1.1 Irreversibility and arrow of time

Irreversibility belongs to the unsolved fundamental problems in recent physics. Nonequilibrium processes are omnipresent in our daily experience. However, a fundamental, microscopic description of such processes is missing yet.

Our microscopic description of physical phenomena is expressed by equations of motion, well known in mechanics, electrodynamics, quantum mechanics, and field theory. We model a physical system, we determine the degrees of freedom and the forces, and we introduce a Lagrangian. The equations of motion are differential equations. If we know the initial state, the future of the system can be predicted solving the equations of motion. Anything is determined. The equations of motion are invariant with respect to time reversion. The time evolution is reversible. No arrow of time is selected out, nothing happens what is not prescribed by the initial state.

This picture was created by celestial dynamics. It is very successful, very presumptuous, and many processes are described with high precision. However, it is in contradiction to daily experience. We know birth and death, decay, destruction, and many other phenomena that are irreversible, selecting out the arrow of time.

A qualitative new discipline in physics is thermodynamics. It considers not a model but any real system. The laws of thermodynamics define new quantities, the state variables. The second law determines the entropy S as state variable (and the temperature T) via

$$d\mathbf{S} = \frac{1}{T} \delta \mathbf{Q}\_{\text{reversible}} \tag{1}$$

in thermal equilibrium. This problem was solved with the Langevin equation: instead of the trajectory vð Þt as solution of a differential equation, the stochastic

The random acceleration Rð Þt (or the stochastic force mRð Þt ) is a stochastic process, which is characterized by special properties. For instance, white noise is a Gaussian process that is characterized by the mean value h i Rð Þt ¼ 0 and the auto-

D is the diffusion coefficient. An interesting result is the Einstein relation

which relates the friction coefficient γ (dissipation) to the fluctuations φ in the system (stochastic forces), characterized by the parameter D; see [5] for more

Within statistical mechanics, the thermodynamic state of an ensemble of manyparticle systems at time t is described by the statistical operator ρð Þt . We assume that the time evolution of the quantum state of the system is given by the Hamiltonian H<sup>t</sup> which may contain time-dependent external fields. The von Neumann

The von Neumann equation describes reversible dynamics. The equation of motion is based on the Schrödinger equation. Time inversion and conjugate complex means that the first term on the left-hand side as well as the second one change the sign, since i ! �i and both the Hamiltonian and the statistical operator are Hermitean. However, the von Neumann equation is not sufficient to determine ρð Þt because it is a first-order differential equation, and an initial value ρð Þ t<sup>0</sup> at time t<sup>0</sup> is necessary to specify a solution. This problem emerges clearly in equilibrium.

By definition, in thermodynamic equilibrium, the thermodynamic state of the system is not changing with time. Both, H<sup>t</sup> and <sup>ρ</sup>ð Þ<sup>t</sup> , are not depending on <sup>t</sup> so that

The solution of the von Neumann equation in thermodynamic equilibrium

<sup>ℏ</sup> <sup>H</sup>; <sup>ρ</sup>eq h i <sup>¼</sup> <sup>0</sup>: The time-independent statistical operator <sup>ρ</sup>eq

∂ ∂t <sup>H</sup><sup>t</sup> ½ �¼ ; <sup>ρ</sup>ð Þ<sup>t</sup> <sup>0</sup>: (8)

ρeqðÞ¼ t 0: (9)

D <sup>γ</sup> <sup>¼</sup> <sup>k</sup>B<sup>T</sup> m

equation follows as equation of motion for the statistical operator,

∂ ∂t ρðÞþt i ℏ

1.4 Thermodynamic equilibrium and entropy

becomes trivial, <sup>i</sup>

5

dt <sup>V</sup>ðÞ¼� <sup>t</sup> <sup>γ</sup> <sup>V</sup>ðÞ�<sup>t</sup> <sup>v</sup>rel <sup>½</sup> ð Þ<sup>t</sup> � þ <sup>R</sup>ð Þ<sup>t</sup> : (5)

� � <sup>¼</sup> <sup>φ</sup>ijð Þ¼ <sup>t</sup><sup>2</sup> � <sup>t</sup><sup>1</sup> <sup>2</sup>Dδijδð Þ <sup>t</sup><sup>2</sup> � <sup>t</sup><sup>1</sup> : (6)

(7)

process Vð Þt is considered. It obeys the stochastic differential equation

d

Rið Þ t<sup>1</sup> Rjð Þ t<sup>2</sup>

(fluctuation-dissipation theorem, FDT)

1.3 Von Neumann equation

correlation function

Nonequilibrium Statistical Operator

DOI: http://dx.doi.org/10.5772/intechopen.84707

details.

where δQ is the heat imposed to the system within a reversible process, together with the third law which fixes the value S Tð Þ¼ ¼ 0 0 independent on other state variables. For irreversible processes holds

$$
\frac{dS}{dt} > \frac{\delta Q}{T}.\tag{2}
$$

In particular, for isolated system, δQ ¼ 0, irreversible processes are possible so that dS=dt>0. Typical examples are friction that transforms mechanical energy to thermal energy, temperature equilibration without production of mechanical work, diffusion processes to balance concentration gradients. An arrow of time is selected out, time reversion describes a phenomenon which is not possible. How can irreversible evolution in time be obtained from the fundamental microscopic equations of motion which are reversible in time?

For equilibrium thermodynamics, a microscopic approach is given by statistical physics. Additional concepts are introduced such as probability and distribution function, ensembles in thermodynamic equilibrium, and information theory. New thermodynamic quantities are introduced, basically the entropy, which have no direct relation to mechanical quantities describing the equation of motion. However, nonequilibrium processes are described in a phenomenological way, and no fundamental solution of the problem of irreversibility is found until now. A substantial step to develop a theory of irreversible evolution is the Zubarev method of the nonequilibrium statistical operator (NSO) [1–6] to be described in the following section. It is a consistent theory to describe different nonequilibrium processes what is indispensable for a basic approach.

### 1.2 Langevin equation

To give an example for a microscopic approach to a nonequilibrium process, let us consider the Brownian motion. A particle suspended in a liquid, moving with velocity <sup>v</sup>medium, experiences a friction force <sup>F</sup>fricð Þ<sup>t</sup> ,

$$\frac{d}{dt}\mathbf{v}(t) = \frac{1}{m}\mathbf{F}^{\text{fric}}(t) = -\gamma[\mathbf{v}(t) - \mathbf{v}\_{\text{medium}}],\tag{3}$$

with the coefficient of friction γ. The solution

$$\mathbf{v}(t) = \mathbf{v}(t\_0)e^{-\gamma(t-t\_0)} + \mathbf{v}\_{\text{medium}} \left[\mathbf{1} - e^{-\gamma(t-t\_0)}\right] \tag{4}$$

describes the relaxation from the initial state vð Þ t<sup>0</sup> at t<sup>0</sup> to the final state vmedium for t � t<sup>0</sup> ! ∞. Independent of the initial state, the particle rests in equilibrium with the medium. In the general case not considered here, an external force can be added.

As it is well known, this simple relaxation behavior cannot be correct because it does not describe the Brownian motion, showing the existence of fluctuations also

A qualitative new discipline in physics is thermodynamics. It considers not a model but any real system. The laws of thermodynamics define new quantities, the state variables. The second law determines the entropy S as state variable (and the

where δQ is the heat imposed to the system within a reversible process, together with the third law which fixes the value S Tð Þ¼ ¼ 0 0 independent on other state

In particular, for isolated system, δQ ¼ 0, irreversible processes are possible so that dS=dt>0. Typical examples are friction that transforms mechanical energy to thermal energy, temperature equilibration without production of mechanical work, diffusion processes to balance concentration gradients. An arrow of time is selected out, time reversion describes a phenomenon which is not possible. How can irreversible evolution in time be obtained from the fundamental microscopic equations

For equilibrium thermodynamics, a microscopic approach is given by statistical physics. Additional concepts are introduced such as probability and distribution function, ensembles in thermodynamic equilibrium, and information theory. New thermodynamic quantities are introduced, basically the entropy, which have no direct relation to mechanical quantities describing the equation of motion. However, nonequilibrium processes are described in a phenomenological way, and no fundamental solution of the problem of irreversibility is found until now. A substantial step to develop a theory of irreversible evolution is the Zubarev method of the nonequilibrium statistical operator (NSO) [1–6] to be described in the following section. It is a consistent theory to describe different nonequilibrium processes what

To give an example for a microscopic approach to a nonequilibrium process, let us consider the Brownian motion. A particle suspended in a liquid, moving with

�γð Þ <sup>t</sup>�t<sup>0</sup> <sup>þ</sup> <sup>v</sup>medium <sup>1</sup> � <sup>e</sup>

describes the relaxation from the initial state vð Þ t<sup>0</sup> at t<sup>0</sup> to the final state vmedium for t � t<sup>0</sup> ! ∞. Independent of the initial state, the particle rests in equilibrium with the medium. In the general case not considered here, an external force can be

As it is well known, this simple relaxation behavior cannot be correct because it does not describe the Brownian motion, showing the existence of fluctuations also

<sup>F</sup>fricðÞ¼� <sup>t</sup> <sup>γ</sup>½ � <sup>v</sup>ðÞ�<sup>t</sup> <sup>v</sup>medium , (3)

�γð Þ t�t<sup>0</sup> h i

(4)

<sup>T</sup> <sup>δ</sup>Qreversible (1)

<sup>T</sup> : (2)

dS <sup>¼</sup> <sup>1</sup>

dS dt <sup>&</sup>gt; δQ

temperature T) via

Non-Equilibrium Particle Dynamics

variables. For irreversible processes holds

of motion which are reversible in time?

is indispensable for a basic approach.

velocity <sup>v</sup>medium, experiences a friction force <sup>F</sup>fricð Þ<sup>t</sup> ,

with the coefficient of friction γ. The solution

vðÞ¼ t vð Þ t<sup>0</sup> e

1 m

d dt <sup>v</sup>ðÞ¼ <sup>t</sup>

1.2 Langevin equation

added.

4

in thermal equilibrium. This problem was solved with the Langevin equation: instead of the trajectory vð Þt as solution of a differential equation, the stochastic process Vð Þt is considered. It obeys the stochastic differential equation

$$\frac{d}{dt}\mathbf{V}(t) = -\gamma[\mathbf{V}(t) - \mathbf{v}\_{\rm rel}(t)] + \mathbf{R}(t). \tag{5}$$

The random acceleration Rð Þt (or the stochastic force mRð Þt ) is a stochastic process, which is characterized by special properties. For instance, white noise is a Gaussian process that is characterized by the mean value h i Rð Þt ¼ 0 and the autocorrelation function

$$
\left< R\_i(t\_1) R\_j(t\_2) \right> = \rho\_{ij}(t\_2 - t\_1) = \mathcal{D} \delta\_{ij} \delta(t\_2 - t\_1). \tag{6}
$$

D is the diffusion coefficient. An interesting result is the Einstein relation (fluctuation-dissipation theorem, FDT)

$$\frac{D}{\gamma} = \frac{k\_B T}{m} \tag{7}$$

which relates the friction coefficient γ (dissipation) to the fluctuations φ in the system (stochastic forces), characterized by the parameter D; see [5] for more details.

### 1.3 Von Neumann equation

Within statistical mechanics, the thermodynamic state of an ensemble of manyparticle systems at time t is described by the statistical operator ρð Þt . We assume that the time evolution of the quantum state of the system is given by the Hamiltonian H<sup>t</sup> which may contain time-dependent external fields. The von Neumann equation follows as equation of motion for the statistical operator,

$$\frac{\partial}{\partial t}\rho(t) + \frac{\mathbf{i}}{\hbar}[\mathbf{H}^t, \rho(t)] = \mathbf{0}.\tag{8}$$

The von Neumann equation describes reversible dynamics. The equation of motion is based on the Schrödinger equation. Time inversion and conjugate complex means that the first term on the left-hand side as well as the second one change the sign, since i ! �i and both the Hamiltonian and the statistical operator are Hermitean. However, the von Neumann equation is not sufficient to determine ρð Þt because it is a first-order differential equation, and an initial value ρð Þ t<sup>0</sup> at time t<sup>0</sup> is necessary to specify a solution. This problem emerges clearly in equilibrium.

### 1.4 Thermodynamic equilibrium and entropy

By definition, in thermodynamic equilibrium, the thermodynamic state of the system is not changing with time. Both, H<sup>t</sup> and <sup>ρ</sup>ð Þ<sup>t</sup> , are not depending on <sup>t</sup> so that

$$\frac{\partial}{\partial t} \rho\_{\text{eq}}(t) = \mathbf{0}.\tag{9}$$

The solution of the von Neumann equation in thermodynamic equilibrium becomes trivial, <sup>i</sup> <sup>ℏ</sup> <sup>H</sup>; <sup>ρ</sup>eq h i <sup>¼</sup> <sup>0</sup>: The time-independent statistical operator <sup>ρ</sup>eq

commutes with the Hamiltonian. We conclude that ρeq depends only on constants of motion C<sup>n</sup> that commute with H. But, the von Neumann equation is not sufficient to determine how ρeq depends on constants of motion Cn. We need a new additional principle, not included in Hamiltonian dynamics.

Equilibrium statistical mechanics is based on the following principle to determine the statistical operator ρeq: consider the functional (information entropy)

$$\mathcal{S}\_{\text{inf}}[\rho] = -\text{Tr}\{\rho \ln \rho\} \tag{10}$$

The entropy Seqð Þ Ω; β; μ follows from Eq. (12). The dependence of extensive quantities on the volume Ω is trivial for homogeneous systems. After a thermodynamic potential is calculated, all thermodynamic variables are derived in a consistent manner. The method to construct statistical ensembles from the maximum of entropy at given conditions, which take into account the different contacts with the surrounding bath, is well accepted in equilibrium statistical mechanics and is applied successfully to different phenomena, including phase transitions.

Can we extend the definition of equilibrium entropy (12) also for ρð Þt that describes the evolution in nonequilibrium? Time evolution is given by an unitary transformation that leaves the trace invariant. Thus, the expression Trf g ρð Þt ln ρð Þt

The entropy for a system in nonequilibrium, however, may increase with time, according to the second law of thermodynamics. The equations of motion, including the Schrödinger equation and the Liouville-von Neumann equation, describe reversible motion and are not appropriate to describe irreversible processes. Therefore, the entropy concept (12) elaborated in equilibrium statistical physics together with the Liouville-von Neumann equation cannot be used as fundamental approach

½Trf g ρð Þt ln ρð Þt � ¼ 0: (15)

is constant for a solution ρð Þt of the von Neumann equation

to nonequilibrium statistical physics.

Nonequilibrium Statistical Operator

DOI: http://dx.doi.org/10.5772/intechopen.84707

discussed controversially.

be given here, see [1–5].

7

2.1 Construction of the Zubarev NSO

d dt

2. The method of nonequilibrium statistical operator (NSO)

After the laws of thermodynamics have been formulated in the nineteenth century, in particular, the definition of entropy for systems in thermodynamic equilibrium and the increase of intrinsic entropy in nonequilibrium processes, a microscopic approach to nonequilibrium evolution was first given by Ludwig Boltzmann who formulated the kinetic theory of gases [7] using the famous Stoßzahlansatz. The question how irreversible evolution in time can be obtained from reversible microscopic equations has been arisen immediately and was

The rigorous derivation of the kinetic equations from a microscopic description of a system was given only a long time afterward by Bogoliubov [8] introducing a

In the first step, we interrogate the concept of thermodynamic equilibrium. This is an idealization, because slow processes are always possible. As example, we may take the nuclear decay of long-living isotopes, hindered chemical reactions, or the

new additional theorem, the principle of weakening of initial correlation.

A generalization of this principle has been given by Zubarev [1, 2], who invented the method of the nonequilibrium statistical operator (NSO). This approach has been applied to various problems in nonequilibrium statistical physics, see [3, 4] and may be considered as a unified, fundamental approach to nonequilibrium systems which includes different theories such as quantum master equations, kinetic theory, and linear response theory to be outlined below. An exhaustive review of the Zubarev NSO method and its manifold applications cannot

for arbitrary ρ that is consistent with the given conditions Trf g¼ ρ 1 (normalization) and

$$\operatorname{Tr}\{\rho \mathbf{C}\_{\mathfrak{n}}\} = \langle \mathbf{C}\_{\mathfrak{n}} \rangle \tag{11}$$

(self-consistency conditions). Respecting these conditions, we vary ρ and determine the maximum of the information entropy for the optimal distribution ρeq so that δSinf½ ρeq� ¼ 0. As it is well-known, the method of Lagrange multipliers can be used to account for the self-consistency conditions (11). The corresponding maximum value for Sinf½ � ρ

$$\mathcal{S}\_{\rm eq} \left[ \rho\_{\rm eq} \right] = -k\_{\rm B} \text{Tr} \left\{ \rho\_{\rm eq} \ln \rho\_{\rm eq} \right\} \tag{12}$$

is the equilibrium entropy of the system at given constraints Ch i<sup>n</sup> and k<sup>B</sup> is the Boltzmann constant. The solution of this variational principle leads to the Gibbs ensembles for thermodynamic equilibrium, see also Section 4.

As an example, we consider an open system which is in thermal contact and particle exchange with reservoirs. The corresponding equilibrium statistical operator has to obey the given constraints: normalization Trf gρ ¼ 1, thermal contact with the bath so that Trf g ρH ¼ U (internal energy), particle exchange with a reservoir so that for the particle number operator N<sup>c</sup> of species c, the average is given by Trf g ρN<sup>c</sup> ¼ ncΩ, where Ω denotes the volume of the system (we do not use V to avoid confusion with the potential), and nc is the particle density of species c. Looking for the maximum of the information entropy functional with these constraints, one obtains the grand canonical distribution

$$\rho\_{\rm eq} = \frac{e^{-\beta\left(\mathcal{H} - \sum\_{\varepsilon} \mu\_{\varepsilon} \mathcal{N}\_{\varepsilon}\right)}}{\mathrm{Tr}\,\mathbf{e}^{-\beta\left(\mathcal{H} - \sum\_{\varepsilon} \mu\_{\varepsilon} \mathcal{N}\_{\varepsilon}\right)}}\,. \tag{13}$$

The normalization is explicitly accounted for by the denominator (partition function). The second condition means that the energy of a system in heat contact with a thermostat fluctuates around an averaged value Hh i¼ U ¼ uΩ with the given density of internal energy u. This condition is taken into account by the Lagrange multiplier β that must be related to the temperature, a more detailed discussion leads to β ¼ 1=ð Þ kBT . Similarly, the contact with the particle reservoir fixes the particle density nc, introduced by the Lagrange multiplier μc, which has the meaning of the chemical potential of species c.

Within the variational approach, the Lagrange parameters β, μ<sup>c</sup> have to be eliminated. This leads to the equations of state (h i … eq ¼ Trfρeq…g) which relate, e.g., the chemical potentials μ<sup>c</sup> to the particle densities nc,

$$
\langle \mathsf{H} \rangle\_{\mathsf{eq}} = U(\mathfrak{Q}, \beta, \mu\_c), \qquad \langle \mathsf{N}\_c \rangle\_{\mathsf{eq}} = \mathfrak{Q} \mathfrak{n}\_c(T, \mu\_c). \tag{14}
$$
