**5. The temperature of radiation** [Ragazas 2010g]

*E E*

Consider the energy *E t*( ) at a fixed point at time *t* . We define the *temperature of radiation* to

be given by <sup>1</sup> T T where is a scalar constant. Though in defining *temperature*  this way the *accumulation of energy* can be any value, when considering a *temperature scale* is fixed and used as a *standard for measurement*. To distinguish *temperature* and *temperature scale* we will use T and T respectively. We assume that *temperature* is characterized by the following property:

*Characterization of temperature: For a fixed , the temperature is inversely proportional to the duration of time for an accumulation of energy to occur.* 

number of microstates of the system at time t, then *Et A t* () () , for some constant *A* . Thus,

*av <sup>E</sup> <sup>k</sup> E* =

0

*<sup>E</sup> <sup>E</sup> e* 

> .

is a scalar constant. Though in defining *temperature* 

*, the temperature is inversely proportional to the* 

can be any value, when considering a *temperature scale*

respectively. We assume that *temperature* is characterized by the

. If ( )*t* represents the

<sup>0</sup>

0

ln *<sup>E</sup> <sup>k</sup> E* .

and so

0

ln

in (3) above.

*E E E E* 

*S k* ln

*av*

is the *rate of evolution* of the system and

ln *<sup>E</sup> <sup>k</sup> E* 

*T* <sup>=</sup>

we can rewrite this equivalently as

our explanation of the double-slit experiment. [Ragazas 2010j]

**5. The temperature of radiation** [Ragazas 2010g]

 

. From this we have, *Planck's Law,* <sup>0</sup> <sup>1</sup> *E Eav*

where

 *The Fundamental Thermodynamic Relation:* It is a well known fact that the internal energy U, entropy S , temperature T, pressure P and volume V of a system are related by the equation *dU TdS PdV* . By using increments rather than differentials, and using the fact that work performed by the system is given by *W PdV* this can be re-written as

. All the terms in this equation are various entropy quantities. The

fundamental thermodynamic relation can be interpreted thus as saying, *"the total change of entropy of a system equals the sum of the change in the internal (unmanifested) plus the change in the external (manifested) entropy of the system"*. Considering the *entropy-time relationship* above, this can be rephrased more intuitively as saying *"the total lapsed time for a physical process equals the time for the 'accumulation of energy' plus the time for the 'manifestation of energy' for the process"*. This relationship along with *The Second Law of Thermodynamics* establish a *duration of time* over which there is *accumulation of energy before manifestation of energy* – one of our main results in this Chapter and a premise to

Consider the energy *E t*( ) at a fixed point at time *t* . We define the *temperature of radiation* to

is fixed and used as a *standard for measurement*. To distinguish *temperature* and *temperature* 

*to occur.* 

and *E*

*Conversely*, starting with *Boltzmann's Entropy Equation*,

*t* is the *time duration* of evolution, since *Eav <sup>t</sup>*

thermodynamic entropy we get *<sup>E</sup> <sup>S</sup>*

0 0

*E E*

*Entropy-Time Relationship: Skt*

Since *<sup>E</sup> <sup>S</sup> T*

*U W <sup>S</sup> T T*

be given by <sup>1</sup>

*scale* we will use T and T

following property:

this way the *accumulation of energy*

 T T where

*Characterization of temperature: For a fixed* 

*duration of time for an accumulation of energy* 

*e*

<sup>1</sup> *E Eav E E*

we get *Boltzmann's Entropy Equation*, *S k* ln .

Thus if *temperature* is twice as high, the accumulation of energy will be twice as fast, and visa-versa. This *characterization of temperature* agrees well with our physical sense of temperature. It is also in agreement with *temperature* as being the average kinetic energy of the motion of molecules.

For fixed , we can define <sup>1</sup> <sup>T</sup> , which will be unique up to an arbitrary scalar

constant . *Conversely*, for a given <sup>T</sup> as characterized above, we will have <sup>1</sup> T , where

 is a proportionality constant. By setting we get 1 T T . We have the following *temperature-eta* correspondence:

*Temperature-eta Correspondence: Given , we have* <sup>1</sup> <sup>T</sup> *, where is some arbitrary scalar* 

*constant. Conversely, given* <sup>T</sup> *we have* <sup>1</sup> T T *, for some fixed and arbitrary scalar constant . Any temperature scale. therefore, will have some fixed and arbitrary scalar constant associated with it.* 
