adiabatic expansion is ERR, because even in an irreversible adiabatic expansion only energy and not entropy is extracted from the gas. But in an irreversible adiabatic expansion some thermal (momentum-space) entropy is created within the gas, so that the decrease in thermal (momentum-space) entropy only partially offsets the increase in configurational (position-space) entropy owing to expansion, and hence refrigeration is less efficient, with less cooling per given expansion ratio, than in the perfect (reversible) case. It might be argued that no vacuum that the gas expands into is perfect and hence its expansion cannot continue indefinitely. But in an ever-expanding Universe (with no steady-state-theory-type "replacement") the surrounding vacuum becomes ever more perfect. But (except perhaps for an ever-expanding Universe), a one-time infinite expansion is as physically impossible and physically unrealizable as an infinite number of finite operations.

Thus it may be of interest to consider absorption TSRR, even if only in principle, because [24]: (a) It is thermodynamically less costly to supply a *given* quantity of energy input as *heat*, which is sufficient for absorption TSRR, than as *work*, which is required for standard TSRR. (b) If Inequality (11) is fulfilled, then absorption TSRR requires a *smaller* quantity of energy input as *heat* than standard TSRR does as *work*. (c) Some absorption-TSRR systems, notably the Munters/von-Platen system [24] and the Einstein/Szilárd system [25] (which however at least in their original forms cannot attain cryogenic temperatures, let alone approach 0 K), have no moving parts, which minimizes waste of negentropy and free energy via friction while maximizing reliability; also, they operate essentially silently, thus wasting essentially no negentropy and free energy as sound. By Inequality (11), the upper limit of *TI* consistent with *COP*abs *> COP*std, i.e., with absorption TSRR requiring less heat input than standard TSRR does work input for given *TH* and *TC*, never falls below *TH*/2 even in the limit *TC* → 0 K. Hence even after *TC* has been reduced sufficiently that the last lines of Eqs. (2), (9), and (10) and Inequality (11) are applicable, if *TI < TH*/2 then the quantities of *QH*, *dQH*/*dt*,

*dt dt* required for absorption TSRR are *smaller* than those of *W*, *P*, and *W*

respectively, required in accordance with Eqs. (4), (5), and (7), respectively, for standard TSRR — besides being thermodynamically less costly *per given quantity*. Thus it seems that perhaps we should not *a priori* rule out that approaching or even attaining *TC* = 0 K may be easier, at least in principle even if not in practice, via absorption TSRR, or perhaps via some variant or

Nevertheless, for *TI* finitely greater than 0 K (of course *TI > TC*) the advantage of absorption TSRR over standard TSRR is finite. Hence we should restate the first paragraph of Sect. 2.2. with respect to absorption TSRR: The heat input *QH* required to cool any finite sample of matter (and/or of energy such as equilibrium blackbody radiation) maintained within a fixed finite volume *V* or at fixed finite pressure *P* from any initial finite fixed, relatively hot ambient temperature *TH* to *TC* = 0 K via absorption TSRR is *finite* — indeed for typical room-temperature *TH* and for typical laboratory-size samples typically *small*. Hence the unattainability formulation of the Third Law of Thermodynamics does *not* forbid attainment of 0 K via absorption TSRR *by requiring infinite QH* for the process. Rather, it forbids *attainment* of 0 K via absorption TSRR by forbidding the utilization of the required *finite*, typically *small*, *QH*. While *COP*abs as per Eq. (9) is the theoretical maximum allowed by the Second Law of Thermodynamics, for well-designed real-world absorption TSRR systems the actual *COP*abs is not more than a few times smaller this theoretical maximum, and this small numerical

But *even if TC* = 0 K *could* be precisely *attained*, whether via standard, absorption, or other TSRR, the question of *maintaining TC* = 0 K discussed in Sect. 2.3. is still open. *Even if TC* = 0 K *could* be precisely *attained*, whether via standard, absorption, or other TSRR, whether or not it is *maintainable* for finitely longer than the infinitesimally short time allowed in accordance with Eqs. (6) and (7) and the associated discussions is still open. But least in principle even if not in practice *if TC* = 0 K can be *attained*, *then maintaining TC* = 0 K, whether this is possible only for infinitesimally short time or for finite time, may be more

One-time-expansion ERR can, at least in principle, be achieved via a sample of gas at ambient pressure at Earth's surface (or the surface of any other planet with an

,

and *Q*

*<sup>H</sup>* <sup>=</sup> *dQH*

274 Recent Advances in Thermo and Fluid Dynamics

modification thereof, than via standard TSRR.

factor in no way contravenes our result.

easily achievable via absorption TSRR than via standard TSRR.

**2.5. Brief remarks concerning one–time-expansion ERR**

It might also be argued that a one-time infinite operation, for example a one-time infinite adiabatic expansion of a gas after infinite time in an ever-expanding Universe (with no steady-state-theory-type "replacement"), or of the photon gas comprising cosmic background radiation after infinite time in an ever-expanding Universe (with no steady-state-theory-type "replacement"), *can* via ERR attain 0 K. But this requires *infinite* resources — *infinite* volume and *infinite* time. Hence it does not contravene the conclusion that absolute zero 0 K can*not* be attained with *finite* resources — not only classically but even in the quantum regime [6–8] that we will consider in Sect. 3. Hence we do not employ one–time-expansion ERR in this chapter. [In Sect. 3. we will very briefly discuss but not employ another type of ERR that does not require infinite volume (and that is part of a QCR method) for cooling to 0 K, but which still encounters another difficulty with respect to cooling to 0 K.]

We note that, not unlike an irreversible adiabatic expansion of a gas, a polytropic expansion thereof intermediate between adiabatic and isothermal can achieve refrigeration, albeit less efficiently, with less cooling per given expansion ratio, than a perfect (reversible) adiabatic one. A polytropic expansion intermediate between adiabatic and isothermal can be construed as ERR, because only energy and not entropy is *extracted from* the gas. Thermal (momentum-space) entropy is *imported into* the gas during a polytropic expansion, so that the decrease in thermal (momentum-space) entropy only partially offsets the increase in configurational (position-space) entropy owing to expansion, and hence refrigeration is less efficient than in the perfect (reversible) adiabatic case. The only difference between an irreversible adiabatic expansion and a polytropic one intermediate between adiabatic and isothermal is that thermal (momentum-space) entropy is generated within the expanding gas in the former case and imported into it in the latter. [Of course, additional irreversibilities can result in thermal (momentum-space) entropy being generated within the expanding gas during a polytropic expansion intermediate between adiabatic and isothermal, thus rendering refrigeration still less efficient.]

Perfect (reversible) one-time-expansion adiabatic ERR of our gas, or even imperfect (irreversible) adiabatic or even polytropic (intermediate between adiabatic and isothermal) ERR thereof, yields rather than costs work. Of course, this does not count the work that it costs to evacuate its vacuum chamber or to transport it to the vacuum of space.

Of course, except perhaps for the cooling of or in an ever-expanding Universe (with no steady-state-theory-type "replacement"), the difficulties of *maintaining* cold as opposed to merely *attaining* it apply with respect to one–time-expansion ERR as with respect to standard and absorption TSRR. These difficulties also apply with respect to CSRR, QCR, and another type of ERR that is part of a QCR method, all to be considered in Sect. 3.

#### **3. The quest for absolute zero via configurational-entropy-reduction refrigeration (CSRR)**

In Sect. 3. we consider the quest for absolute zero, *TC* = 0 K, via configurational-entropy-reduction refrigeration (CSRR), which localizes a refrigerated system in the position part of phase space (in position space for short), as opposed to thermal-entropy-reduction refrigeration (TSRR), which localizes it in the momentum part of phase space (in momentum space for short). Standard TSRR requires extraction of energy from a refrigerated system via *heat* during at least *some* step of the refrigeration process. It may also entail extraction of energy from a refrigerated system via work (recall Sect. .2.1. In absorption TSRR energy is extracted from a refrigerated system continuously via heat, but never via work (recall Sect. 2.4.). By contrast, CSRR entails *no* extraction of energy from a refrigerated system *either via heat or via work*. CSRR requires finite work input to *attain TC* = 0 K, even if this work input is employed differently than the work input in standard TSRR as per Sect. 2.2, or than high-temperature heat input in absorption TSRR as per Sect. 2.4. CSRR shares with TSRR the difficulties of *maintaining TC* = 0 K as per Sect. 2.3. and the last paragraphs of Sects. 2.4. and 2.5. But in Sect. 3. let us focus mainly on prospects for and limitations on the quest for *attaining TC* = 0 K via CSRR, comparing these prospects and limitations with those via TSRR. We postpone remarking on the difficulties of *maintaining TC* = 0 K via CSRR until the last two paragraphs of Sect. 3.5.

#### **3.1. Questioning the unattainability formulation of the Third Law of Thermodynamics in toto**

The unattainability formulation of the Third Law of Thermodynamics *in toto* — not merely any particular limit(s) imposed thereby — has been questioned [1]. Above all, the question of the attainability of 0 K in a finite number of finite operations (perhaps even in one) by *any method* whatsoever, and hence the status of the unattainability formulation of the Third Law of Thermodynamics in its *strongest* mode, according to which this is impossible, remains open [1,26–28]. Even so, the question of whether or not 0 K is attainable by *any method* whatsoever is sometimes stated to be only of academic interest [27], and it is also sometimes stated that there may be "profound problems [22]" concerning attaining "absolute thermal isolation [22]," i.e., perfect insulation [23], and that infinitely precise measurements [22] may be required to *perfectly verify* [22] that *precisely* 0 K has actually been attained [22].

Yet it has been shown that 0 K may be attainable in a finite number of finite operations (perhaps even in one) via quantum-control-refrigeration (QCR) methods, specifically, employing quantum coherence [1]. This challenges the *strongest-mode* unattainability formulation of the Third Law of Thermodynamics, which forbids the attainment of 0 K by *any method* whatsoever [1].

ERR thereof, yields rather than costs work. Of course, this does not count the work that it

Of course, except perhaps for the cooling of or in an ever-expanding Universe (with no steady-state-theory-type "replacement"), the difficulties of *maintaining* cold as opposed to merely *attaining* it apply with respect to one–time-expansion ERR as with respect to standard and absorption TSRR. These difficulties also apply with respect to CSRR, QCR, and another

In Sect. 3. we consider the quest for absolute zero, *TC* = 0 K, via configurational-entropy-reduction refrigeration (CSRR), which localizes a refrigerated system in the position part of phase space (in position space for short), as opposed to thermal-entropy-reduction refrigeration (TSRR), which localizes it in the momentum part of phase space (in momentum space for short). Standard TSRR requires extraction of energy from a refrigerated system via *heat* during at least *some* step of the refrigeration process. It may also entail extraction of energy from a refrigerated system via work (recall Sect. .2.1. In absorption TSRR energy is extracted from a refrigerated system continuously via heat, but never via work (recall Sect. 2.4.). By contrast, CSRR entails *no* extraction of energy from a refrigerated system *either via heat or via work*. CSRR requires finite work input to *attain TC* = 0 K, even if this work input is employed differently than the work input in standard TSRR as per Sect. 2.2, or than high-temperature heat input in absorption TSRR as per Sect. 2.4. CSRR shares with TSRR the difficulties of *maintaining TC* = 0 K as per Sect. 2.3. and the last paragraphs of Sects. 2.4. and 2.5. But in Sect. 3. let us focus mainly on prospects for and limitations on the quest for *attaining TC* = 0 K via CSRR, comparing these prospects and limitations with those via TSRR. We postpone remarking on the difficulties of *maintaining*

**3. The quest for absolute zero via configurational-entropy-reduction**

costs to evacuate its vacuum chamber or to transport it to the vacuum of space.

type of ERR that is part of a QCR method, all to be considered in Sect. 3.

*TC* = 0 K via CSRR until the last two paragraphs of Sect. 3.5.

**3.1. Questioning the unattainability formulation of the Third Law of**

be required to *perfectly verify* [22] that *precisely* 0 K has actually been attained [22].

The unattainability formulation of the Third Law of Thermodynamics *in toto* — not merely any particular limit(s) imposed thereby — has been questioned [1]. Above all, the question of the attainability of 0 K in a finite number of finite operations (perhaps even in one) by *any method* whatsoever, and hence the status of the unattainability formulation of the Third Law of Thermodynamics in its *strongest* mode, according to which this is impossible, remains open [1,26–28]. Even so, the question of whether or not 0 K is attainable by *any method* whatsoever is sometimes stated to be only of academic interest [27], and it is also sometimes stated that there may be "profound problems [22]" concerning attaining "absolute thermal isolation [22]," i.e., perfect insulation [23], and that infinitely precise measurements [22] may

Yet it has been shown that 0 K may be attainable in a finite number of finite operations (perhaps even in one) via quantum-control-refrigeration (QCR) methods, specifically, employing quantum coherence [1]. This challenges the *strongest-mode* unattainability

**refrigeration (CSRR)**

276 Recent Advances in Thermo and Fluid Dynamics

**Thermodynamics in toto**

In Sect. 3.2, we will first consider CSRR via positional isolation by means of weighing of entities that happen to be in the ground state. Perhaps in principle, even if not in practice, at least *prima facie* this seems to be the simplest possible method of CSRR. So perhaps it may elucidate at least some of the problems of attaining 0 K, and if 0 K can be attained of *verifying* [22] that 0 K has been attained, more easily than the more technically advanced and more practical QCR methods discussed in Ref. [1], which are much more amenable to realization using currently-available technology [1]. We then consider CSRR via positional isolation, by means of a Stern-Gerlach apparatus, of entities that happen to be in the ground state. Our consideration of CSRR via a Stern-Gerlach apparatus will be with reference to a specific one of the QCR methods [1], but we will not employ this or any other QCR method per se. In this regard, in the sixth paragraph of Sect. 3.2. we will briefly discuss but not employ another type of ERR than that discussed in Sect. 2.1 and especially in Sect. 2.5, which entails reduction of a refrigerated system's *non*thermal energy but not of its entropy.

Thus irrespective of the status of TSRR with respect to the unattainability formulation of the Third Law of Thermodynamics, there also exist CSRR methods, which we will consider in this Sect. 3. Even if, as will turn out to at least apparently be the case, even CSRR methods are limited by the strongest-mode unattainability formulation of the Third Law of Thermodynamics, they at least seem to be closer to breaking through this limit than TSRR methods. The ultimate limitation that the unattainability formulation of the Third Law of Thermodynamics can wield in its strongest mode seems to be *purely* dynamic as opposed to *thermo*dynamic — the energy-time uncertainty principle. Thus *exact* attainment of 0 K may be protected against *any* type of refrigeration: TSRR, CSRR, ERR, QCR, or otherwise (or any combination thereof). But the only slightly less ambitious goal of attainment of 0 K *for all practical purposes* seems to be within reach.

#### **3.2. Absolute zero via CSRR (for example isolation in position space by weighing or by Stern-Gerlach apparatus)?**

Consider System A comprised of *N* identical harmonic oscillators, in thermodynamic equilibrium with a heat reservoir at temperature *T*, an average *n* of which are in the ground state. (Averaging is denoted by enclosure within angular brackets.) Let ∆*E* be the gap between adjacent energy states of any given oscillator. Let *T* be low enough so that the probability of even one of the harmonic oscillators being in its second or higher excited states is negligible. In accordance with the Boltzmann distribution, the probability **P***A*<sup>1</sup> of any given System-A oscillator being in its first excited state is *e*−∆*E*/*kT* times the probability **P***A*<sup>0</sup> of being in its ground state. (Note: Probability **P** should not be confused with power *P*.) Hence, normalizing yields **P***A*<sup>0</sup> . <sup>=</sup> *<sup>n</sup>* /*<sup>N</sup>* <sup>=</sup> 1/ 1 + *e*−∆*E*/*kT* and **P***A*<sup>1</sup> . = (*N* − *n*) /*N* = 1 − (*n* /*N*) = 1 − 1/ 1 + *e*−∆*E*/*kT* = *e*−∆*E*/*kT*/ <sup>1</sup> <sup>+</sup> *<sup>e</sup>*−∆*E*/*kT* . = *e*−∆*E*/*kT*. [The dot-equal sign ( .

=) means "very nearly equal to."] Of course, *T* being small enough so that the probability of even one of the harmonic oscillators being in its second or higher excited states is negligible typically implies that **P***A*<sup>1</sup> 1. But if *N* is moderately but not excessively large this can obtain consistently with *N***P***A*<sup>1</sup> = *N* − *n*, the average number of oscillators in the first excited state, exceeding unity.

An oscillator in the first excited state has a mass exceeding that of one in the ground state by ∆*E*/*c*2, and, letting *g* be the local acceleration due to gravity, a weight exceeding that of one in the ground state by *g*∆*E*/*c*2. (From now on unless otherwise noted we take *c* to be the speed of light in vacuum, not the prefactor defined in Sect. 2.3.) Thus (in principle!) the oscillators in the ground state in our original System A at temperature *T* can be positionally isolated by weighing from those in the first excited state therein — creating in only one operation (albeit consisting of *n* weighing steps) Subsystem B comprised of *n* oscillators (*n* ≤ *N*), *all* of which are in the ground state. *Prima facie* it seems that Subsystem B is therefore indeed at the absolute zero of temperature, 0 K. Moreover such positional isolation can in principle be executed via employment only of work interactions and hence with zero heat transfer, either into our ground-state-only Subsystem B or otherwise.

The required work is modest. The entropy — more correctly, *neg*entropy — cost of isolating the first of the *n* ground-state oscillators is ∆*S*isol,1 = *k* ln *<sup>N</sup> <sup>n</sup>* . The negentropy cost of isolating the second of the *<sup>n</sup>* ground-state oscillators is <sup>∆</sup>*S*isol,2 <sup>=</sup> *<sup>k</sup>* ln *<sup>N</sup>*−<sup>1</sup> *<sup>n</sup>*−<sup>1</sup> , with 1 subtracted from *N* in the numerator of the argument of the logarithm because after the first oscillator has been isolated there are 1 fewer total oscillators left in our original System A and in the denominator thereof because there are 1 fewer ground-state oscillators left therein. The negentropy cost of isolating the third of the *<sup>n</sup>* ground-state oscillators is <sup>∆</sup>*S*isol,3 <sup>=</sup> *<sup>k</sup>* ln *<sup>N</sup>*−<sup>2</sup> *<sup>n</sup>*−<sup>2</sup> , of isolating the *<sup>j</sup>*th (1 <sup>≤</sup> *<sup>j</sup>* <sup>≤</sup> *<sup>n</sup>*) <sup>∆</sup>*S*isol,*<sup>j</sup>* <sup>=</sup> *<sup>k</sup>* ln *<sup>N</sup>*−(*j*−1) *<sup>n</sup>*−(*j*−1) <sup>=</sup> *<sup>k</sup>* ln *<sup>N</sup>*−*j*+<sup>1</sup> *<sup>n</sup>*−*j*+<sup>1</sup> , of isolating the *<sup>n</sup>*th and last <sup>∆</sup>*S*isol,*<sup>n</sup>* <sup>=</sup> *<sup>k</sup>* ln *<sup>N</sup>*−*n*+<sup>1</sup> *<sup>n</sup>*−*n*+<sup>1</sup> <sup>=</sup> *<sup>k</sup>* ln (*<sup>N</sup>* <sup>−</sup> *<sup>n</sup>* <sup>+</sup> <sup>1</sup>). Note that the negentropy cost of isolating ground-state oscillators increases with each one isolated and is highest for the last one isolated. Recalling that *T* is the temperature of our original System A, the work required to isolate the *<sup>j</sup>*th of the *<sup>n</sup>* ground-state oscillators is *<sup>W</sup>*isol,*<sup>j</sup>* <sup>=</sup> *<sup>T</sup>*∆*S*isol,*<sup>j</sup>* <sup>=</sup> *kT* ln *<sup>N</sup>*−*j*+<sup>1</sup> *<sup>n</sup>*−*j*+<sup>1</sup> . Thus, if at temperature *T* on average *n* of the *N* harmonic oscillators comprising our original System A are in their ground states, the expectation values of the total negentropy cost ∆*S*isol,total and total work cost *W*isol,total = *T*∆*S*isol,total of isolating all ground-state oscillators into Subsystem B are, to sufficient accuracy, given by and bounded from above in accordance with:

$$
\left< \Delta S\_{\text{isol,total}} \right> = \sum\_{j=1}^{\langle n \rangle} \left< \Delta S\_{\text{isol,j}} \right> = k \sum\_{j=1}^{\langle n \rangle} \ln \frac{N - j + 1}{\langle n \rangle - j + 1} < \left< n \right> k \ln \left( N - \left< n \right> + 1 \right)
$$

$$
\implies \left< W\_{\text{isol,total}} \right> = T \left< \Delta S\_{\text{isol,total}} \right> = kT \sum\_{j=1}^{\langle n \rangle} \Delta S\_{\text{isol,j}} = kT \sum\_{j=1}^{\langle n \rangle} \ln \frac{N - j + 1}{\langle n \rangle - j + 1} < \left< n \right> kT \ln \left( N - \left< n \right> + 1 \right). \tag{12}
$$

(If *n* is not an integer, then the sums in Eq. (12) are, to sufficient accuracy, construed as encompassing all integers *j* from 1 up through and including the one immediately below *n* and then also encompassing the noninteger *n*.) The inequalities in Eq. (12), bounding ∆*S*isol,total and *W*isol,total = *T* ∆*S*isol,total from above, are justified because the negentropy cost of isolating ground-state oscillators increases with each one isolated and is highest for the last one isolated. Thus even the upper bounds on the negentropy and work costs are modest. The negentropy and work costs computed in Eq. (12) assume thermodynamic perfection (reversibility). But even given typical imperfection (irreversibility), which is inevitable in practice as opposed to in principle, the upper bounds on the actual negentropy and work costs would typically be only a few times larger, and hence still modest. The Second Law of Thermodynamics requires that the decrease in entropy associated with localizing ground-state oscillators into Subsystem B be paid for by an increase in entropy elsewhere. The payment for any irreversibilities is most typically via waste heat, which must be dumped anywhere except into Subsystem B. It is best dumped into System A's heat reservoir (not into System A itself). The temperature of this reservoir and hence also of System A itself need not be measurably raised if this heat reservoir is very large and/or is comprised of a substance in its two-phase regime. Of course, this waste heat payment will be larger given imperfect (irreversible) than perfect (reversible) operation, but typically only a few times larger.

An oscillator in the first excited state has a mass exceeding that of one in the ground state by ∆*E*/*c*2, and, letting *g* be the local acceleration due to gravity, a weight exceeding that of one in the ground state by *g*∆*E*/*c*2. (From now on unless otherwise noted we take *c* to be the speed of light in vacuum, not the prefactor defined in Sect. 2.3.) Thus (in principle!) the oscillators in the ground state in our original System A at temperature *T* can be positionally isolated by weighing from those in the first excited state therein — creating in only one operation (albeit consisting of *n* weighing steps) Subsystem B comprised of *n* oscillators (*n* ≤ *N*), *all* of which are in the ground state. *Prima facie* it seems that Subsystem B is therefore indeed at the absolute zero of temperature, 0 K. Moreover such positional isolation can in principle be executed via employment only of work interactions and hence with zero heat transfer, either

The required work is modest. The entropy — more correctly, *neg*entropy — cost of isolating

the second of the *<sup>n</sup>* ground-state oscillators is <sup>∆</sup>*S*isol,2 <sup>=</sup> *<sup>k</sup>* ln *<sup>N</sup>*−<sup>1</sup> *<sup>n</sup>*−<sup>1</sup> , with 1 subtracted from *N* in the numerator of the argument of the logarithm because after the first oscillator has been isolated there are 1 fewer total oscillators left in our original System A and in the denominator thereof because there are 1 fewer ground-state oscillators left therein. The negentropy cost of isolating the third of the *<sup>n</sup>* ground-state oscillators is <sup>∆</sup>*S*isol,3 <sup>=</sup> *<sup>k</sup>* ln *<sup>N</sup>*−<sup>2</sup> *<sup>n</sup>*−<sup>2</sup> ,

last <sup>∆</sup>*S*isol,*<sup>n</sup>* <sup>=</sup> *<sup>k</sup>* ln *<sup>N</sup>*−*n*+<sup>1</sup> *<sup>n</sup>*−*n*+<sup>1</sup> <sup>=</sup> *<sup>k</sup>* ln (*<sup>N</sup>* <sup>−</sup> *<sup>n</sup>* <sup>+</sup> <sup>1</sup>). Note that the negentropy cost of isolating ground-state oscillators increases with each one isolated and is highest for the last one isolated. Recalling that *T* is the temperature of our original System A, the work required

if at temperature *T* on average *n* of the *N* harmonic oscillators comprising our original System A are in their ground states, the expectation values of the total negentropy cost ∆*S*isol,total and total work cost *W*isol,total = *T*∆*S*isol,total of isolating all ground-state oscillators into Subsystem B are, to sufficient accuracy, given by and bounded from above in accordance

*<sup>n</sup>* <sup>−</sup> *<sup>j</sup>* <sup>+</sup> <sup>1</sup> *<sup>&</sup>lt; <sup>n</sup> <sup>k</sup>* ln (*<sup>N</sup>* <sup>−</sup> *<sup>n</sup>* <sup>+</sup> <sup>1</sup>)

*n* ∑ *j*=1

∆*S*isol,total

ln *<sup>N</sup>* <sup>−</sup> *<sup>j</sup>* <sup>+</sup> <sup>1</sup>

to isolate the *<sup>j</sup>*th of the *<sup>n</sup>* ground-state oscillators is *<sup>W</sup>*isol,*<sup>j</sup>* <sup>=</sup> *<sup>T</sup>*∆*S*isol,*<sup>j</sup>* <sup>=</sup> *kT* ln *<sup>N</sup>*−*j*+<sup>1</sup>

ln *<sup>N</sup>* <sup>−</sup> *<sup>j</sup>* <sup>+</sup> <sup>1</sup>

*n* ∑ *j*=1

*W*isol,total

∆*S*isol,*<sup>j</sup>* = *kT*

= *T*

(If *n* is not an integer, then the sums in Eq. (12) are, to sufficient accuracy, construed as encompassing all integers *j* from 1 up through and including the one immediately below *n* and then also encompassing the noninteger *n*.) The inequalities in Eq. (12),

the negentropy cost of isolating ground-state oscillators increases with each one isolated and is highest for the last one isolated. Thus even the upper bounds on the negentropy and work costs are modest. The negentropy and work costs computed in Eq. (12) assume thermodynamic perfection (reversibility). But even given typical imperfection (irreversibility), which is inevitable in practice as opposed to in principle, the upper bounds on the actual negentropy and work costs would typically be only a few times larger, and

*<sup>n</sup>*−(*j*−1) <sup>=</sup> *<sup>k</sup>* ln *<sup>N</sup>*−*j*+<sup>1</sup>

*<sup>n</sup>* . The negentropy cost of isolating

*<sup>n</sup>*−*j*+<sup>1</sup> , of isolating the *<sup>n</sup>*th and

*<sup>n</sup>* <sup>−</sup> *<sup>j</sup>* <sup>+</sup> <sup>1</sup> *<sup>&</sup>lt; <sup>n</sup> kT* ln (*<sup>N</sup>* <sup>−</sup> *<sup>n</sup>* <sup>+</sup> <sup>1</sup>). (12)

from above, are justified because

*<sup>n</sup>*−*j*+<sup>1</sup> . Thus,

into our ground-state-only Subsystem B or otherwise.

278 Recent Advances in Thermo and Fluid Dynamics

of isolating the *<sup>j</sup>*th (1 <sup>≤</sup> *<sup>j</sup>* <sup>≤</sup> *<sup>n</sup>*) <sup>∆</sup>*S*isol,*<sup>j</sup>* <sup>=</sup> *<sup>k</sup>* ln *<sup>N</sup>*−(*j*−1)

with:

∆*S*isol,total =

bounding

*n* ∑ *j*=1 ∆*S*isol,*<sup>j</sup>* = *k n* ∑ *j*=1

=⇒ *W*isol,total = *T* ∆*S*isol,total = *kT*

∆*S*isol,total

and

the first of the *n* ground-state oscillators is ∆*S*isol,1 = *k* ln *<sup>N</sup>*

Note, first, that this is a CSRR (as opposed to TSRR) operation, entailing only *positional* isolation of the oscillators by weight. The isolation and hence *localization* of the *n* ground-state oscillators into Subsystem B is in the position, not the momentum, part of phase space. Entropy is the logarithmic measure of delocalization and negentropy the logarithmic measure of localization. The negentropy cost for reversible CSRR by weighing is a *localization* cost paid for by work not heat (although the entropy cost exacted owing to any irreversibilities is typically via waste heat, which must be dumped anywhere except into Subsystem B preferably into System A's heat reservoir). Moreover no energy — neither heat nor work — is extracted from either System A or Subsystem B at *any* point during our CSRR process. (Indeed since the oscillators to comprise Subsystem B are in their ground states, energy can*not* be extracted from them.) This is in contrast with both standard and absorption TSRR (recall Sect. 2. and the first paragraph of this Sect.3). Second, since the difference in masses and therefore also weights between an oscillator being in its ground or first excited state is finite, we circumvent the objection that infinitely precise measurements [22] would be required to verify [22] that *precisely* 0 K has been attained. Third, while the unattainability formulation of the Third Law of Thermodynamics forbids the expenditure of the typically small amount of work required to attain 0 K via standard TSRR and the expenditure of the typically small amount of high-temperature heat required to attain 0 K via absorption TSRR, it does *not* forbid the expenditure of the typically small amount of work required to attain 0 K via CSRR. Fourth, we stated "in principle!" — i.e., as a thought experiment — no currently-available or even currently-foreseeable practical weighing technology is sensitive enough. This is in contrast to the QCR systems investigated in Ref. [1], which although more complex, are realizable in practice using currently-available technology.

So does our positional isolation of the *n* ground-state oscillators into Subsystem B at least *prima facie* seem to challenge the strongest-mode unattainability formulation of the Third Law of Thermodynamics [2–5]? If the unattainability formulation of the Third Law of Thermodynamics in its strongest mode *does* forbid attaining 0 K via CSRR, then it must be for *another reason*. As will be discussed in Sects. 3.3.–3.5, this other reason is *purely* dynamic rather than *thermo*dynamic — the energy-time uncertainty principle.

Of the methods discussed in Ref. [1], the one closest to our weighing thought-experiment discussed in the five immediately preceding paragraphs seems to be that discussed in Sect. 3 of Ref. [1] — but employing only the *first step* of that method. Similarly to the weighing thought-experiment example discussed in the five immediately preceding paragraphs, the proposed real system discussed in Sect. 3 of Ref. [1] (System A by our notation) consists of a mixture of atoms, some of which are in the ground state and some in the first excited state. Also as in our weighing thought-experiment, the temperature is assumed low enough so that the probability of occupancy of the second or higher excited states is negligible. This first step of the method employed in Sect. 3 of Ref. [1] entails positional isolation of atoms on the ground state from those in the first excited state by a Stern-Gerlach apparatus. The subsystem comprised of atoms in the ground state after positional isolation via the Stern-Gerlach apparatus constitutes Subsystem B, our subsystem at the absolute zero of temperature, 0 K. This positional isolation of atoms is a CSRR process. This Stern-Gerlach-apparatus version of our thought experiment may, in accordance with Sect. 3 of Ref. [1], be more realizable experimentally than the weighing version thereof.

In contrast with Sect. 3 of Ref. [1], in the Stern-Gerlach modification of our weighing CSRR method no attempt is made to thence also de-excite the atoms in the first excited state down to the ground state, the *second step* of the QCR method discussed in Sect. 3 of Ref. [1]. It is important to recognize that this *second step* is *neither* a TSRR process *nor* a CSRR process. The entropy of a set of atoms is zero if they are *all in the same quantum state*, irrespective of whether this quantum state is the ground state or not. (But see the last paragraph of the Appendix concerning this point.) In the particular case currently under consideration, we have a set of atoms *all in the first excited state*. Their de-excitation from the first excited state to the ground state maintains their entropy constant at zero. Thus it is *not* an entropy-reduction (SR) process — it is neither a TSRR process nor a CSRR process. It is rather *another type* of refrigeration process, which we have dubbed as *energy*-reduction refrigeration (ERR): *energy E* but *not* entropy *S* is extracted to de-excite these atoms from the first excited state to the ground state. Thus, as was the case with one-time-expansion ERR which we considered in Sect. 2, especially in Sect. 2.5, this version of ERR yields rather than costs work. But unlike one-time-expansion ERR which we considered in Sect. 2.1, especially in Sect. 2.5, this version of ERR process does *not* require infinite *volume* to attain 0 K. Moreover, the extracted energy *E* is *non*thermal, because the oscillators are initially all in the first excited state, *not* in a Boltzmann distribution among states. Similarly as is the case with respect to the expenditure of the typically small amount of work required to attain 0 K via CSRR, the unattainability formulation of the Third Law of Thermodynamics does *not* forbid ERR by forbidding the extraction of the typically small amount of *non*thermal energy required to de-excite these atoms from the first excited state to the ground state. Thus if the unattainability formulation of the Third Law of Thermodynamics in its strongest mode *does* forbid ERR, then, as with CSRR, it must be for *another reason*. As will be discussed in Sects. 3.3.–3.5, this other reason is, as with CSRR, *purely* dynamic rather than *thermo*dynamic — the energy-time uncertainty principle, which imposes the requirement of infinite *time* to attain 0 K.

But, for the moment, not considering the energy-time uncertainty principle, once it has been established and proven that via a CSRR process, entailing isolation in position space rather than in momentum space, e.g., via weighing or employment of a Stern-Gerlach apparatus, that we can be *perfectly* sure that *all n* oscillators in our new ground-state-only Subsystem B really are in the ground state, then this system is indeed at *precisely* 0 K. By contrast, *even* not considering the energy-time uncertainty principle, we can never be *perfectly* sure that all *N* oscillators in our original System A really are in the ground state so long as System A's temperature *T >* 0 is positive, however slightly positive [2–5]. No matter how slightly positive, we can never be *perfectly* sure that all *N* System-A oscillators are in their ground states. In explanation, let the *N* oscillators in our original System A be in thermal equilibrium with a heat reservoir at positive temperature *T* so small that the probability of any one given System-A oscillator being in its first excited state is **P***A*<sup>1</sup> . = *e*−∆*E*/*kT* ≪ 1, and we can neglect the probability of even one of them being in its second excited or higher excited state. Thus, the probability that not even one System-A oscillator is in the first excited state, and hence

that all *N* of them are in the ground state, is **P***<sup>N</sup> <sup>A</sup>*<sup>0</sup> = (1 − **P***A*1) *N* . = <sup>1</sup> <sup>−</sup> *<sup>e</sup>*−∆*E*/*kT<sup>N</sup>* , which for arbitrarily small positive *T* and *N not* arbitrarily large simplifies to **P***<sup>N</sup> A*0 . <sup>=</sup> <sup>1</sup> <sup>−</sup> *Ne*−∆*E*/*kT*. For arbitrarily small positive *T*, if *N* is *not* arbitrarily large, **P***<sup>N</sup> <sup>A</sup>*<sup>0</sup> can be arbitrarily close to 1, but it can never be *precisely* 1 as is required to attain *precisely* 0 K. If *N is* arbitrarily large, then the situation is even worse. For then, however large ∆*E* and however small *T*, it is certain that at least one System-A oscillator is in its first excited state [28]. But this is a limitation *only* of our *original* System A of *N* oscillators, *not* of our Subsystem B of *n* oscillators in their ground state that we have positionally isolated by weighing, by a Stern-Gerlach apparatus, or by any other CSRR method. For arbitrarily small positive *T*, if *N* is *not* arbitrarily large, *n* can be considerably closer to *N* than to *N* − 1, so that if we form Subsystem B it would likely — but *not* for sure — contain all *N* oscillators of System A. The "*not*" in "*not* for sure" is why, if *T* is positive, however slightly positive, System A is *not* our new ground-state-only Subsystem B.

ground state from those in the first excited state by a Stern-Gerlach apparatus. The subsystem comprised of atoms in the ground state after positional isolation via the Stern-Gerlach apparatus constitutes Subsystem B, our subsystem at the absolute zero of temperature, 0 K. This positional isolation of atoms is a CSRR process. This Stern-Gerlach-apparatus version of our thought experiment may, in accordance with Sect. 3 of Ref. [1], be more realizable

In contrast with Sect. 3 of Ref. [1], in the Stern-Gerlach modification of our weighing CSRR method no attempt is made to thence also de-excite the atoms in the first excited state down to the ground state, the *second step* of the QCR method discussed in Sect. 3 of Ref. [1]. It is important to recognize that this *second step* is *neither* a TSRR process *nor* a CSRR process. The entropy of a set of atoms is zero if they are *all in the same quantum state*, irrespective of whether this quantum state is the ground state or not. (But see the last paragraph of the Appendix concerning this point.) In the particular case currently under consideration, we have a set of atoms *all in the first excited state*. Their de-excitation from the first excited state to the ground state maintains their entropy constant at zero. Thus it is *not* an entropy-reduction (SR) process — it is neither a TSRR process nor a CSRR process. It is rather *another type* of refrigeration process, which we have dubbed as *energy*-reduction refrigeration (ERR): *energy E* but *not* entropy *S* is extracted to de-excite these atoms from the first excited state to the ground state. Thus, as was the case with one-time-expansion ERR which we considered in Sect. 2, especially in Sect. 2.5, this version of ERR yields rather than costs work. But unlike one-time-expansion ERR which we considered in Sect. 2.1, especially in Sect. 2.5, this version of ERR process does *not* require infinite *volume* to attain 0 K. Moreover, the extracted energy *E* is *non*thermal, because the oscillators are initially all in the first excited state, *not* in a Boltzmann distribution among states. Similarly as is the case with respect to the expenditure of the typically small amount of work required to attain 0 K via CSRR, the unattainability formulation of the Third Law of Thermodynamics does *not* forbid ERR by forbidding the extraction of the typically small amount of *non*thermal energy required to de-excite these atoms from the first excited state to the ground state. Thus if the unattainability formulation of the Third Law of Thermodynamics in its strongest mode *does* forbid ERR, then, as with CSRR, it must be for *another reason*. As will be discussed in Sects. 3.3.–3.5, this other reason is, as with CSRR, *purely* dynamic rather than *thermo*dynamic — the energy-time uncertainty

experimentally than the weighing version thereof.

280 Recent Advances in Thermo and Fluid Dynamics

principle, which imposes the requirement of infinite *time* to attain 0 K.

System-A oscillator being in its first excited state is **P***A*<sup>1</sup>

that all *N* of them are in the ground state, is **P***<sup>N</sup>*

But, for the moment, not considering the energy-time uncertainty principle, once it has been established and proven that via a CSRR process, entailing isolation in position space rather than in momentum space, e.g., via weighing or employment of a Stern-Gerlach apparatus, that we can be *perfectly* sure that *all n* oscillators in our new ground-state-only Subsystem B really are in the ground state, then this system is indeed at *precisely* 0 K. By contrast, *even* not considering the energy-time uncertainty principle, we can never be *perfectly* sure that all *N* oscillators in our original System A really are in the ground state so long as System A's temperature *T >* 0 is positive, however slightly positive [2–5]. No matter how slightly positive, we can never be *perfectly* sure that all *N* System-A oscillators are in their ground states. In explanation, let the *N* oscillators in our original System A be in thermal equilibrium with a heat reservoir at positive temperature *T* so small that the probability of any one given

the probability of even one of them being in its second excited or higher excited state. Thus, the probability that not even one System-A oscillator is in the first excited state, and hence

.

*<sup>A</sup>*<sup>0</sup> = (1 − **P***A*1)

*N* . = 

= *e*−∆*E*/*kT* ≪ 1, and we can neglect

<sup>1</sup> <sup>−</sup> *<sup>e</sup>*−∆*E*/*kT<sup>N</sup>*

, which

Positional isolation via pure CSRR is not the only method by which absolute zero might be attained. The attainment of absolute zero via QCR methods [1], of which the specific one discussed in Sect. 3 of Ref. [1] employs first CSRR and then ERR, has been investigated [1]. But positional isolation via pure CSRR seems simpler and easier in principle, even if, via weighing, it may not be realizable in practice. But perhaps as discussed three paragraphs previously, via a Stern-Gerlach apparatus it may be [1]. Its simplicity in principle allows us to focus on attainment of absolute zero per se rather on experimental technical issues. Moreover, as noted three paragraphs previously, the QCR method discussed in Sect. 3 of Ref. [1] employs purely-CSRR positional isolation as its first step; as noted two paragraphs previously, only the de-excitation of atoms still in the first excited state down to the ground state in its second step is an ERR process.

#### **3.3. The energy-time uncertainty principle: a** *purely* **dynamic (***not thermo***dynamic) Third-Law limitation under quantum mechanics**

We re-emphasize (recall Sect. 3.2, especially the second-to-last paragraph thereof) that the attainment of absolute zero requires *perfect* certainty that our *entire* new *n*-oscillator Subsystem B is in its ground state — that *all n* oscillators of Subsystem B are in the ground state. But the energy-time uncertainty principle may contravene [29–41]. [Dr. Bernard L. Cohen [29] employs the energy-time uncertainty principle in discussing quantum fluctuations. Dr. Robert Gomer [30] (cited by Dr. Cohen [29]) shows how the position-momentum uncertainty principle can be employed in more limited circumstances. Dr. Mark J. Hagmann [31–35] extends and evaluates Dr. Cohen's work, and compares it with other works. Drs. Donald H. Kobe and V. C. Aguilera-Navarro [38] provide a derivation from first principles of the energy-time uncertainty relation [38], which they and Drs. Hiromi Iwamoto and Mario Goto employ in a study of tunneling times [39]. Drs. V. V. Dodonov and A. V. Dodonov provide extensive considerations concerning the energy-time uncertainty principle [40]. A heuristic overview is provided by the current author [41].] In order to ensure that *all n* oscillators in Subsystem B really are in the ground state, the best that the energy-time uncertainty principle allows us to do is to isolate each of these oscillators for a sufficiently long time interval ∆*t* pursuant to its being incorporated into Subsystem B, or equivalently to isolate Subsystem B for a sufficiently long time ∆*t*. Let us estimate how long ∆*t* must be. Recall that we let ∆*E* denote an energy *gap* — in the case currently under consideration the energy *gap* between adjacent states of any given one of our harmonic oscillators. Let ∆E denote the magnitude of a *quantum* energy *fluctuation*, and let ∆E denote the magnitude of a *thermal* energy *fluctuation*. The minimum possible root-mean-square *quantum* fluctuation magnitude that the energy-time uncertainty principle allows in energy ∆Erms during a time interval ∆*t* is ∆Erms = *h*¯ /2∆*t* [29–41]. (Spontaneously-occurring quantum fluctuations are, or at least tend to be, of minimal possible magnitude, as is required for the macroscopic world being maximally close to classical [29–41].) We require ∆Erms to be much smaller than the typical *upper* limiting root-mean-square magnitude ∆Erms ≈ *kT* of *thermal* energy fluctuations in our original System A. ∆Erms ≈ *kT* is a typical *upper* limiting thermal-energy-fluctuation root-mean-square magnitude because if *N* − *n N* then most oscillators in System A will usually be in the ground state and hence at least ∆*E kT* and hence at most ∆Erms *kT* and more to be expected ∆Erms *kT*. But we also require ∆*E kT* for the energy gap between harmonic-oscillator energy levels to ensure that probability that the second or higher excited states are occupied can be neglected compared to the already small probability that the first excited state is occupied (recall the first paragraph of Sect. 3.2). Thus all told we require ∆Erms = *h*¯ /2∆*t* ∆Erms *kT* ∆*E*. This implies that the strong inequality ∆*t h*¯ /2∆Erms *h*¯ /2*kT* and the even stronger one ∆*t* ≫ *h*¯ /2∆*E* must be fulfilled [29–41]. But even such a long ∆*t* is not long enough to allow us to be *perfectly* certain that all *n* oscillators in our new ground-state-only Subsystem B really are in the ground state, but only to be *almost* perfectly certain that they are [29–41]. Thus, our caveat is as follows: In order to be *perfectly* certain that all *n* oscillators in our new ground-state-only Subsystem B really are in the ground state, each oscillator must be isolated for ∆*t* → ∞ — for infinite time, forever — pursuant to its being incorporated into our new ground-state-only oscillator Subsystem B, or equivalently Subsystem B must be isolated for ∆*t* → ∞. The quantum-mechanical probability **P***B*<sup>1</sup> that any one given oscillator isolated for inclusion in our new Subsystem B is in its first excited state decays exponentially or at least quasi-exponentially with increasing ∆*t* in accordance with [29–41]

$$\mathbb{P}\left(\Delta\mathcal{E}\Delta t\right) \sim e^{-\Delta\mathcal{E}/\Delta\mathcal{E}\_{\rm rms}} = e^{-2\Delta\mathcal{E}\Delta t/\hbar}$$

$$\stackrel{\Delta\mathcal{E}=\Delta E}{\Longrightarrow} \mathbb{P}\_{\rm B1} = \mathbb{P}\left(\Delta E\Delta t\right) \sim e^{-\Delta E/\Delta\mathcal{E}\_{\rm rms}} = e^{-2\Delta E\Delta t/\hbar}.\tag{13}$$

(This exponential or at least quasi-exponential decay is brought out in Ref. [29] and is important implicitly and/or explicitly in Refs. [30–35] and [41]. It is not specifically mentioned but is not inconsistent with Refs. [36–40].) The first line of Eq. (13) expresses the general approximate probability of a quantum energy fluctuation of magnitude ∆E persisting for time ∆*t*. In the second line of Eq. (13) ∆E is set equal to the energy gap ∆*E* between adjacent harmonic-oscillator energy states (the gap between the ground and first excited states being of current interest). Thus the probability that any one given Subsystem-B oscillator is in its ground state after isolation for ∆*t* is **P***B*<sup>0</sup> = 1 − **P***B*<sup>1</sup> = 1 − **P** (∆*E*∆*t*) ∼ <sup>1</sup> <sup>−</sup> *<sup>e</sup>*−2∆*E*∆*t*/¯*h*. Hence the probability that all *<sup>n</sup>* Subsystem-B oscillators are in their ground states after isolation for ∆*t* is **P***<sup>n</sup> <sup>B</sup>*<sup>0</sup> = (1 − **P***B*1) *<sup>n</sup>* <sup>=</sup> [<sup>1</sup> <sup>−</sup> **<sup>P</sup>** (∆*E*∆*t*)]*<sup>n</sup>* <sup>∼</sup> <sup>1</sup> <sup>−</sup> *<sup>e</sup>*−2∆*E*∆*t*/¯*<sup>h</sup> n* , which for *n* not too large and sufficiently large ∆*t* simplifies to **P***<sup>n</sup> <sup>B</sup>*<sup>0</sup> <sup>∼</sup> <sup>1</sup> <sup>−</sup> *ne*−2∆*E*∆*t*/¯*h*. Thus <sup>1</sup> <sup>−</sup> **<sup>P</sup>***<sup>n</sup> <sup>B</sup>*<sup>0</sup> <sup>∼</sup> *ne*−2∆*E*∆*t*/¯*h*, which depends only *linearly* on *<sup>n</sup>* but decreases *exponentially* (or at least quasi-exponentially) with increasing ∆*t*, soon becomes negligible. But however strongly negligible it becomes, it *never* becomes *precisely* 0 except in the limit <sup>∆</sup>*<sup>t</sup>* <sup>→</sup> <sup>∞</sup>. And 1 <sup>−</sup> **<sup>P</sup>***<sup>n</sup> <sup>B</sup>*<sup>0</sup> is required to be *precisely* 0 — equivalently **P***<sup>n</sup> <sup>B</sup>*<sup>0</sup> is required to be *precisely* 1 — if *precisely* 0 K is to be attained and if we are to have *perfect* verification [22] that *precisely* 0 K has been attained. The difference in masses and therefore also weights of an oscillator being in its ground state as opposed to in its first excited state is finite, thereby, as per the first five paragraphs of Sect. 3.2, circumventing the objection that infinitely precise measurements [22] would be required to verify [22] that *precisely* 0 K has been attained. This objection is similarly circumvented if instead of weighing we employ a Stern-Gerlach apparatus as per the sixth paragraph of Sect. 3.2, in accordance with the first step of the method employed in Sect. 3 of Ref. [1]. But the objection posed by the energy-time uncertainty principle seems to be uncircumventable: *Exact* attainment of 0 K and *perfect* verification [22] that *precisely* 0 K has been attained seems to require infinite time. Thus, the energy-time uncertainty principle may provide additional — quantum-mechanical and hence *purely* dynamic as opposed to *thermo*dynamic — protection against *exact* attainment of 0 K and *perfect* verifiability [22] that *precisely* 0 K has been attained, and hence against the unattainability formulation of the Third Law of Thermodynamics in its strongest mode being *precisely* violated. It is not clear whether or not the energy-time uncertainty principle imposes a similar limitation on the QCR systems and methods discussed in Ref. [1]. But owing to the universality of quantum mechanics and hence of the energy-time uncertainty principle, this seems likely to be the case. Indeed owing to the universality of quantum mechanics and hence of the energy-time uncertainty principle, this seems likely to be the case in general, irrespective of the refrigeration method — TSRR, CSRR, ERR, QCR, etc., or any combination thereof — that is employed. This is in accordance with the conclusion reached in Refs. [6–8] via far more technical and mathematical analyses.

the magnitude of a *thermal* energy *fluctuation*. The minimum possible root-mean-square *quantum* fluctuation magnitude that the energy-time uncertainty principle allows in energy ∆Erms during a time interval ∆*t* is ∆Erms = *h*¯ /2∆*t* [29–41]. (Spontaneously-occurring quantum fluctuations are, or at least tend to be, of minimal possible magnitude, as is required for the macroscopic world being maximally close to classical [29–41].) We require ∆Erms to be much smaller than the typical *upper* limiting root-mean-square magnitude ∆Erms ≈ *kT* of *thermal* energy fluctuations in our original System A. ∆Erms ≈ *kT* is a typical *upper* limiting thermal-energy-fluctuation root-mean-square magnitude because if *N* − *n N* then most oscillators in System A will usually be in the ground state and hence at least ∆*E kT* and hence at most ∆Erms *kT* and more to be expected ∆Erms *kT*. But we also require ∆*E kT* for the energy gap between harmonic-oscillator energy levels to ensure that probability that the second or higher excited states are occupied can be neglected compared to the already small probability that the first excited state is occupied (recall the first paragraph of Sect. 3.2). Thus all told we require ∆Erms = *h*¯ /2∆*t* ∆Erms *kT* ∆*E*. This implies that the strong inequality ∆*t h*¯ /2∆Erms *h*¯ /2*kT* and the even stronger one ∆*t* ≫ *h*¯ /2∆*E* must be fulfilled [29–41]. But even such a long ∆*t* is not long enough to allow us to be *perfectly* certain that all *n* oscillators in our new ground-state-only Subsystem B really are in the ground state, but only to be *almost* perfectly certain that they are [29–41]. Thus, our caveat is as follows: In order to be *perfectly* certain that all *n* oscillators in our new ground-state-only Subsystem B really are in the ground state, each oscillator must be isolated for ∆*t* → ∞ — for infinite time, forever — pursuant to its being incorporated into our new ground-state-only oscillator Subsystem B, or equivalently Subsystem B must be isolated for ∆*t* → ∞. The quantum-mechanical probability **P***B*<sup>1</sup> that any one given oscillator isolated for inclusion in our new Subsystem B is in its first excited state decays exponentially or at least

quasi-exponentially with increasing ∆*t* in accordance with [29–41]

<sup>∆</sup>E=∆*<sup>E</sup>* <sup>=</sup><sup>⇒</sup> **<sup>P</sup>***B*<sup>1</sup> <sup>=</sup> **<sup>P</sup>** (∆*E*∆*t*) <sup>∼</sup> *<sup>e</sup>*

<sup>−</sup>∆E/∆Erms = *e*

*<sup>B</sup>*<sup>0</sup> = (1 − **P***B*1)

*<sup>B</sup>*<sup>0</sup> <sup>∼</sup> *ne*−2∆*E*∆*t*/¯*h*, which depends only *linearly* on *<sup>n</sup>* but decreases *exponentially* (or at least quasi-exponentially) with increasing ∆*t*, soon becomes negligible. But however strongly negligible it becomes, it *never* becomes *precisely* 0 except in the limit <sup>∆</sup>*<sup>t</sup>* <sup>→</sup> <sup>∞</sup>. And 1 <sup>−</sup> **<sup>P</sup>***<sup>n</sup>*

to be attained and if we are to have *perfect* verification [22] that *precisely* 0 K has been attained.

which for *n* not too large and sufficiently large ∆*t* simplifies to **P***<sup>n</sup>*

(This exponential or at least quasi-exponential decay is brought out in Ref. [29] and is important implicitly and/or explicitly in Refs. [30–35] and [41]. It is not specifically mentioned but is not inconsistent with Refs. [36–40].) The first line of Eq. (13) expresses the general approximate probability of a quantum energy fluctuation of magnitude ∆E persisting for time ∆*t*. In the second line of Eq. (13) ∆E is set equal to the energy gap ∆*E* between adjacent harmonic-oscillator energy states (the gap between the ground and first excited states being of current interest). Thus the probability that any one given Subsystem-B oscillator is in its ground state after isolation for ∆*t* is **P***B*<sup>0</sup> = 1 − **P***B*<sup>1</sup> = 1 − **P** (∆*E*∆*t*) ∼ <sup>1</sup> <sup>−</sup> *<sup>e</sup>*−2∆*E*∆*t*/¯*h*. Hence the probability that all *<sup>n</sup>* Subsystem-B oscillators are in their ground

−2∆E∆*t*/¯*h*

<sup>−</sup>∆*E*/∆Erms = *e*

*<sup>n</sup>* <sup>=</sup> [<sup>1</sup> <sup>−</sup> **<sup>P</sup>** (∆*E*∆*t*)]*<sup>n</sup>* <sup>∼</sup>

*<sup>B</sup>*<sup>0</sup> is required to be *precisely* 1 — if *precisely* 0 K is

<sup>−</sup>2∆*E*∆*t*/¯*h*. (13)

<sup>1</sup> <sup>−</sup> *<sup>e</sup>*−2∆*E*∆*t*/¯*<sup>h</sup>*

*<sup>B</sup>*<sup>0</sup> <sup>∼</sup> <sup>1</sup> <sup>−</sup> *ne*−2∆*E*∆*t*/¯*h*. Thus

*n* ,

*<sup>B</sup>*<sup>0</sup> is

**P** (∆E∆*t*) ∼ *e*

states after isolation for ∆*t* is **P***<sup>n</sup>*

282 Recent Advances in Thermo and Fluid Dynamics

required to be *precisely* 0 — equivalently **P***<sup>n</sup>*

<sup>1</sup> <sup>−</sup> **<sup>P</sup>***<sup>n</sup>*

Note the *qualitative* — not merely *quantitative* — distinction between the *thermo*dynamic (Boltzmann-distribution) probability **P***<sup>A</sup>* discussed in Sect. 3.2. as opposed to the *purely* dynamic (quantum-mechanical) probability **P***<sup>B</sup>* discussed in this Sect. 3.3. *Even if*, *thermo*dynamically, *exact* attainment of 0 K and *perfect* verification [22] that *precisely* 0 K has been attained *could be* achieved for Subsystem B, the *pure* dynamics of quantum mechanics, specifically the energy-time uncertainty principle, seems to impose the requirement that infinite time must elapse first. [This distinction between *thermo*dynamic probabilities as opposed to *purely* dynamic (quantum-mechanical) probabilities should not be confused with the distinction between the derivation of the *thermo*dynamic Boltzmann distribution per se in classical as opposed to quantum statistical mechanics. The latter distinction, which we do not consider in this chapter, obtains largely owing to the postulate of random phases being required in quantum but not classical statistical mechanics [42,43].]

Nevertheless, given the *exponential* (or at least quasi-exponential) decay of 1 <sup>−</sup> **<sup>P</sup>***<sup>n</sup> B*0 ∼ *ne*−2∆*E*∆*t*/¯*<sup>h</sup>* [29–41], fulfillment of the *very* strong inequality ∆*t* ≫ *h*¯ /2∆*E does* seem to imply that we can be close enough to perfectly certain that all *n* oscillators in our new Subsystem B really are in the ground state *for all practical purposes*. Hence it seems that we must be content with attainment for all practical purposes as opposed to exact attainment of 0 K and verification for all practical purposes as opposed to perfect verification [22] that *precisely* 0 K has been attained. Thus perhaps the energy-time uncertainty principle provides the ultimate protection against *perfect* violation of the unattainability formulation of the Third Law of Thermodynamics in its strongest mode. But given the *exponential* (or at least quasi-exponential) decay of 1 <sup>−</sup> **<sup>P</sup>***<sup>n</sup> <sup>B</sup>*<sup>0</sup> <sup>∼</sup> *ne*−2∆*E*∆*t*/¯*<sup>h</sup>* [29–41], perhaps, while not *perfectly* violating the strongest-mode unattainability formulation of the Third Law of Thermodynamics, CSRR as opposed to TSRR at least challenges it in the strongest manner that the laws of physics allow. Recall from the sixth and seventh paragraphs of Sect. 3.2. that the specific QCR method discussed in Sect. 3 of Ref. [1] employs first CSRR and then ERR [1], and that we employed the first step of this method for CSRR positional isolation via Stern-Gerlach apparatus. Recall also that in the last two paragraphs of Sect. 2.3. we have already employed exponential decay in consideration of rendering insulation for a refrigerated system perfect *for all practical purposes*, even if it cannot be *exactly* perfect [23].

#### **3.4. The quest for absolute zero under classical versus quantum mechanics**

The energy-time uncertainty principle is of purely quantum-mechanical origin. It does not exist in classical mechanics, whether Newtonian or relativistic. Thus under classical mechanics, whether Newtonian or relativistic, it might seem, at least *prima facie*, that, at least in principle, CSRR via positional isolation by means of weighing as discussed in the first five paragraphs of Sect. 3.2, by means of a Stern-Gerlach apparatus as discussed in the sixth paragraph of Sect. 3.2 (whether or not enhanced via ERR as per the seventh paragraph of Sect. 3.2), or via QCR in general [1], *can* not only attain *precisely* 0 K but also provide *perfect* verification [22] that *precisely* 0 K has been attained — and both in *finite*, even *arbitrarily short*, time ∆*t*. Hence it may seem, at least *prima facie*, that under classical mechanics, at least in principle, the unattainability statement of the Third Law of Thermodynamics even in its strongest mode *can* be *precisely* violated via CSRR. However, experimentally realizable proposals for attaining 0 K are quantum-mechanical [1]. Indeed the entire Universe is ultimately quantum-mechanical; classical mechanics, Newtonian or even relativistic, being only a limiting approximation. Hence attainment of 0 K *for all practical purposes* and verification [22] that 0 K has been attained *for all practical purposes* is probably the best that can be achieved. Our conclusion seems unalterable even if one accepts the viewpoint expressed by Dr. David Bohm that classical mechanics should be considered in its own right and as prerequisite for quantum mechanics, rather than as a limiting case of quantum mechanics [44]. This is opposed to the more generally accepted viewpoint that classical mechanics should be considered as a limiting case of quantum mechanics. Moreover, even Dr. Bohm expresses the latter viewpoint in his recognition of the Universe as being ultimately quantum-mechanical [45].

Besides, classical mechanics imposes its own burden on the quest for absolute zero. While there are instances of quantization even in classical mechanics, for example the discrete allowed frequencies and wavelengths of a vibrating string of finite length [46], of electromagnetic waves within a finite volume [47–49], and of sound waves within a finite volume [47–49], so far as is known energy is always continuous and never quantized in classical mechanics.4 Quantization of frequency *ν* and hence also of wavelength *λ*, for example the discrete allowed frequencies and wavelengths of a vibrating string of finite length [46], of electromagnetic waves within a finite volume [47–49], and of sound waves within a finite volume [47–49], implies quantization of energy *E* in *quantum* mechanics in accordance with the *quantum*-mechanical relation *E* = *hν* = *hc*/*λ* [50–52]. (Here c is the speed of wave prorogation, whether of waves on a string, of light waves, of sound waves, etc. Note for example that for photons *ν* in a material medium equals *ν* in a vacuum; both *c* and *λ* are smaller in a material medium than in a vacuum by a ratio equal to the index of refraction of the medium.) The relation *E* = *mc*<sup>2</sup> = *hν* = *hc*/*λ* is at the heart of the very closely related Einstein [50–52] and deBroglie [50–52] postulates [50–52]. The relation *E* = *mc*<sup>2</sup> obtains

<sup>4</sup> (Re: Entries [48] and [49], Refs. [48] and [49]) Photons of equilibrium blackbody radiation are discussed in Sect. 10.6 and phonons of sound waves in solids in Problem 10.8 of Chap. 10 on pp. 369–371 of Ref. [48]. Both photons and phonons are discussed in Chap. 6 of Ref. [49].

in both classical and quantum relativistic mechanics. But quantization of frequency *ν* and hence also of wavelength *λ* does *not* imply quantization of energy *E* in *classical* mechanics because the relation *E* = *hν* = *hc*/*λ* is *strictly quantum*-mechanical [50–52]. There exists *no classical*-mechanical relation such as *E* = *hν* = *hc*/*λ* [50–52]. Even in quantum mechanics the relation *E* = *hν* = *hc*/*λ* is necessary but not sufficient for discreteness of energy levels as opposed to an energy continuum, for in an infinite volume *ν* and *λ* can take on a continuum of values. But given only the additional very mild condition of a finite volume — a fixed finite volume or even an unfixed but always finite volume for example corresponding to maintenance of constant pressure — only discrete values of *λ* and hence of *ν* = *c*/*λ* will fit therein, thus ensuring discreteness of energy levels under quantum — but not classical mechanics. This can also be shown via considerations of the Schrödinger equation [53]. Thus so far as is known quantization of energy [50–52] and discrete energy levels [53] can exist under quantum mechanics [50–53], indeed *must* exist under quantum mechanics given finite volume, but can*not* exist under classical mechanics [50–53]. Hence under classical mechanics, owing to continuity of energy, infinitely precise measurements [2,3,22] would be required to *perfectly* verify [2,3,22] that *precisely* 0 K has been attained [2,3,22]: With an infinitesimal gap between the ground and first excited states, weighing of our harmonic oscillators would have to be infinitely precise. With an infinitesimal gap between the ground and first excited states of atoms an infinitely-sensitive Stern-Gerlach apparatus would be required to separate atoms in the two states. By contrast, under quantum mechanics, owing to discreteness of energy levels [50–53] of a system within a finite volume, measurements of merely finite precision suffice for verification [22] that *precisely* 0 K has been attained.

Thus quantum mechanics, via the energy-time uncertainty principle, imposes the requirement of isolation for infinite time for *perfect* verification [22] that *precisely* 0 K has been attained, but by requiring discreteness of energy levels for a system of finite volume lifts the requirement of infinitely precise measurements [22] to *perfectly* verify [22] that *precisely* 0 K has been attained. By contrast, classical mechanics, since it lacks an energy-time uncertainty principle, lifts the requirement of isolation for infinite time for *perfect* verification [22] that *precisely* 0 K has been attained, but by requiring an energy continuum imposes the requirement of infinitely precise measurements [22] to *perfectly* verify [22] that *precisely* 0 K has been attained. Of these two requirements, the first seems less onerous than the second, because as discussed in Sect. 3.3. the *un*certainty that *precisely* 0 K has been attained decays *exponentially* (or at least quasi-exponentially) with time [29–41]. This exponential or at least quasi-exponential decay mitigates (albeit does not remove) the requirement of isolation for infinite time under quantum mechanics for *perfect* verification [22] that *precisely* 0 K has been attained. No such decay, exponential, quasi-exponential, or otherwise, mitigates the requirement under classical mechanics for an energy continuum, hence implying that infinitely precise measurements [22] are requisite for perfect verification [22] that *precisely* 0 K has been attained. So at least *prima facie* it seems that quantum mechanics at least brings us closer than classical mechanics to *perfect* verifiability [22] that *precisely* 0 K can be attained.

#### **3.5. Summary: TSRR versus CSRR**

via Stern-Gerlach apparatus. Recall also that in the last two paragraphs of Sect. 2.3. we have already employed exponential decay in consideration of rendering insulation for a refrigerated system perfect *for all practical purposes*, even if it cannot be *exactly* perfect [23].

The energy-time uncertainty principle is of purely quantum-mechanical origin. It does not exist in classical mechanics, whether Newtonian or relativistic. Thus under classical mechanics, whether Newtonian or relativistic, it might seem, at least *prima facie*, that, at least in principle, CSRR via positional isolation by means of weighing as discussed in the first five paragraphs of Sect. 3.2, by means of a Stern-Gerlach apparatus as discussed in the sixth paragraph of Sect. 3.2 (whether or not enhanced via ERR as per the seventh paragraph of Sect. 3.2), or via QCR in general [1], *can* not only attain *precisely* 0 K but also provide *perfect* verification [22] that *precisely* 0 K has been attained — and both in *finite*, even *arbitrarily short*, time ∆*t*. Hence it may seem, at least *prima facie*, that under classical mechanics, at least in principle, the unattainability statement of the Third Law of Thermodynamics even in its strongest mode *can* be *precisely* violated via CSRR. However, experimentally realizable proposals for attaining 0 K are quantum-mechanical [1]. Indeed the entire Universe is ultimately quantum-mechanical; classical mechanics, Newtonian or even relativistic, being only a limiting approximation. Hence attainment of 0 K *for all practical purposes* and verification [22] that 0 K has been attained *for all practical purposes* is probably the best that can be achieved. Our conclusion seems unalterable even if one accepts the viewpoint expressed by Dr. David Bohm that classical mechanics should be considered in its own right and as prerequisite for quantum mechanics, rather than as a limiting case of quantum mechanics [44]. This is opposed to the more generally accepted viewpoint that classical mechanics should be considered as a limiting case of quantum mechanics. Moreover, even Dr. Bohm expresses the latter viewpoint in his recognition of the Universe as being

Besides, classical mechanics imposes its own burden on the quest for absolute zero. While there are instances of quantization even in classical mechanics, for example the discrete allowed frequencies and wavelengths of a vibrating string of finite length [46], of electromagnetic waves within a finite volume [47–49], and of sound waves within a finite volume [47–49], so far as is known energy is always continuous and never quantized in classical mechanics.4 Quantization of frequency *ν* and hence also of wavelength *λ*, for example the discrete allowed frequencies and wavelengths of a vibrating string of finite length [46], of electromagnetic waves within a finite volume [47–49], and of sound waves within a finite volume [47–49], implies quantization of energy *E* in *quantum* mechanics in accordance with the *quantum*-mechanical relation *E* = *hν* = *hc*/*λ* [50–52]. (Here c is the speed of wave prorogation, whether of waves on a string, of light waves, of sound waves, etc. Note for example that for photons *ν* in a material medium equals *ν* in a vacuum; both *c* and *λ* are smaller in a material medium than in a vacuum by a ratio equal to the index of refraction of the medium.) The relation *E* = *mc*<sup>2</sup> = *hν* = *hc*/*λ* is at the heart of the very closely related Einstein [50–52] and deBroglie [50–52] postulates [50–52]. The relation *E* = *mc*<sup>2</sup> obtains

<sup>4</sup> (Re: Entries [48] and [49], Refs. [48] and [49]) Photons of equilibrium blackbody radiation are discussed in Sect. 10.6 and phonons of sound waves in solids in Problem 10.8 of Chap. 10 on pp. 369–371 of Ref. [48]. Both photons and

**3.4. The quest for absolute zero under classical versus quantum mechanics**

ultimately quantum-mechanical [45].

284 Recent Advances in Thermo and Fluid Dynamics

phonons are discussed in Chap. 6 of Ref. [49].

In summary, the *thermo*dynamic difficulties in *attaining precisely* 0 K via TSRR [2–5] seem to be circumventable via CSRR. By contrast, the *purely* dynamic (quantum-mechanical) limitation imposed by the energy-time uncertainty principle as per Sects. 3.3. and 3.4. is, strictly, not circumventable via either TSRR or CSRR, but this limitation may not be crucial if we do not insist on *exact* attainment of 0 K and *perfect* verification [22] that *precisely* 0 K has been attained, but are content with attainment of 0 K and verification [22] of its attainment that is perfect *for all practical purposes*.

Thus it seems that CSRR, based on localization of refrigerated entities in position space, as discussed in Sect. 3, at least brings us *closer* to *exact* attainment of *TC* = 0 K and to *perfect* verification [22] that 0 K has been attained than does TSRR, based on localization of refrigerated entities in momentum space, as discussed in Sect. 2. And CSRR under quantum mechanics (e.g., via weighing or via Stern-Gerlach apparatus), or via QCR methods [1], as opposed to under classical (Newtonian or relativistic) mechanics, seems to bring us the *closest*. (Recall from the sixth and seventh paragraphs of Sect. 3.2. that the specific QCR method discussed in Sect. 3 of Ref. [1] employs first CSRR via Stern-Gerlach apparatus and then ERR [1]; we employed the first, CSRR, step thereof in the sixth paragraph of Sect. 3.2.)

Moreover, *even if TC* = 0 K *could* be precisely *attained* and also its attainment *could* be *perfectly* verified [22], whether via standard or absorption TSRR, via CSRR, via one-time-expansion ERR as discussed in Sect. 2, via ERR as part of a QCR method as discussed in this Sect. 3, or via other ERR methods, via QCR, etc., or via any combination thereof, the question of *maintaining TC* = 0 K as per Sect. 2.3. and the last paragraphs of Sects. 2.4. and 2.5. is still open [23]. *Even if TC* = 0 K *could* be precisely *attained* and also its attainment could be *perfectly* verified [22], the question of whether or not it can be *maintained* for finite time, as opposed to merely the infinitesimally short time allowed in accordance with Eqs. (6) and (7) and the associated discussions, is still open. But least in principle even if not in practice *if TC* = 0 K can be *attained*, whether *maintainable* only for an instant or longer, then this may be more easily achievable via CSRR, especially under quantum rather than classical mechanics, than via standard or even absorption TSRR (or any other TSRR). The same is probably true with respect to QCR as opposed to TSRR.

But the issue of *maintenance* [23] seems *inseparable* from that of *verifiability* [22]. For, as discussed in Sects. 3.3. and 3.4, the energy-time uncertainty principle requires ∆*t* → ∞ for perfect verifiability that *TC* = 0 K has been attained, which is obviously incompatible with *TC* = 0 K being maintained only for an instant, or even for any finite number of instants, in accordance with Eqs. (6) and (7) and the associated discussions. If *TC* = 0 K can be maintained only for an instant, (or any finite number of instants), then the energy-time uncertainty principle seems to preclude verification *even for all practical purposes* that *TC* = 0 K has actually been attained. Thus verification [22] *even for all practical purposes* that *TC* = 0 K has actually been attained seems to require insulation [23] that is perfect *for all practical purposes* as per Sect. 2.3.

#### **3.6. What if: Better-than-perfect refrigeration?**

The Second Law of Thermodynamics forbids better-than-perfect refrigeration, in which the total entropy change is negative. Yet the universal validity of the Second Law of Thermodynamics has been seriously questioned [54–58], albeit with the understanding that even if not universally valid at the very least it has a very wide range of validity [54–58]. Thus *what if* better-than-perfect refrigeration *is* possible, whether via TSRR, CSRR, ERR, QCR, etc., or any combination thereof? (ERR entails zero entropy change; hence it could be part, but not the entirety, of a better-than-perfect refrigeration process, if such can exist [54–58]. Recall from the sixth and seventh paragraphs of Sect. 3.2. that the specific QCR method discussed in Sect. 3 of Ref. [1] employs first CSRR and then ERR [1]: we employed the first, CSRR, step thereof in the sixth paragraph of Sect. 3.2.) For example, what if CSRR-isolation of our Subsystem-B oscillators could be achieved at smaller negentropy and work cost than in accordance with Eq. (12)? Unfortunately, *even if* better-than-perfect refrigeration in contravention of the Second Law of *Thermo*dynamics is possible, the *purely* dynamic limitations discussed in Sects. 3.3.–3.5. still obtain for both TSRR and CSRR (as well as for ERR, QCR, etc., or any combination of refrigeration methods). Thus at least *prima facie* it seems that *even if* the second law can be contravened [54–58] and hence better-than-perfect refrigeration is possible, owing to the energy–time uncertainty principle the strongest-mode unattainability statement of the third law could still be violated only for all practical purposes and not perfectly. And this considers *only* the difficulties of *attaining TC* = 0 K. *Even if* the Second Law of Thermodynamics can be contravened [54–58] and hence better-than-perfect refrigeration is possible, the difficulties in *maintaining TC* = 0 K for longer than an infinitessimal time, discussed in Sect. IIC, the last paragraphs of Sects. 2.4. and 2.5, and the last two paragraphs of Sect. 3.5, may preclude verification [22] *even for all practical purposes* that *TC* = 0 K has actually been attained — unless insulation [23] that is perfect *for all practical purposes* as per Sect. 2.3. is possible.

#### **4. Hot true and cold effective negative Kelvin temperatures**

#### **4.1. Hot true negative Kelvin temperatures**

not insist on *exact* attainment of 0 K and *perfect* verification [22] that *precisely* 0 K has been attained, but are content with attainment of 0 K and verification [22] of its attainment that is

Thus it seems that CSRR, based on localization of refrigerated entities in position space, as discussed in Sect. 3, at least brings us *closer* to *exact* attainment of *TC* = 0 K and to *perfect* verification [22] that 0 K has been attained than does TSRR, based on localization of refrigerated entities in momentum space, as discussed in Sect. 2. And CSRR under quantum mechanics (e.g., via weighing or via Stern-Gerlach apparatus), or via QCR methods [1], as opposed to under classical (Newtonian or relativistic) mechanics, seems to bring us the *closest*. (Recall from the sixth and seventh paragraphs of Sect. 3.2. that the specific QCR method discussed in Sect. 3 of Ref. [1] employs first CSRR via Stern-Gerlach apparatus and then ERR [1]; we employed the first, CSRR, step thereof in the sixth paragraph of Sect. 3.2.) Moreover, *even if TC* = 0 K *could* be precisely *attained* and also its attainment *could* be *perfectly* verified [22], whether via standard or absorption TSRR, via CSRR, via one-time-expansion ERR as discussed in Sect. 2, via ERR as part of a QCR method as discussed in this Sect. 3, or via other ERR methods, via QCR, etc., or via any combination thereof, the question of *maintaining TC* = 0 K as per Sect. 2.3. and the last paragraphs of Sects. 2.4. and 2.5. is still open [23]. *Even if TC* = 0 K *could* be precisely *attained* and also its attainment could be *perfectly* verified [22], the question of whether or not it can be *maintained* for finite time, as opposed to merely the infinitesimally short time allowed in accordance with Eqs. (6) and (7) and the associated discussions, is still open. But least in principle even if not in practice *if TC* = 0 K can be *attained*, whether *maintainable* only for an instant or longer, then this may be more easily achievable via CSRR, especially under quantum rather than classical mechanics, than via standard or even absorption TSRR (or any other TSRR). The same is probably true

But the issue of *maintenance* [23] seems *inseparable* from that of *verifiability* [22]. For, as discussed in Sects. 3.3. and 3.4, the energy-time uncertainty principle requires ∆*t* → ∞ for perfect verifiability that *TC* = 0 K has been attained, which is obviously incompatible with *TC* = 0 K being maintained only for an instant, or even for any finite number of instants, in accordance with Eqs. (6) and (7) and the associated discussions. If *TC* = 0 K can be maintained only for an instant, (or any finite number of instants), then the energy-time uncertainty principle seems to preclude verification *even for all practical purposes* that *TC* = 0 K has actually been attained. Thus verification [22] *even for all practical purposes* that *TC* = 0 K has actually been attained seems to require insulation [23] that is perfect *for all practical*

The Second Law of Thermodynamics forbids better-than-perfect refrigeration, in which the total entropy change is negative. Yet the universal validity of the Second Law of Thermodynamics has been seriously questioned [54–58], albeit with the understanding that even if not universally valid at the very least it has a very wide range of validity [54–58]. Thus *what if* better-than-perfect refrigeration *is* possible, whether via TSRR, CSRR, ERR, QCR, etc., or any combination thereof? (ERR entails zero entropy change; hence it could be part, but not the entirety, of a better-than-perfect refrigeration process, if such can exist [54–58]. Recall from the sixth and seventh paragraphs of Sect. 3.2. that the specific

perfect *for all practical purposes*.

286 Recent Advances in Thermo and Fluid Dynamics

with respect to QCR as opposed to TSRR.

**3.6. What if: Better-than-perfect refrigeration?**

*purposes* as per Sect. 2.3.

Negative Kelvin temperatures certainly exist [59–63]. But *true* negative Kelvin temperatures are *hotter* than *T* = ∞ K, not colder than *T* = 0 K [59–62]. *True* negative Kelvin temperatures exist only in systems with an upper bound in energy, wherein *T* = (*∂E*/*∂S*)*V*,*<sup>N</sup> <* 0 obtains if enough energy is pumped into such a system so that its high-energy state is more populated than its low one — a population inversion [59–62]. [As per standard notation, the subscript "*V*, *N*" denotes fixed volume and number of entities (most typically atoms or molecules).] The temperature of *any* system can, at least in principle, be raised to *T* = ∞ K via energy pumped into the system as heat and/or as work. But unless a heat reservoir at a negative Kelvin temperature is available, energy must be pumped into a system as work rather than as heat, i.e., *non*thermally, if the system's temperature is to be raised to negative Kelvin values, because heat input from a heat reservoir at a positive Kelvin temperature can never raise a system's temperature above *T* = ∞ K, which corresponds to all of the system's states being uniformly populated — just short of a population inversion [59–62].

Consider, for simplicity, a 2-energy-level system with both levels nondegenerate. A total of *N* entities (typically atoms whose nuclei can manifest spin aligned either parallel or antiparallel to an external magnetic field) can be distributed among these 2 energy levels. At *T* = +0 K, the probability is unity that all *N* entities are in the lower level and hence the system's entropy is minimized at *S* = 0. As energy is pumped into the system (as heat and/or as work), its temperature *T* = (*∂E*/*∂S*)*V*,*<sup>N</sup>* increases through increasing positive values and its entropy increases. At *T* = +∞ K = −∞ K, each entity has a probability of 1/2 of being in either level and hence the system's entropy is maximized at *S* = *Nk* ln 2. As more energy is pumped into the system (as work only unless a heat reservoir at *T* = −0 K is available), its temperature *T* = (*∂E*/*∂S*)*V*,*<sup>N</sup>* increases through decreasing negative values from *T* = −∞ K to *T* = −0 K and its entropy decreases, until at *T* = −0 K the probability is unity that all *N* entities are in the upper level and hence the system's entropy is again minimized at *S* = 0.

It should be noted that the concept of hot negative Kelvin temperature can meaningfully be applied for 2-energy-level systems [63], whether or not either level or both are degenerate but *only* for 2-energy-level systems [63]. For systems with 3 or more energy levels, wherein population need not be a monotonic function of level (multiple population inversions are possible with 4 or more levels), the concept of hot negative Kelvin temperature becomes unwieldy and contrived [63].5

It has been remarked [59–62] that there are advantages in defining temperature via 1/*T* = − (*∂S*/*∂E*)*V*,*N*, because by this definition the numerical value of a system's temperature always increases monotonically with its increasing ability to spontaneously deliver heat to its surroundings or equivalently with its decreasing ability to spontaneously accept heat from its surroundings, whether temperature defined via *T* = (*∂E*/*∂S*)*V*,*<sup>N</sup>* is positive or negative [59–62]. But temperature defined via *T* = (*∂E*/*∂S*)*V*,*<sup>N</sup>* has the advantages of numerical proportionality to temperature as measured by an ideal-gas thermometer and to average thermal kinetic energy per degree of freedom of ideal-gas molecules in the classical (nonquantum) regime. So we employ the definition *T* = (*∂E*/*∂S*)*V*,*<sup>N</sup>* in this chapter.

#### **4.2. Cold effective negative Kelvin temperatures**

Insofar as is known, *true* negative Kelvin temperatures that are *colder* than *T* = 0 K do not exist. Nevertheless, we can still consider *effective* negative Kelvin temperatures that are *colder* than *T* = 0 K — linearly extrapolating the Kelvin temperature scale downwards through *T* = 0 K to negative *effective* values. Such cold *effective* negative Kelvin temperatures *do* exist. Consider, for example, the effective wind-chill temperature W on Neptune, at the level in Neptune's atmosphere where the pressure is 1 bar, approximately 1 atm. The wind-chill temperature W is the temperature that calm air must have to produce the same chilling effect as moving air — wind — at speed V, all other things being equal. The *true* mean temperature (without wind chill) at the 1 bar level on Neptune is approximately *T* = 72 K = −201 ◦C = −330 ◦F [65]. The standard *formula* for wind-chill temperature W employed by the U. S. A. National Weather Service is [66]

$$\mathcal{W} = \left[ 0.6215 T\_{\odot} + \left( 0.4275 T\_{\odot} - 35.75 \right) \mathcal{V}\_{\text{mi}}^{0.16} + 35.74 \right] \,^{\circ} \text{F} . \tag{14}$$

In Eq. (14), the wind speed V is that at the 5 ft (typical face) level, based on reduction owing to surface friction of wind speed measured at the standard 10 m or 33 ft level to the 5 ft level [66]. But over flat open ground or over open water, this reduction in wind speed is

<sup>5</sup> (Re: Entry [63], Ref. [63]) In Ref. [63], Dr. Peter Atkins doesn't seem to explicitly state that negative Kelvin temperatures are hotter than ∞ K, not colder than 0 K. He admits the possibility of attaining 0 K via noncyclic processes, but as we showed in Sect. 3. of this chapter *purely* dynamic — as opposed to *thermo*dynamic — limitations may contravene. On pp. 103–104 of Ref. [63], he correctly states that the third law of thermodynamics is "not really in the same league" as the zeroth, first, and second laws, and that "hints of the Third Law of Thermodynamics are already present in the consequences of the second law," but that the Third Law of Thermodynamics is "the final link in the confirmation that Boltzmann's and Clausius's definitions refer to the same property." But his statement that "we need to do an ever increasing, and ultimately infinite, amount of work to remove energy from a body as heat as its temperature approaches absolute zero" neglects the rapid decrease in specific heat as absolute zero is approached as discussed in Sect. 2. of this chapter.

typically small. So for simplicity let us neglect this typically small difference in wind speeds.6 For the given temperature *T* = 72 K = −201 ◦C = −330 ◦F at the 1 bar level on Neptune [65], even with a slow (by Neptune standards) V = 50 mi / h wind, Eq. (14) yields W = −500 ◦F = −296 ◦C = −22 K. [A wind speed of V = 50 mi / h is chosen so that our example is more "Earthlike." Typical wind speeds on Neptune are considerably higher than 50 mi / h (See Ref. [65].) But according to Eq. (14), the chilling effect of wind increases at a decreasing rate with increasing wind speed: (*∂*W/*∂*V)*<sup>T</sup>* = 0.16 (0.4375*T*◦<sup>F</sup> <sup>−</sup> 35.75) <sup>V</sup>−0.84 mi / h ◦F / (mi / h). Hence W decreases only very slowly at wind speeds above 50 mi / h, at least assuming that if not Eq. (14) in its entirety then at least this aspect of Eq. (14) retains at least approximate validity at Neptune-like temperatures. The singularity in (*∂*W/*∂V*)*<sup>T</sup>* at V = 0 mi / h is sufficiently weak that it has no effect on values of W itself.] Since standard atmospheric pressure at sea level on Earth is approximately 1 bar, for illustrative purposes and for argument's sake let us assume that the standard wind chill *formula* [Eq. (14)] retains at least approximate validity at the 1 bar level on Neptune, especially since the atmospheric density of 0.45 kg / m<sup>3</sup> at the 1 bar level on Neptune is at least comparable to that at the 1 bar level on Earth. (We will appraise this assumption later in this Sect. 4.2, especially in the second-to-last paragraph thereof.) The temperature in Neptune's atmosphere at the 0.1 bar level is *T* = 55 K = −218 ◦C = −361 ◦F [65]. Since Eq. (14) was derived for standard conditions (1 bar atmospheric pressure on Earth), its accuracy may be reduced if it is applied at the 0.1 bar level on Neptune. If we nevertheless apply it at the 0.1 bar level on Neptune, we obtain, even with a slow (by Neptune standards) V = 50 mi / h wind, W = −544 ◦F = −320 ◦C = −47 K.

It should be noted that the concept of hot negative Kelvin temperature can meaningfully be applied for 2-energy-level systems [63], whether or not either level or both are degenerate but *only* for 2-energy-level systems [63]. For systems with 3 or more energy levels, wherein population need not be a monotonic function of level (multiple population inversions are possible with 4 or more levels), the concept of hot negative Kelvin temperature becomes

It has been remarked [59–62] that there are advantages in defining temperature via 1/*T* = − (*∂S*/*∂E*)*V*,*N*, because by this definition the numerical value of a system's temperature always increases monotonically with its increasing ability to spontaneously deliver heat to its surroundings or equivalently with its decreasing ability to spontaneously accept heat from its surroundings, whether temperature defined via *T* = (*∂E*/*∂S*)*V*,*<sup>N</sup>* is positive or negative [59–62]. But temperature defined via *T* = (*∂E*/*∂S*)*V*,*<sup>N</sup>* has the advantages of numerical proportionality to temperature as measured by an ideal-gas thermometer and to average thermal kinetic energy per degree of freedom of ideal-gas molecules in the classical

(nonquantum) regime. So we employ the definition *T* = (*∂E*/*∂S*)*V*,*<sup>N</sup>* in this chapter.

0.6215*T*◦<sup>F</sup> <sup>+</sup> (0.4275*T*◦<sup>F</sup> <sup>−</sup> 35.75) <sup>V</sup>0.16

In Eq. (14), the wind speed V is that at the 5 ft (typical face) level, based on reduction owing to surface friction of wind speed measured at the standard 10 m or 33 ft level to the 5 ft level [66]. But over flat open ground or over open water, this reduction in wind speed is

<sup>5</sup> (Re: Entry [63], Ref. [63]) In Ref. [63], Dr. Peter Atkins doesn't seem to explicitly state that negative Kelvin temperatures are hotter than ∞ K, not colder than 0 K. He admits the possibility of attaining 0 K via noncyclic processes, but as we showed in Sect. 3. of this chapter *purely* dynamic — as opposed to *thermo*dynamic — limitations may contravene. On pp. 103–104 of Ref. [63], he correctly states that the third law of thermodynamics is "not really in the same league" as the zeroth, first, and second laws, and that "hints of the Third Law of Thermodynamics are already present in the consequences of the second law," but that the Third Law of Thermodynamics is "the final link in the confirmation that Boltzmann's and Clausius's definitions refer to the same property." But his statement that "we need to do an ever increasing, and ultimately infinite, amount of work to remove energy from a body as heat as its temperature approaches absolute zero" neglects the rapid decrease in specific heat as absolute zero is approached

mi / h + 35.74

◦F . (14)

Insofar as is known, *true* negative Kelvin temperatures that are *colder* than *T* = 0 K do not exist. Nevertheless, we can still consider *effective* negative Kelvin temperatures that are *colder* than *T* = 0 K — linearly extrapolating the Kelvin temperature scale downwards through *T* = 0 K to negative *effective* values. Such cold *effective* negative Kelvin temperatures *do* exist. Consider, for example, the effective wind-chill temperature W on Neptune, at the level in Neptune's atmosphere where the pressure is 1 bar, approximately 1 atm. The wind-chill temperature W is the temperature that calm air must have to produce the same chilling effect as moving air — wind — at speed V, all other things being equal. The *true* mean temperature (without wind chill) at the 1 bar level on Neptune is approximately *T* = 72 K = −201 ◦C = −330 ◦F [65]. The standard *formula* for wind-chill temperature W employed by the U. S. A.

**4.2. Cold effective negative Kelvin temperatures**

unwieldy and contrived [63].5

288 Recent Advances in Thermo and Fluid Dynamics

National Weather Service is [66]

as discussed in Sect. 2. of this chapter.

W =  The standard wind-chill *formula* [Eq. (14)] should not be confused with the standard wind-chill *table* [66]. The standard wind-chill *table* is based on a standard of calm of 3 mi / h (typical walking speed), rather than on the *true* standard of calm V = 0 mi / h in *true* accordance with the standard wind-chill *formula* [Eq. (14)] that we adopt in this Sect. 4.2. Also, the recommended ranges of applicability of the standard wind-chill *table* are −50 ◦F *< T* ≤ 50 ◦F and 3 mi / h *<* V *<* 110 mi / h [66]. But we base our calculations of W on the standard wind-chill *formula* [Eq. (14)], for which no limits on the range of applicability are stated for either *T* or V [66]. If there is a sufficiently strong wind on Neptune, then Eq. (14) yields a *cold* negative Kelvin *effective* wind-chill temperature W.

A physical interpretation is this: In order to produce the same chilling effect as air at temperature *T* = 72 K = −201 ◦C = −330 ◦F at the 1 bar level on Neptune [65] with a 50 mi / h wind [65,66], calm air would have to be at temperature W = −22 K = −296 ◦C = −500 ◦F — colder than absolute zero, sub-0 K. [The 0.1 bar level on Neptune is colder, but as noted in the first paragraph of this Sect. 4.2, Eq. (14) is likely more accurate if applied at the 1 bar level on Neptune.] The average thermal translational kinetic energy of the air molecules would have to be negative, and hence their average thermal speed imaginary. Our physical interpretation assumes that this super-cold, or hyper-cold, sub-0 K air of our imagination remains an ideal gas, for which the restricted definition of temperature as twice

<sup>6</sup> (Re: Entries [66] and [67], Refs. [66] and [67]) An online brochure accessible at Ref. [66] provides more information. Reference [67] augments Ref. [66] with still more information, including references and a few alternative formulas for wind-chill temperature W. (In Australia the wind-chill temperature W is dubbed as the apparent temperature AT .) In this Sect. 4.2. we always calculate W based on the formula employed by the U. S. A. National Weather Service [Eq. (14)].

the average thermal kinetic energy *E*kin per molecular translational degree of freedom divided by Boltzmann's constant, i.e., *T* = 2 *E*kin /*k* [68–81], is valid — and is extrapolated as remaining valid even for negative values of *E*kin and *T*. A necessary, but probably not sufficient, property of our hypothetical super-cold, or hyper-cold, air molecules is that they exert no attractive forces, however weak, on each other, so that they could never condense into a liquid or solid. (This could obtain, at least for all practical purposes, if the average distance between real air molecules is more than a few orders of magnitude larger than typical molecular sizes of <sup>∼</sup> <sup>10</sup>−<sup>10</sup> m to <sup>∼</sup> <sup>10</sup>−<sup>9</sup> m — but then of course the density would be much lower than the 0.45 kg / m<sup>3</sup> obtaining at the 1 bar <sup>≈</sup> 1 atm level on Neptune.) Our physical interpretation seems limited to this restricted definition of temperature. There seems to be no obvious way of extending our physical interpretation of cold negative *effective* (wind-chill) Kelvin temperature W in terms of the most general definition of *true* Kelvin temperature, i.e., *T* = (*∂E*/*∂S*)*V*,*<sup>N</sup>* [68–81]. Even for *true* (not effective) *non*negative Kelvin temperatures, the restricted definition of temperature *T* = 2 *E*kin /*k* [68–81] is valid if and only if, as is the case of ideal gases, *E*kin is directly proportional to *T* [68–81]. There exist excellent in-depth discussions of the concept of temperature, especially concerning the point that "Temperature is deeper than average kinetic energy." [78,79]. Nevertheless, although taking temperature as proportional to average thermal kinetic energy per degree of freedom is not the most general concept [78,79], it suffices to serve as one of the elements in an important derivation of Boltzmann's principle relating entropy and probability [80,81] and in an important generalization of the relation between entropy and heat [82].7

It has been argued [83], that Eq. (14) for wind-chill temperature W is only an approximation [83], that even as an approximation it is valid only at Earth-like or "human"

<sup>7</sup> (Re: Entries [71–77], Refs. [2], [3], [4], [48], [49], [76], and [77]) It is usually stated that the definition of temperature in terms of the Carnot efficiency of a reversible heat engine yields only a ratio of the two temperatures of the hot and cold reservoirs, not the one temperature of either reservoir considered individually, as does *T* = (*∂E*/*∂S*)*V*,*N*, and as does even the more restricted *T* = 2 *E*kin /*k* for the special case of ideal-gas reservoirs. To obtain actual values of temperature by this method rather than just the ratio of two temperatures, it is usually stated that the temperature of at least one of the two reservoirs must be ascertained by other means, most typically by allowing one reservoir to attain thermodynamic equilibrium with water at its triple point. This is discussed in Entries [71–75]. Thermodynamic equilibrium with water at its triple point is likewise employed to fix the temperature scale of ideal-gas thermometers, as discussed in Entry [75]. However, Refs. [76] and [77] describe how this requirement for a water-triple-point (or any other heat reservoir) is overcome, at least in principle even if not in practice, by employing a sequence of ideal, reversible, Carnot engines, the cold reservoir for engine **N** serving as the hot reservoir for engine **N** + 1, with the heat input for engine **N** + 1 to that for engine **N** being in a fixed ratio *r* (0 *< r <* 1), and with each engine doing an equal amount of work. Then the last engine in the sequence *must* have a cold reservoir at *T* = 0 K, thus dispensing with the requirement of a water-triple-point or other standardizing heat reservoir. Of course, this considers only the Second-Law aspect of the problem; according to the unattainability formulation of the third law, especially in its strongest mode, a cold reservoir at *precisely T* = 0 K is impossible. But perhaps a cold reservoir at *T arbitrarily close to* 0 K or even *sufficiently close to* 0 K suffices to thus dispense, even if not perfectly then at least for all practical purposes, with the requirement for a water-triple-point or other standardizing heat reservoir. [One point concerning Sect. 58 of Ref. [77]: Consider, as in Sect. 58 of Ref. [77], a heat engine operating between a heat source at positive Kelvin temperature and a heat sink at *true* (*not* merely effective) *cold* negative Kelvin temperature *if* true (not merely effective) *cold* negative Kelvin temperatures could exist — *if* the Kelvin temperature scale *could* be linearly extrapolated downwards through *T* = 0 K to *true* (not merely effective) negative values. Contrary to what is stated in Sect. 58 of Ref. [77], such a heat engine, *if* it could exist, would *not* discard more heat to its heat sink than it received from its heat source, thereby violating the First Law of Thermodynamics (conservation of energy). It would discard *negative* heat, i.e., it would *extract* heat, from its heat sink — in addition to extracting heat from its heat source as does a standard heat engine. Its work output (neglecting friction and other irreversibilities) would equal the heat it extracts from its heat source *plus* the heat it extracts from its heat sink. Thus its efficiency as usually defined = (work output) ÷ (heat extracted from heat source *alone*) *>* 100%, consistent with the First Law of Thermodynamics, but of course *in*consistent with the second and third laws.

temperatures [83]. Thus, even though Eq. (14) is likely more accurate if applied at the 1 bar level on Neptune than at the 0.1 bar level on Neptune, we cannot be sure of its accuracy even at the 1 bar level on Neptune [83]. Moreover, it has also been argued [83] that W is more correctly expressed as W / m<sup>2</sup> of heat loss flux rather than as the temperature of calm air that would have the same chilling effect as moving air — wind — at speed V [83]. Indeed, many if not most national weather services *do* express <sup>W</sup> as W / m<sup>2</sup> of heat loss flux rather than as the temperature of calm air that would have the same chilling effect. (The national weather services of the U. S. A. [66], Canada [67], and Australia [67] employ wind-chill formulas for the temperature of calm air that would have the same chilling effect.) But *if* wind chill *is* expressed as the temperature of calm air that would have the same chilling effect [66,67], *then* irrespective of the equation for W that even if not exactly correct is at least a good approximation at the 1 bar level on Neptune, be that Eq. (14) or otherwise, it seems inescapable that in order to produce the same chilling effect as a sufficiently strong wind at sufficiently cold but still positive Kelvin temperatures, calm air *must* be colder than 0 K. Thus it seems inescapable that the *effective* (wind-chill) Kelvin temperature W must then be colder than 0 K even if no *actual* temperature can be colder than 0 K.

This would obtain even more strongly for a helium atmosphere, which remains gaseous at a pressure of 1 bar at Kelvin temperature *T* which, while still positive, is nevertheless much colder than the value *T* = 72 K = −201 ◦C = −330 ◦F obtaining at the 1 bar level on Neptune or even than the value *T* = 55 K = −218 ◦C = −361 ◦F obtaining at the 0.1 bar level on Neptune [65,84,85].8 While recognizing the caveats discussed in the immediately preceding paragraph, nevertheless for illustrative purposes and for argument's sake let us assume that the standard wind chill *formula* [Eq. (14)] retains at least approximate validity for gaseous helium at a pressure of 1 bar. At a pressure of 1 bar, the common isotope of naturally-occurring helium, 2He4, is gaseous at *<sup>T</sup>* <sup>=</sup> 5 K <sup>=</sup> <sup>−</sup><sup>268</sup> ◦<sup>C</sup> <sup>=</sup> <sup>−</sup><sup>450</sup> ◦F, and the rare isotope of naturally-occurring helium (which fortunately can be produced artificially [84]), 2He3, is gaseous at *<sup>T</sup>* <sup>=</sup> 4 K <sup>=</sup> <sup>−</sup><sup>269</sup> ◦<sup>C</sup> <sup>=</sup> <sup>−</sup><sup>452</sup> ◦F [84,85]. Again taking <sup>V</sup> <sup>=</sup> 50 mi / h, for *T* = 5 K = −268 ◦C = −451 ◦F Eq. (14) yields W = −118 K = −391 ◦C = −671 ◦F, and for *T* = 4 K = −269 ◦C = −452 ◦F Eq. (14) yields W = −119 K = −392 ◦C = −674 ◦F.

#### **4.3. Limits of the possible**

the average thermal kinetic energy *E*kin per molecular translational degree of freedom divided by Boltzmann's constant, i.e., *T* = 2 *E*kin /*k* [68–81], is valid — and is extrapolated as remaining valid even for negative values of *E*kin and *T*. A necessary, but probably not sufficient, property of our hypothetical super-cold, or hyper-cold, air molecules is that they exert no attractive forces, however weak, on each other, so that they could never condense into a liquid or solid. (This could obtain, at least for all practical purposes, if the average distance between real air molecules is more than a few orders of magnitude larger than typical molecular sizes of <sup>∼</sup> <sup>10</sup>−<sup>10</sup> m to <sup>∼</sup> <sup>10</sup>−<sup>9</sup> m — but then of course the density would be much lower than the 0.45 kg / m<sup>3</sup> obtaining at the 1 bar <sup>≈</sup> 1 atm level on Neptune.) Our physical interpretation seems limited to this restricted definition of temperature. There seems to be no obvious way of extending our physical interpretation of cold negative *effective* (wind-chill) Kelvin temperature W in terms of the most general definition of *true* Kelvin temperature, i.e., *T* = (*∂E*/*∂S*)*V*,*<sup>N</sup>* [68–81]. Even for *true* (not effective) *non*negative Kelvin temperatures, the restricted definition of temperature *T* = 2 *E*kin /*k* [68–81] is valid if and only if, as is the case of ideal gases, *E*kin is directly proportional to *T* [68–81]. There exist excellent in-depth discussions of the concept of temperature, especially concerning the point that "Temperature is deeper than average kinetic energy." [78,79]. Nevertheless, although taking temperature as proportional to average thermal kinetic energy per degree of freedom is not the most general concept [78,79], it suffices to serve as one of the elements in an important derivation of Boltzmann's principle relating entropy and probability [80,81] and

290 Recent Advances in Thermo and Fluid Dynamics

in an important generalization of the relation between entropy and heat [82].7

course *in*consistent with the second and third laws.

It has been argued [83], that Eq. (14) for wind-chill temperature W is only an approximation [83], that even as an approximation it is valid only at Earth-like or "human"

<sup>7</sup> (Re: Entries [71–77], Refs. [2], [3], [4], [48], [49], [76], and [77]) It is usually stated that the definition of temperature in terms of the Carnot efficiency of a reversible heat engine yields only a ratio of the two temperatures of the hot and cold reservoirs, not the one temperature of either reservoir considered individually, as does *T* = (*∂E*/*∂S*)*V*,*N*, and as does even the more restricted *T* = 2 *E*kin /*k* for the special case of ideal-gas reservoirs. To obtain actual values of temperature by this method rather than just the ratio of two temperatures, it is usually stated that the temperature of at least one of the two reservoirs must be ascertained by other means, most typically by allowing one reservoir to attain thermodynamic equilibrium with water at its triple point. This is discussed in Entries [71–75]. Thermodynamic equilibrium with water at its triple point is likewise employed to fix the temperature scale of ideal-gas thermometers, as discussed in Entry [75]. However, Refs. [76] and [77] describe how this requirement for a water-triple-point (or any other heat reservoir) is overcome, at least in principle even if not in practice, by employing a sequence of ideal, reversible, Carnot engines, the cold reservoir for engine **N** serving as the hot reservoir for engine **N** + 1, with the heat input for engine **N** + 1 to that for engine **N** being in a fixed ratio *r* (0 *< r <* 1), and with each engine doing an equal amount of work. Then the last engine in the sequence *must* have a cold reservoir at *T* = 0 K, thus dispensing with the requirement of a water-triple-point or other standardizing heat reservoir. Of course, this considers only the Second-Law aspect of the problem; according to the unattainability formulation of the third law, especially in its strongest mode, a cold reservoir at *precisely T* = 0 K is impossible. But perhaps a cold reservoir at *T arbitrarily close to* 0 K or even *sufficiently close to* 0 K suffices to thus dispense, even if not perfectly then at least for all practical purposes, with the requirement for a water-triple-point or other standardizing heat reservoir. [One point concerning Sect. 58 of Ref. [77]: Consider, as in Sect. 58 of Ref. [77], a heat engine operating between a heat source at positive Kelvin temperature and a heat sink at *true* (*not* merely effective) *cold* negative Kelvin temperature *if* true (not merely effective) *cold* negative Kelvin temperatures could exist — *if* the Kelvin temperature scale *could* be linearly extrapolated downwards through *T* = 0 K to *true* (not merely effective) negative values. Contrary to what is stated in Sect. 58 of Ref. [77], such a heat engine, *if* it could exist, would *not* discard more heat to its heat sink than it received from its heat source, thereby violating the First Law of Thermodynamics (conservation of energy). It would discard *negative* heat, i.e., it would *extract* heat, from its heat sink — in addition to extracting heat from its heat source as does a standard heat engine. Its work output (neglecting friction and other irreversibilities) would equal the heat it extracts from its heat source *plus* the heat it extracts from its heat sink. Thus its efficiency as usually defined = (work output) ÷ (heat extracted from heat source *alone*) *>* 100%, consistent with the First Law of Thermodynamics, but of

How impossible is the super-cold, or hyper-cold, sub-0 K air of our imagination — but with a true as opposed to merely effective sub-0 K temperature? It is (at the very least, almost) certainly physically impossible, but, at least *prima facie*, it seems not to be logically impossible. The physically impossible at least does not exist and possibly even cannot exist in physical reality, but can exist in the imagination and hence in virtual reality (imagination displayed via a computer). The logically impossible cannot exist — rather than merely does not exist — not only in physical reality, but to boot not even in the imagination and hence not even in virtual reality.

A Euclidean (planar) right triangle that violates the Pythagorean Theorem is not merely physically impossible but logically impossible. Such a triangle cannot exist — rather than

<sup>8</sup> (Re: Entry [85], Ref. [17]) In Fig. 14-19 (a) on p. 381 of Ref. [17], the normal boiling point of 2He3 is incorrectly shown at a pressure of approximately 1.3 atm instead of at 1 atm.

merely does not exist — not only in physical reality, but to boot not even in the imagination: it cannot even be imagined; it cannot exist even in virtual reality.

By contrast, for example, a violation of the first law of thermodynamics (conservation of energy) [86,87] is (at least so far as is known [86,87]) physically but not logically impossible — perpetual motion of the first kind can at least be *imagined*; it *can* exist at least in *virtual* reality [86,87]. Indeed even concerning the *physical* impossibility (or possibility?) of violation of energy conservation, we should note that energy conservation has never been rigorously proven in general relativity, and that there have been serious proposals for its possible violation at cosmological distance and time scales [86,87]. But: Any proposed violation of energy conservation should address the difficulty posed by Noether's Theorem [88]. According to Noether's Theorem [88], nonconservation of energy implies that the time-invariance of the fundamental laws of physics must be broken (and vice versa). There isn't much wiggle room — even small changes in the (at least apparent) fine-tuning of at least some of the laws of physics would render life (at least carbon-based life as we know it on Earth) impossible [89–92]. Energy, even free energy or equivalently negentropy, is far from being the only requirement for life. But could Noether's Theorem be satisfied by considering nascent energy to be a new boundary (initial) condition upon the Universe, thereby preserving the time-invariance of the laws of physics? For example, consider the following thought experiment: What if a mass *m* subject to local gravitational acceleration *g* could spontaneously rise through a height ∆*y* to the ceiling — not spontaneously get cooler and rise to the ceiling (on demand rather than via unpredictable and uncontrollable fluctuation), thereby violating the Second Law of Thermodynamics, but just spontaneously rise to the ceiling, thereby violating the First Law of Thermodynamics (energy conservation)? Could the nascent gravitational potential energy *mg*∆*y* simply be a new initial condition upon the Universe, leaving the time-invariance of the laws of physics intact? Might Noether's Theorem accept payment in the cheap currency of boundary (initial) conditions instead of the expensive currency of the laws of physics, and hence not pose any difficulty? [Note: Proposals such as those cited for genuine creation of nascent energy [86,87] should not be confused with proposals for creation of positive mass-energy at the expense of negative energy, typically at the expense of negative gravitational energy [93–101], but in some versions of the steady-state theory [102–107] at the expense of a negative-energy creation field (the C field) [105–107]. (There are difficulties associated with the C field [106,107].] The former proposals [86,87] but not the latter ones [93–107] contravene the First Law of Thermodynamics (conservation of energy).]

Thus since knowledge is imperfect and incomplete, perhaps one should not *a priori* rule out any nonzero probability, however remote, that a logically possible phenomenon might also be physically possible [86,87]. Hence the "Insofar as is known" in the first sentence of Sect. 4.2, the "(at the very least, almost)" in the second sentence of the first paragraph of Sect. 4.3, and the "(at least so far as is known [86,87])" in the first sentence of the third paragraph of Sect. 4.3. Unlike our Pythagorean-Theorem-violating Euclidean (planar) right triangle but like a violation of energy conservation [86,87], our super-cold, or hyper-cold, sub-0 K air can at least be *imagined*.

## **5. Brief concluding remarks**

merely does not exist — not only in physical reality, but to boot not even in the imagination:

By contrast, for example, a violation of the first law of thermodynamics (conservation of energy) [86,87] is (at least so far as is known [86,87]) physically but not logically impossible — perpetual motion of the first kind can at least be *imagined*; it *can* exist at least in *virtual* reality [86,87]. Indeed even concerning the *physical* impossibility (or possibility?) of violation of energy conservation, we should note that energy conservation has never been rigorously proven in general relativity, and that there have been serious proposals for its possible violation at cosmological distance and time scales [86,87]. But: Any proposed violation of energy conservation should address the difficulty posed by Noether's Theorem [88]. According to Noether's Theorem [88], nonconservation of energy implies that the time-invariance of the fundamental laws of physics must be broken (and vice versa). There isn't much wiggle room — even small changes in the (at least apparent) fine-tuning of at least some of the laws of physics would render life (at least carbon-based life as we know it on Earth) impossible [89–92]. Energy, even free energy or equivalently negentropy, is far from being the only requirement for life. But could Noether's Theorem be satisfied by considering nascent energy to be a new boundary (initial) condition upon the Universe, thereby preserving the time-invariance of the laws of physics? For example, consider the following thought experiment: What if a mass *m* subject to local gravitational acceleration *g* could spontaneously rise through a height ∆*y* to the ceiling — not spontaneously get cooler and rise to the ceiling (on demand rather than via unpredictable and uncontrollable fluctuation), thereby violating the Second Law of Thermodynamics, but just spontaneously rise to the ceiling, thereby violating the First Law of Thermodynamics (energy conservation)? Could the nascent gravitational potential energy *mg*∆*y* simply be a new initial condition upon the Universe, leaving the time-invariance of the laws of physics intact? Might Noether's Theorem accept payment in the cheap currency of boundary (initial) conditions instead of the expensive currency of the laws of physics, and hence not pose any difficulty? [Note: Proposals such as those cited for genuine creation of nascent energy [86,87] should not be confused with proposals for creation of positive mass-energy at the expense of negative energy, typically at the expense of negative gravitational energy [93–101], but in some versions of the steady-state theory [102–107] at the expense of a negative-energy creation field (the C field) [105–107]. (There are difficulties associated with the C field [106,107].] The former proposals [86,87] but not the latter ones [93–107] contravene the First Law of

Thus since knowledge is imperfect and incomplete, perhaps one should not *a priori* rule out any nonzero probability, however remote, that a logically possible phenomenon might also be physically possible [86,87]. Hence the "Insofar as is known" in the first sentence of Sect. 4.2, the "(at the very least, almost)" in the second sentence of the first paragraph of Sect. 4.3, and the "(at least so far as is known [86,87])" in the first sentence of the third paragraph of Sect. 4.3. Unlike our Pythagorean-Theorem-violating Euclidean (planar) right triangle but like a violation of energy conservation [86,87], our super-cold, or hyper-cold, sub-0 K air can

it cannot even be imagined; it cannot exist even in virtual reality.

292 Recent Advances in Thermo and Fluid Dynamics

Thermodynamics (conservation of energy).]

at least be *imagined*.

Hopefully our considerations of and related to absolute zero 0 K have been helpful. In Sect. 2.2, we showed that in principle 0 K can be *attained* at the expense of only a finite, typically small, cost of work via standard TSRR, in Sect. 2.5. at the expense of an even smaller cost of high-temperature heat via absorption TSRR, and in Sect. 3.1. at the expense of a small cost of work via CSRR, employing weighing or a Stern-Gerlach apparatus. (Recall from the sixth and seventh paragraphs of Sect. 3.2. that the specific QCR method discussed in Sect. 3 of Ref. [1] employs first CSRR and then ERR [1]: we employed the first, Stern-Gerlach-apparatus CSRR, step thereof in the sixth paragraph of Sect. 3.2.) In the standard and absorption TSRR cases, the unattainability formulation of the Third Law of Thermodynamics of thermodynamics does not require infinite expenditure of work and heat, respectively to attain 0 K, but forbids the expenditure, respectively, of the required small cost of work and even smaller cost of heat. But in the CSRR cases, it does *not*, even in its strongest mode, forbid the expenditure of the required small cost of work.

But there are also the difficulties of *maintaining* 0 K and of *verifying* [22] that 0 K has even been *attained*, which we discussed in Sect. 2.3, the last paragraphs of Sects. 2.4. and 2.5, and Sect. 3. *Perfectly maintaining* 0 K for more than infinitessimal time requires perfect insulation [23], and *perfectly verifying* [22] that 0 K has even been *attained* requires infinite time. Even given perfect insulation [23] (recall Sect. 2.3.) and hence that 0 K can be perfectly *maintained*, the unattainability formulation of the Third Law of Thermodynamics in its *strongest mode*, which forbids attainment of 0 K by *any means* whatsoever, seems inviolable with respect to *perfect* verification [22] that 0 K has been *attained*, because of the infinite-time requirement imposed by the energy-time uncertainty principle. But if we do not insist on *exactly* perfect verification [22] and are willing to accept verification that is perfect *for all practical purposes*, then to this extent the unattainability formulation of the third law even in its strongest mode *is* challenged. The limitation to "for all practical purposes" is further imposed because as per Sect. 2.3. insulation [23] can be perfect only for all practical purposes. At least in principle and possibly also in practice, CSRR and QCR [1] seem superior to standard TSRR or even absorption TSRR in effecting the challenge to the unattainability formulation of the Third Law of Thermodynamics in its strongest mode, albeit for all practical purposes and not with exact perfection.

Hopefully also our considerations in Sect. 4. of negative Kelvin temperatures, both true ones hotter than ∞ K and effective ones colder than 0 K, have been helpful.

#### **6. Appendix: A few fine points concerning the Third Law of Thermodynamics**

It is generally stated that the Nernst formulation of the Third Law of Thermodynamics, according to which all entropy changes vanish at 0 K, and the unattainability formulation thereof, according to which 0 K is unattainable in a finite number of finite operations, are equivalent. But we should note that there are dissensions to this viewpoint [108–113].9

<sup>9</sup> (Re: Entry [109], Ref. [109] Footnote 5 on p. 494 of Ref. [109] concerns "a residual inequivalence" between the Nernst heat theorem and unattainability principle, with the former construed as more fundamental.

Also, in considering the discreteness of energy eigenstates required by quantum mechanics in any system constrained within a fixed finite volume, or even within an unfixed but always finite volume for example corresponding to maintenance of constant pressure, we did not mention the role of quantum-mechanical Bose-Einstein symmetry or Fermi-Dirac antisymmetry requirements on the allowed wave functions [114]. The gaps between energy eigenstates at very low temperatures in light of these requirements can be much larger than would be the case in the absence of these requirements [114]. For a typical laboratory-type macroscopic system, the energy gap ∆*E* between the ground and first excited state is ∆*E* ∼ <sup>10</sup>−<sup>20</sup> <sup>K</sup> *<sup>k</sup>* <sup>−</sup> <sup>10</sup>−<sup>19</sup> <sup>K</sup> *<sup>k</sup>* [114,115]. Yet the entropy and heat capacity of a typical laboratory-type macroscopic system is typically already only a very small fraction of the value predicted by classical (as opposed to quantum) statistical mechanics at *T* ∼ 10 K [114,115]. It has been noted that at *T* ∼ 10 K the energy *per particle* in a typical laboratory-type macroscopic system is <sup>∼</sup> <sup>∆</sup>*<sup>E</sup>* <sup>∼</sup> <sup>10</sup>−<sup>20</sup> <sup>K</sup> *<sup>k</sup>* <sup>−</sup>10−<sup>19</sup> <sup>K</sup> *<sup>k</sup>* [114,115]. But because the characteristic temperatures of quantum statistical mechanics, for example, the Debye, Fermi-Dirac, and Bose-Einstein temperatures [116], are independent of the size of a system [116], this is a fortuitous result owing to the typical sizes of laboratory-type macroscopic systems [114,115].

A third formulation of the Third Law of Thermodynamics has also been stated [117], according to which the zero of entropy with a system in its ground energy eigenstate (assumed nondegenerate), is as unattainable as 0 K itself [117].<sup>10</sup> This hird formulation of the Third Law of Thermodynamics has been stated with respect to *thermo*dynamics. But, in fact, it ultimately obtains owing to the *pure* (quantum) dynamics of the energy-time uncertainty principle, and with respect to fixing a system *exactly* into *any* of its energy eigenstates in general (not just specifically its ground state), degenerate or not. The energy-time uncertainty principle requires infinite time to *exactly* — with *strictly zero* uncertainty — fix the energy of any system into *any* of its energy eigenstates in general (not just specifically its ground state), degenerate or not.

#### **Acknowledgments**

I gratefully acknowledge Dr. Marlan O. Scully for insightful and thoughtful discussions and communications concerning Ref. [1]. I also am very grateful to Dr. Donald H. Kobe for expressing interest in and providing helpful remarks concerning Ref. [1], as well as for acquainting me with Refs. [80–82]. Their contributions helped to inspire the ideas for this chapter. Additionally, I am indebted to a reviewer for the American Journal of Physics for valuable insights concerning an earlier version of this manuscript that was not accepted, and I am grateful to Dr. Daniel P. Sheehan for helpful comments and suggestions concerning earlier versions of this manuscript. I am thankful to Dr. Paolo Grigolini for very helpful and thoughtful considerations concerning both earlier versions and the most recent versions of this manuscript in Special Problems courses. I thank Dr. S. Mort Zimmerman for engaging in general scientific discussions over many years, and both Dan Zimmerman and Dr. Kurt W. Hess and for brief yet helpful discussions concerning this chapter and for engaging in general

<sup>10</sup> (Re: Entries [17] and [117], Refs. [17] and [117]) The detailed discussions concerning the third law of thermodynamics in Ref. [117] are largely deleted in Ref. [17], which provides only a brief mention of the third law on p. 217. Reference [117] dubs the Nernst formulation of the Third Law of Thermodynamics as the Nernst-Simon formulation thereof. Reference [17] does not render Ref. [117] obsolete, because Ref. [117] discusses aspects not discussed in Ref. [17], and vice versa.

scientific discussions at times. I also thank Dr. Iva Simcic, Publishing Process Manager, for much very helpful advice in preparing this chapter and for much extra time to prepare it, and Technical Support at MacKichan Software for their very helpful advice concerning Scientific WorkPlace 5.5.

#### **Author details**

Jack Denur

Also, in considering the discreteness of energy eigenstates required by quantum mechanics in any system constrained within a fixed finite volume, or even within an unfixed but always finite volume for example corresponding to maintenance of constant pressure, we did not mention the role of quantum-mechanical Bose-Einstein symmetry or Fermi-Dirac antisymmetry requirements on the allowed wave functions [114]. The gaps between energy eigenstates at very low temperatures in light of these requirements can be much larger than would be the case in the absence of these requirements [114]. For a typical laboratory-type macroscopic system, the energy gap ∆*E* between the ground and first excited state is ∆*E* ∼ <sup>10</sup>−<sup>20</sup> <sup>K</sup> *<sup>k</sup>* <sup>−</sup> <sup>10</sup>−<sup>19</sup> <sup>K</sup> *<sup>k</sup>* [114,115]. Yet the entropy and heat capacity of a typical laboratory-type macroscopic system is typically already only a very small fraction of the value predicted by classical (as opposed to quantum) statistical mechanics at *T* ∼ 10 K [114,115]. It has been noted that at *T* ∼ 10 K the energy *per particle* in a typical laboratory-type macroscopic system is <sup>∼</sup> <sup>∆</sup>*<sup>E</sup>* <sup>∼</sup> <sup>10</sup>−<sup>20</sup> <sup>K</sup> *<sup>k</sup>* <sup>−</sup>10−<sup>19</sup> <sup>K</sup> *<sup>k</sup>* [114,115]. But because the characteristic temperatures of quantum statistical mechanics, for example, the Debye, Fermi-Dirac, and Bose-Einstein temperatures [116], are independent of the size of a system [116], this is a fortuitous result

owing to the typical sizes of laboratory-type macroscopic systems [114,115].

degenerate or not.

[17], and vice versa.

**Acknowledgments**

294 Recent Advances in Thermo and Fluid Dynamics

A third formulation of the Third Law of Thermodynamics has also been stated [117], according to which the zero of entropy with a system in its ground energy eigenstate (assumed nondegenerate), is as unattainable as 0 K itself [117].<sup>10</sup> This hird formulation of the Third Law of Thermodynamics has been stated with respect to *thermo*dynamics. But, in fact, it ultimately obtains owing to the *pure* (quantum) dynamics of the energy-time uncertainty principle, and with respect to fixing a system *exactly* into *any* of its energy eigenstates in general (not just specifically its ground state), degenerate or not. The energy-time uncertainty principle requires infinite time to *exactly* — with *strictly zero* uncertainty — fix the energy of any system into *any* of its energy eigenstates in general (not just specifically its ground state),

I gratefully acknowledge Dr. Marlan O. Scully for insightful and thoughtful discussions and communications concerning Ref. [1]. I also am very grateful to Dr. Donald H. Kobe for expressing interest in and providing helpful remarks concerning Ref. [1], as well as for acquainting me with Refs. [80–82]. Their contributions helped to inspire the ideas for this chapter. Additionally, I am indebted to a reviewer for the American Journal of Physics for valuable insights concerning an earlier version of this manuscript that was not accepted, and I am grateful to Dr. Daniel P. Sheehan for helpful comments and suggestions concerning earlier versions of this manuscript. I am thankful to Dr. Paolo Grigolini for very helpful and thoughtful considerations concerning both earlier versions and the most recent versions of this manuscript in Special Problems courses. I thank Dr. S. Mort Zimmerman for engaging in general scientific discussions over many years, and both Dan Zimmerman and Dr. Kurt W. Hess and for brief yet helpful discussions concerning this chapter and for engaging in general

<sup>10</sup> (Re: Entries [17] and [117], Refs. [17] and [117]) The detailed discussions concerning the third law of thermodynamics in Ref. [117] are largely deleted in Ref. [17], which provides only a brief mention of the third law on p. 217. Reference [117] dubs the Nernst formulation of the Third Law of Thermodynamics as the Nernst-Simon formulation thereof. Reference [17] does not render Ref. [117] obsolete, because Ref. [117] discusses aspects not discussed in Ref. Address all correspondence to: jackdenur@my.unt.edu

Electric & Gas Technology, Inc., Rowlett, Texas, USA

#### **References** ontent...

[1] Scully MO, Aharonov Y, Kapale KT, Tannor DJ, Süssmann G, Walther H. Sharpening accepted thermodynamic wisdom via quantum control: or cooling to an internal temperature of zero by external coherent control fields without spontaneous emission. Journal of Modern Optics. 2002; **49**: 2297–2307. DOI: 10.1080/0950034021000011392

[2] Callen HB. Thermodynamics. New York: John Wiley & Sons; 1960, p. 27, Chap. 10 (especially Sect. 10.4), and Sect. 11.2.

[3] Callen HB. Thermodynamics and an Introduction to Thermostatistics. 2nd ed. New York: John Wiley & Sons; 1985, p. 30, Chap. 11 (especially Sect. 11-2), and Sect. 12-2.

[4] King, AL. Thermophysics. San Francisco: W. H. Freeman; 1962, Sects. 6.5 and 7.3, and p. 280.

[5] Guggenheim EA. Thermodynamics: An Advanced Treatment for Chemists and Physicists. 7th ed. Amsterdam: North-Holland; 1985, Sects. 1.66, 2.17, 3.53, 3.57–3.59, and 11.13–11.17 (especially 11.17).

[6] Kosloff R, Amikam L. Quantum Refrigerators and the III-Law of Thermodynamics. 12th Joint European Thermodynamics Conference; 1–5 July 2013; Brescia.

[7] Kosloff R, Amikam L. Quantum Heat Engines and Refrigerators: Continuous Devices. Ann. Rev. Phys. Chem. 2014; **65**: 365–393. DOI: 10.1146/annurev-physchem-040513-103724

[8] Allahverdyan AE, Hovhannisyan K, Mahler G. Optimal refrigerator. Phys. Rev. E. 2010; **81**. 051129-1–12.

[9] Reference [2], Chap. 4, especially Sects. 4.4– 4.7, and most especially Sect. 4-7.

[10] Reference [3], Chap. 4, especially Sects. 4.5–4-6, and most especially Sect. 4-6.

[11] Reference [4], pp. 57–82, especially Sects. 4.8, 5.3, 6.1, and 6.2.

[12] Reference [2], p. 75.

[13] Reference [3], pp. 115–116.

[14] Reference [2], Sects. 10.2 and 11.7.

[15] Reference [3], Chap. 11 (especially Sect. 11-2), Sects. 15-2, 15-3, 16-7, 16-8, 18-4, and 18-6.

[16] Reference [3], Sects. 15-3 and 16-2.

[17] Zemansky MW, Dittman RH. Heat and Thermodynamics. 7th ed. Boston: McGraw-Hill; 1997, Sects. 10.2 and 10.8.

[18] Wark K Jr. Thermodynamics. 6th ed. Boston: McGraw-Hill; 1999, Sects. 3-8 and 12-4.

[19] Reference [4], pp. 36, 237–240, and 316–326.

[20] Bohren CF, Albrecht BA. Atmospheric Thermodynamics. New York: Oxford University Press; 1998, pp. 335–366.

[21] Reference [4], Sect. 20.2.

[22] Reference [3], Sect. 11-3, especially the third sentence.

[23] Baturina TI, Vinokur VM. Superinsulator-Superconductor Duality in Two Dimensions. Ann. Phys. 2013; **331**: 236–257. DOI: 10.1016/j.aop.2012.12.007

[24] Reference [4], Sects. 5.8 and 6.1.

[25] Einstein A, Szilárd L. Refrigeration. U. S. A. Patent No. 1,781,541; 1930.

[26] Reference [2], p. 190.

[27] Reference [2], Sect. 10.4.

[28] Private communication, reviewer for the American Journal of Physics. 2010.

[29] Cohen BL. A simple treatment of potential barrier penetration. Am. J. Phys. 1965; **33**: 97–98. DOI: 10.1119/1.1971334

[30] Gomer R. Field Emission and Field Ionization. Cambridge, Mass: Harvard University Press; 1961, pp. 6–11, 66–70, and 181–183. The application of the position-momentum uncertainty principle per se is discussed on pp. 6–8.

[31] Hagmann MJ. Transit time for quantum tunneling. Solid State Comm. 1992; **82**: 867–870. DOI: 10.1016/0038-1098(92)90710-Q

[32] Hagmann MJ. Quantum tunneling times: A new solution compared to 12 other methods. Int. J. Quant. Chem. Suppl. 1992; **26**: 299–309. DOI: 10.1002/qua.560440826

[33] Hagmann MJ. Distribution of times for barrier traversal caused by energy fluctuations. J. App. Phys. 1993; **74**: 7302–7305. DOI: 10.1063/1.354995

[34] Hagmann MJ. Effects of the finite duration of quantum tunneling in laser-assisted scanning tunneling microscopy. Int. J. Quant. Chem. Suppl. 1994; **28**: 271–282. DOI: 10.1002/qua.560520829

[35] Hagmann MJ. Reduced effects of laser illumination due to the finite duration of quantum tunneling. J. Vac. Sci. Technol. B. 1994; **12**: 3191–3195. DOI: 10.1116/1.587498

[36] Landau LD, Lifshitz EM. Statistical Physics. 3rd ed. Oxford: Pergamon; 1980 (2005 Printing), Sect. 110.

[37] Landau LD, Lifshitz EM. Quantum Mechanics. 2nd revised ed. Oxford: Butterworth-Heinemann; 1991 (2005 Printing), Sects. 1, 16, and 44.

[38] Kobe DH, Aguilera-Navarro VC. Derivation of the energy-time uncertainty relation. Phys. Rev. A. 1994; **50**: 933–938. DOI: 10.1103/PhysRevA.50.933

[39] Kobe DH, Iwamoto H, Goto M, Aguilera-Navarro VC. Derivation of the energy-time uncertainty relation. Phys. Rev. A. 2001; **64**: Article ID: 022104: 8 pages. DOI: 10.1103/PhysRevA.64.022104

[40] Dodonov VV, Dodonov AV. Energy-time and frequency-time uncertainty relations: exact inequalities. Physica Scripta. 2015; **90**: Article ID: 074049: 22 pages. DOI: 10.1088/0031-8949/90/7/074049

[41] Denur J. The energy-time uncertainty principle and quantum phenomena. Am. J. Phys. 2010; **78**: 1132–1145. DOI: 10.1119/1.3133084

[42] Hill TL. Statistical Mechanics: Principles and Selected Applications. Original copyright: New York: McGraw-Hill; 1956. Unabridged and unaltered republication: New York: Dover; 1987. Chapters 1 and 2; especially Sects. 4, 5, 10, 11, and 13.

[43] Tolman RC. The Principles of Statistical Mechanics. Original copyright: Oxford, U. K.: Clarendon Press; 1938. Unabridged and unaltered republication: New York: Dover; 1979. Sections 21–25, pp. 344–345, and Sects. 84 and 98(e).

[44] Bohm D. Quantum Theory. Original copyright: Englewood Cliffs, N. J.: Prentice-Hall; 1951. Unaltered and unabridged republication: New York: Dover; 1989. Chapter 23.

[45] Reference [44], Chap. 8 (especially Sects. 8.22 and 8.23) and Chap. 22 (especially Sects. 22.2–22.3).

[46] Eisner RM. Fundamentals of Modern Physics. New York: John Wiley & Sons; 1961, Sect. 7.7.

[47] Reference [46], Sect. 2.5.

[14] Reference [2], Sects. 10.2 and 11.7.

296 Recent Advances in Thermo and Fluid Dynamics

[16] Reference [3], Sects. 15-3 and 16-2.

[19] Reference [4], pp. 36, 237–240, and 316–326.

[22] Reference [3], Sect. 11-3, especially the third sentence.

uncertainty principle per se is discussed on pp. 6–8.

J. App. Phys. 1993; **74**: 7302–7305. DOI: 10.1063/1.354995

Ann. Phys. 2013; **331**: 236–257. DOI: 10.1016/j.aop.2012.12.007

[25] Einstein A, Szilárd L. Refrigeration. U. S. A. Patent No. 1,781,541; 1930.

[28] Private communication, reviewer for the American Journal of Physics. 2010.

Int. J. Quant. Chem. Suppl. 1992; **26**: 299–309. DOI: 10.1002/qua.560440826

tunneling. J. Vac. Sci. Technol. B. 1994; **12**: 3191–3195. DOI: 10.1116/1.587498

1997, Sects. 10.2 and 10.8.

Press; 1998, pp. 335–366. [21] Reference [4], Sect. 20.2.

[26] Reference [2], p. 190. [27] Reference [2], Sect. 10.4.

97–98. DOI: 10.1119/1.1971334

DOI: 10.1016/0038-1098(92)90710-Q

10.1002/qua.560520829

Printing), Sect. 110.

[24] Reference [4], Sects. 5.8 and 6.1.

[15] Reference [3], Chap. 11 (especially Sect. 11-2), Sects. 15-2, 15-3, 16-7, 16-8, 18-4, and 18-6.

[17] Zemansky MW, Dittman RH. Heat and Thermodynamics. 7th ed. Boston: McGraw-Hill;

[20] Bohren CF, Albrecht BA. Atmospheric Thermodynamics. New York: Oxford University

[23] Baturina TI, Vinokur VM. Superinsulator-Superconductor Duality in Two Dimensions.

[29] Cohen BL. A simple treatment of potential barrier penetration. Am. J. Phys. 1965; **33**:

[30] Gomer R. Field Emission and Field Ionization. Cambridge, Mass: Harvard University Press; 1961, pp. 6–11, 66–70, and 181–183. The application of the position-momentum

[31] Hagmann MJ. Transit time for quantum tunneling. Solid State Comm. 1992; **82**: 867–870.

[32] Hagmann MJ. Quantum tunneling times: A new solution compared to 12 other methods.

[33] Hagmann MJ. Distribution of times for barrier traversal caused by energy fluctuations.

[34] Hagmann MJ. Effects of the finite duration of quantum tunneling in laser-assisted scanning tunneling microscopy. Int. J. Quant. Chem. Suppl. 1994; **28**: 271–282. DOI:

[35] Hagmann MJ. Reduced effects of laser illumination due to the finite duration of quantum

[36] Landau LD, Lifshitz EM. Statistical Physics. 3rd ed. Oxford: Pergamon; 1980 (2005

[18] Wark K Jr. Thermodynamics. 6th ed. Boston: McGraw-Hill; 1999, Sects. 3-8 and 12-4.

[48] Baeirlein R. Atoms and Information Theory. San Francisco: W. H. Freeman; 1971, Sect. 10.6, and Problem 10.8 of Chap. 10 on pp. 369–371.

[49] Baeirlein R. Thermal Physics. Cambridge, U. K.: Cambridge University Press; 1999, Chap. 6.

[50] Reference [46], Sects. 3.6–3.7, especially Sect. 3.7.

[51] Reference [46], Sects. 6.3.

[52] Brillouin L. Relativity Reexamined. New York: Academic; 1970, Sects. 1.1 and 3.3–3.4.

[53] Reference [46], Sect. 7.5.

[54] Sheehan DP, editor. Quantum Limits to the Second Law, AIP Conference Proceedings Volume 643. Melville, N. Y.: American Institute of Physics; 2002.

[55] Nukulov AV, Sheehan DP, editors. Special Issue: Quantum Limits to the Second Law of Thermodynamics. Entropy. March 2004: Vol. 6, Issue 1.

[56] Cápek V, Sheehan DP. Challenges to the Second Law of Thermodynamics: Theory and ˇ Experiment. Dordrecht, The Netherlands: Springer; 2005.

[57] Sheehan DP, editor. Special Issue: The Second Law of Thermodynamics: Foundations and Status. Found. Phys. December 2007: Vol. 37, Issue 12.

[58] Sheehan DP, editor. Second Law of Thermodynamics: Status and Challenges, AIP Conference Proceedings Volume 1411. Melville, N. Y.: American Institute of Physics; 2011.

[59] Reference [4], Sect. 18.4 and relevant references cited therein, especially our Ref. [60] immediately following.

[60] Ramsey NF. Thermodynamics and Statistical Mechanics at Negative Absolute Temperatures. Phys. Rev. 1956: **103**; 20–28. DOI: 10.1103/PhysRev.103.20

[61] Reference [48], Sect. 11.5.

[62] Reference [49], Sect. 14.6.

[63] Atkins P. Four Laws that Drive the Universe. Oxford, U. K.: Oxford University Press; 2007, Chap. 5, especially p. 118.

[64] Reference [20], Sects. 2.2–2.3, especially Fig. 2.7 on p. 56.

[65] Williams DR. Neptune Fact Sheet [Internet]. 2015. Available from: nssdc.gsfc.nasa.gov/planetary/factsheet/neptunefact.html. [Accessed: 2015-12-05]

[66] NWS Windchill Chart [Internet]. 2001. Available from: www.nws.noaa.gov/om/winter/windchill.shtml [Accessed: 2015-12-05]

[67] See the Wikipedia article entitled "Wind chill." [Internet]. 2015. Available from: www.wikipedia.org [Accessed: 2015-12-05]

[68] Reference [2], Sects. 2.1–2.6 and 4.10–4.11.

[69] Reference [3], Sects. 2-1–2-6 and 4-8.

[70] Reference [4], Chaps. 1 and 2, and Sects. 9.1 and 11.1–11.6.

[71] Reference [2], Sect. 4.9.

[72] Reference [3], Sect. 4-8.

[73] Reference [4], Sects. 6.4–6.6.


[76] Beep NR. Meteorological Thermodynamics and Atmospheric Statics. In Berry FA, Bollay E, Beers NR, editors. Handbook of Meteorology. New York: McGraw-Hill; 1945, pp. 320–325 and 332–337, especially pp. 321–323 and 335–337.

[77] Faries VM. Applied Thermodynamics. Revised Edition. New York: Macmillan; 1949, Sect. 58.

[78] Reference [48], Sects. 4.3 and 7.5.

[79] Reference [49], Sects. 1.1–1.2, pp. 177–178, and Sect. 14.7.

[80] Campisi M, Kobe DH. Derivation of Boltzmann Principle. arXiv:0911.2070v1 [cond-mat.stat-mech].11 Nov 2009. 9 pages. [Internet]. 2009. Available from: xxx.lanl.gov/abs/0911.2070 or arxiv.org/abs/0911.2070 [Accessed: 2015-12-05]

[81] Campisi M, Kobe DH. Derivation of the Boltzmann principle. Am. J. Phys. 2010; **78**: 608–615. DOI: 10.1119/1.3298372

[82] Campisi M., Bagci GB. Tsallis ensemble as an exact orthode. Phys. Lett. A 2007; **362**: 11–15. DOI: 10.1016/j.physleta.2006.09.081

[83] Private communication, reviewer for the American Journal of Physics. 2010.

[84] Reference [4], Sect. 19.1.

[56] Cápek V, Sheehan DP. Challenges to the Second Law of Thermodynamics: Theory and ˇ

[57] Sheehan DP, editor. Special Issue: The Second Law of Thermodynamics: Foundations

[58] Sheehan DP, editor. Second Law of Thermodynamics: Status and Challenges, AIP Conference Proceedings Volume 1411. Melville, N. Y.: American Institute of Physics; 2011. [59] Reference [4], Sect. 18.4 and relevant references cited therein, especially our Ref. [60]

[60] Ramsey NF. Thermodynamics and Statistical Mechanics at Negative Absolute

[63] Atkins P. Four Laws that Drive the Universe. Oxford, U. K.: Oxford University Press;

[65] Williams DR. Neptune Fact Sheet [Internet]. 2015. Available from:

[66] NWS Windchill Chart [Internet]. 2001. Available from:

[67] See the Wikipedia article entitled "Wind chill." [Internet]. 2015. Available from:

[76] Beep NR. Meteorological Thermodynamics and Atmospheric Statics. In Berry FA, Bollay E, Beers NR, editors. Handbook of Meteorology. New York: McGraw-Hill; 1945, pp. 320–325

[77] Faries VM. Applied Thermodynamics. Revised Edition. New York: Macmillan; 1949,

nssdc.gsfc.nasa.gov/planetary/factsheet/neptunefact.html. [Accessed: 2015-12-05]

www.nws.noaa.gov/om/winter/windchill.shtml [Accessed: 2015-12-05]

Temperatures. Phys. Rev. 1956: **103**; 20–28. DOI: 10.1103/PhysRev.103.20

Experiment. Dordrecht, The Netherlands: Springer; 2005.

and Status. Found. Phys. December 2007: Vol. 37, Issue 12.

[64] Reference [20], Sects. 2.2–2.3, especially Fig. 2.7 on p. 56.

[70] Reference [4], Chaps. 1 and 2, and Sects. 9.1 and 11.1–11.6.

immediately following.

[61] Reference [48], Sect. 11.5. [62] Reference [49], Sect. 14.6.

298 Recent Advances in Thermo and Fluid Dynamics

2007, Chap. 5, especially p. 118.

www.wikipedia.org [Accessed: 2015-12-05] [68] Reference [2], Sects. 2.1–2.6 and 4.10–4.11.

[69] Reference [3], Sects. 2-1–2-6 and 4-8.

and 332–337, especially pp. 321–323 and 335–337.

[78] Reference [48], Sects. 4.3 and 7.5.

[71] Reference [2], Sect. 4.9. [72] Reference [3], Sect. 4-8.

Sect. 58.

[73] Reference [4], Sects. 6.4–6.6. [74] Reference [48], Sect. 4.2. [75] Reference [49], pp. 82–86.

[85] Reference [17], Sect. 14.6.

[86] Sheehan DP, Kriss VG: Energy Emission by Quantum Systems in an Expanding FRW Metric. arXiv:astro-ph/0411299v1. 11 Nov 2004. 12 pages. [Internet]. 2004. Available from: xxx.lanl.gov/abs/astro-ph/0411299 or arxiv.org/abs/astro-ph/0411299 [Accessed: 2015-12-05]

[87] Harrison ER. Mining Energy in an Expanding Universe. Astroph. J. 1995: **446**; 63–66. DOI: 10.1086/175767

[88] Reference [3], Chap. 21, especially Sects. 21-2–21-4.

[89] Vilenkin A. Many Worlds in One. New York: Hill and Wang; 2007, pp. 132–139 and 144–151. DOI: 10.1063/1.2743129

[90] Tegmark M. Our Mathematical Universe. New York: Vintage Books; 2015, pp. 138–145, 150, and 352–365.

[91] Carr B, editor. Universe or Multiverse? Cambridge, U. K.: Cambridge University Press; 2007. See especially Chaps. 1, 2, 3, 5, 22, 23, and 25.

[92] Rindler W. Relativity: Special, General, and Cosmological. 2nd ed. New York: Oxford University Press; 2006, Sect. 18.6 and references cited therein.

[93] Reference [89], pp. 11–12 and 183–186.

[94] Davies PCW. The Physics of Time Asymmetry. Berkeley: University of California Press; 1977, pp. 190–191 and 199–200.

[95] Tryon EP. Is the Universe a Vacuum Fluctuation? Nature 1973; **246**: 396–397. DOI: 10.1038/246396a0

[96] Vilenkin A. Many Worlds in One: The Search for Other Universes. New York: Hall and Wang; 2007. DOI: 10.1063/1.2743129, pp. 183–186.

[97] Filippenko A, Pasachoff JM. A Universe From Nothing. 2 pages. [Internet]. 2001. Available from: https:www.astrosociety.org/publications/a-universe-from-nothing [Accessed: 2015-12-05]

[98] Potter F, Jargodski C. Mad About Modern Physics. Hoboken, NJ; 2005, Question 224, "The Total Energy," on p. 115, and Answer 224, "The Total Energy," along with cited references, on pp. 275–276.

[99] Berman MS. On the Zero-Energy Universe. Int. J. Theor. Phys. 2009; **48**: 3278–3286. DOI: 10.1007/s10773-009-0125-8

[100] Berman MS, Trevisan LA. On Creation of Universe Out of Nothing. Int. J. Modern Phys. B. 2010; **19**: 1309–1313. DOI: 10.1142/S021827181007342

[101] Penrose R. The Road to Reality: A Complete Guide to the Laws of the Universe. New York: Alfred A. Knopf; 2005, Sects. 19.7–19.8 (especially Sect. 19.8 and most especially pp. 468–469). See also references cited therein, especially those cited in Note 19.17 on p. 470, and Notes for Sects. 19.7–19.8 on pp. 469–470.

[102] Vilenkin A. Many Worlds in One: The Search for Other Universes. New York: Hall and Wang; 2007. DOI: 10.1063/1.2743129, pp. 27–28, 37, and 170–171.

[103] Rindler W. Relativity: Special, General, and Cosmological. 2nd ed. New York: Oxford University Press; 2006, pp. 359, 368–369, 387–388, and 411.

[104] Bradley W. Carroll BW, Ostlie DA. An Introduction to Modern Astrophysics. 2nd ed. San Francisco: Pearson Addison Wesley; 2007, pp. 1163–1167.

[105] Reference [94], Sect. 7.2 and references cited therein.

[106] Reference [94], the last sentence on p. 187 and the reference cited therein.

[107] Davies PCW. Is the Universe Transparent or Opaque? J. Phys. A: Math. Gen. 1972; **5**: 1722–1737. DOI: 10.1088/0305-4470/5/12/012

[108] Baierlein R. Thermal Physics. Cambridge, U. K.: Cambridge University Press; 1999, Chap. 14, especially Sects.14.4–14.5, 14.7, 14.9, and relevant references cited under "Further Reading" on pp. 352–353.

[109] Berry RS, Rice SA, Ross J. Physical Chemistry. 2nd ed. New York: Oxford University Press; 2000, Chap. 18, especially Sect. 18.2.

[110] Wheeler JC. Nonequivalence of the Nernst-Simon and unattainability statements of the third law of thermodynamics. Phys. Rev. A 1991; **43**: 5289–5295. DOI: 10.1103/PhysRevA.43.5289

[111] Wheeler JC. Addendum to "Nonequivalence of the Nernst-Simon and untenability statements of the third law of thermodynamics." Phys. Rev. A 1992; **45**: 2637–2640. DOI: 10.1103/PhysRevA.45.2637

[112] Landsberg PT. A comment on Nernst's theorem. J. Phys. A. Math. Gen. 1989; **22**:139–141. DOI: 10.1088/0305-4470/22/1/021

[113] Landsberg PT. Answer to Question #34. What is the third law of thermodynamics trying to tell us? Am. J. Phys. 1997; **65**: 269–270. DOI: 10.1119/1.18483

[114] Garrod C. Statistical Mechanics and Thermodynamics. New York; Oxford University Press; 1995, Sect. 5.16 (especially the discussion concerning Axiom 5), Sect. 5.18, Problem 5.19 on p. 137, and, as auxiliary material, Appendix A.10 and Exercise 5.11 on pp. 441–443.

[115] Reference [108], Sect.14.4.

[98] Potter F, Jargodski C. Mad About Modern Physics. Hoboken, NJ; 2005, Question 224, "The Total Energy," on p. 115, and Answer 224, "The Total Energy," along with cited

[99] Berman MS. On the Zero-Energy Universe. Int. J. Theor. Phys. 2009; **48**: 3278–3286. DOI:

[100] Berman MS, Trevisan LA. On Creation of Universe Out of Nothing. Int. J. Modern

[101] Penrose R. The Road to Reality: A Complete Guide to the Laws of the Universe. New York: Alfred A. Knopf; 2005, Sects. 19.7–19.8 (especially Sect. 19.8 and most especially pp. 468–469). See also references cited therein, especially those cited in Note 19.17 on p. 470,

[102] Vilenkin A. Many Worlds in One: The Search for Other Universes. New York: Hall and

[103] Rindler W. Relativity: Special, General, and Cosmological. 2nd ed. New York: Oxford

[104] Bradley W. Carroll BW, Ostlie DA. An Introduction to Modern Astrophysics. 2nd ed.

[107] Davies PCW. Is the Universe Transparent or Opaque? J. Phys. A: Math. Gen. 1972; **5**:

[108] Baierlein R. Thermal Physics. Cambridge, U. K.: Cambridge University Press; 1999, Chap. 14, especially Sects.14.4–14.5, 14.7, 14.9, and relevant references cited under "Further

[109] Berry RS, Rice SA, Ross J. Physical Chemistry. 2nd ed. New York: Oxford University

[110] Wheeler JC. Nonequivalence of the Nernst-Simon and unattainability statements of the third law of thermodynamics. Phys. Rev. A 1991; **43**: 5289–5295. DOI:

[111] Wheeler JC. Addendum to "Nonequivalence of the Nernst-Simon and untenability statements of the third law of thermodynamics." Phys. Rev. A 1992; **45**: 2637–2640. DOI:

[112] Landsberg PT. A comment on Nernst's theorem. J. Phys. A. Math. Gen. 1989;

[113] Landsberg PT. Answer to Question #34. What is the third law of thermodynamics

[114] Garrod C. Statistical Mechanics and Thermodynamics. New York; Oxford University Press; 1995, Sect. 5.16 (especially the discussion concerning Axiom 5), Sect. 5.18, Problem 5.19 on p. 137, and, as auxiliary material, Appendix A.10 and Exercise 5.11 on pp. 441–443.

trying to tell us? Am. J. Phys. 1997; **65**: 269–270. DOI: 10.1119/1.18483

[106] Reference [94], the last sentence on p. 187 and the reference cited therein.

Phys. B. 2010; **19**: 1309–1313. DOI: 10.1142/S021827181007342

Wang; 2007. DOI: 10.1063/1.2743129, pp. 27–28, 37, and 170–171.

University Press; 2006, pp. 359, 368–369, 387–388, and 411.

San Francisco: Pearson Addison Wesley; 2007, pp. 1163–1167. [105] Reference [94], Sect. 7.2 and references cited therein.

and Notes for Sects. 19.7–19.8 on pp. 469–470.

1722–1737. DOI: 10.1088/0305-4470/5/12/012

Press; 2000, Chap. 18, especially Sect. 18.2.

**22**:139–141. DOI: 10.1088/0305-4470/22/1/021

Reading" on pp. 352–353.

10.1103/PhysRevA.43.5289

10.1103/PhysRevA.45.2637

references, on pp. 275–276.

300 Recent Advances in Thermo and Fluid Dynamics

10.1007/s10773-009-0125-8

[116] Reference [108], Sects. 6.5, 9.1, 9.4, and 14.4.

[117] Zemansky MW, Dittman RH. Heat and Thermodynamics. 6th ed. New York: McGraw-Hill; 1981, Sect. 19-6 and Problems for Chap. 19 on pp. 520–521, especially Problems 19-2–19-4.
