2.1.1 The relevant statistical operator

A solution of the problem to combine equilibrium thermodynamics and nonequilibrium processes was proposed by Zubarev [1, 2]. To characterize the nonequilibrium state of a system, we introduce the set of relevant observables Bf g<sup>n</sup> extending the set of conserved quantities Cf g<sup>n</sup> . At time <sup>t</sup>, the observed values Bh i<sup>n</sup> <sup>t</sup> have to be reproduced by the statistical operator ρð Þt , i.e.,

$$\operatorname{Tr}\{\rho(t)\mathbf{B}\_{\pi}\} = \langle \mathbf{B}\_{\pi} \rangle^{t}. \tag{16}$$

entropy that is based on the single-particle distribution function, but does not take

The solution of the problem how to find the missing signatures of ρð Þt not already described by ρrelð Þt was found by Zubarev [1, 2] generalizing the Bogoliubov principle of weakening of initial correlations [8]. He proposed to use the relevant statistical operator ρrelð Þ t<sup>0</sup> at some initial time t<sup>0</sup> as initial condition to construct

<sup>ρ</sup>t<sup>0</sup> ðÞ¼ <sup>t</sup> <sup>U</sup>ð Þ <sup>t</sup>; <sup>t</sup><sup>0</sup> <sup>ρ</sup>relð Þ <sup>t</sup><sup>0</sup> <sup>U</sup>†

iℏ ∂ ∂t

tions (18) valid at t<sup>0</sup> are not automatically valid also at t.

ρϵðÞ¼ t ϵ

ðt

�∞

The unitary time evolution operator Uð Þ t; t<sup>0</sup> is the solution of the differential

<sup>U</sup>ð Þ¼ <sup>t</sup>; <sup>t</sup><sup>0</sup> <sup>H</sup><sup>t</sup>

<sup>U</sup>ð Þ¼ <sup>t</sup>; <sup>t</sup><sup>0</sup> <sup>e</sup>�<sup>i</sup>

with the initial condition Uð Þ¼ t0; t<sup>0</sup> 1. This unitary operator is known from the solution of the Schrödinger equation. If the Hamiltonian is not time dependent, we

If the Hamiltonian is time dependent, the solution is given by a time-ordered

To get rid of these incorrect initial correlations, according to the Bogoliubov principle of weakening of initial correlations, one can consider the limit t<sup>0</sup> ! �∞. According to Zubarev, it is more efficient to average over the initial time so that no special time instant t<sup>0</sup> is singled out. This is of importance, for instance, if there are long-living oscillations determined by the initial state. According to Abel's theorem, see [1–4], the limit t<sup>0</sup> ! �∞ can be replaced by the limit ϵ ! þ0 in the expression

<sup>e</sup><sup>ϵ</sup>ð Þ <sup>t</sup>1�<sup>t</sup> <sup>U</sup>ð Þ <sup>t</sup>; <sup>t</sup><sup>1</sup> <sup>ρ</sup>relð Þ <sup>t</sup><sup>1</sup> <sup>U</sup>†

This averaging over different initial time instants means a mixing of phases so that long-living oscillations are damped out. Finally, we obtain the nonequilibrium

This way, ρrelð Þ t<sup>1</sup> for all times �∞ < t<sup>1</sup> < t serves as initial condition to solve the

weakening of initial correlations. The missing correlations are formed dynamically

<sup>ρ</sup>NSOðÞ¼ <sup>t</sup> lim<sup>ϵ</sup>!<sup>0</sup>

Liouville-von Neumann equation, according to the Bogoliubov principle of

during the time evolution of the system. The more information about the

Now, it is easily shown that ρ<sup>t</sup><sup>0</sup> ð Þt is a solution of the von Neumann equation. All missing correlations not contained in ρrelð Þ t<sup>0</sup> are formed dynamically during the time evolution of the system. However, incorrect initial correlations contained in ρrelð Þ t<sup>0</sup> may survive for a finite time interval t � t0, and the self-consistency condi-

ð Þ t; t<sup>0</sup> : (22)

Uð Þ t; t<sup>0</sup> , (23)

<sup>ℏ</sup>Hð Þ <sup>t</sup>�t<sup>0</sup> : (24)

ð Þ t; t<sup>1</sup> dt1: (25)

ρϵð Þt : (26)

higher order correlation functions into account.

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

Nonequilibrium Statistical Operator

ρð Þt ,

equation

have

exponent.

statistical operator as

9

2.2 The Zubarev solution of the initial value problem

However, these conditions are not sufficient to fix ρð Þt , and we need an additional principle to find the correct one in between many possible distributions which all fulfill the conditions (16). We ask for the most probable distribution at time t, where the information entropy has a maximum value (see Section 4)

$$\delta[\mathrm{Tr}\{\rho\_{\mathrm{rel}}(t)\mathrm{ln}\,\rho\_{\mathrm{rel}}(t)\}]=\mathbf{0}\tag{17}$$

with the self-consistency conditions

$$\operatorname{Tr}\{\rho\_{\text{rel}}(t)\mathbf{B}\_{\text{n}}\} = \langle \mathbf{B}\_{\text{n}}\rangle^{t} \tag{18}$$

and Tr ρrel f g ð Þt ¼ 1. Once more, we use Lagrange multipliers λnð Þt to account for the self-consistency conditions (18). Since the averages are, in general, time dependent, the corresponding Lagrange multipliers are now time-dependent functions as well. We find the generalized Gibbs distribution

$$\rho\_{\rm rel}(t) = \mathbf{e}^{-\Phi(t) - \sum\_{n} \lambda\_{a}(t)\mathbf{B}\_{n}}, \qquad \Phi(t) = \ln \operatorname{Tr} \left\{ \mathbf{e}^{-\sum\_{n} \lambda\_{a}(t)\mathbf{B}\_{n}} \right\}, \tag{19}$$

where the Lagrange multipliers λnð Þt (thermodynamic parameters) are determined by the self-consistency conditions (18). Φð Þt is the Massieux-Planck function, needed for normalization purposes and playing the role of a thermodynamic potential. Generalizing the equilibrium case, Eq. (12), we can consider the relevant entropy in nonequilibrium

$$\mathcal{S}\_{\rm rel}(t) = -k\_{\rm B} \operatorname{Tr} \left\{ \rho\_{\rm rel}(t) \ln \rho\_{\rm rel}(t) \right\}. \tag{20}$$

Relations similar to the relations known from equilibrium thermodynamics can be derived. In particular, the production of entropy results as

$$\frac{\partial \mathbf{S}\_{\rm rel}(t)}{\partial t} = \sum\_{n} \lambda\_{n}(t) \left< \dot{\mathbf{B}}\_{n} \right>^{\rm f} \tag{21}$$

as known from the thermodynamics of irreversible processes. In contrast to Eq. (15), this expression can have a positive value so that Srelð Þt can increase with time.

The relevant statistical operator ρrelð Þt is not the wanted nonequilibrium statistical operator ρð Þt because it does not obey the Liouville-von Neumann equation. Also, Srelð Þt is not the thermodynamic entropy because it is based on the arbitrary choice of the set Bf g<sup>n</sup> of relevant observables, and not all possible variables are correctly reproduced. As example, we consider below the famous Boltzmann

long-time relaxation of glasses. Concepts introduced for equilibrium have to be generalized to nonequilibrium. An example is the thermodynamics of irreversible

A solution of the problem to combine equilibrium thermodynamics and nonequilibrium processes was proposed by Zubarev [1, 2]. To characterize the nonequilibrium state of a system, we introduce the set of relevant observables Bf g<sup>n</sup> extending the set of conserved quantities Cf g<sup>n</sup> . At time <sup>t</sup>, the observed values Bh i<sup>n</sup> <sup>t</sup>

Trf g <sup>ρ</sup>ð Þ<sup>t</sup> <sup>B</sup><sup>n</sup> <sup>¼</sup> h i <sup>B</sup><sup>n</sup> <sup>t</sup>

However, these conditions are not sufficient to fix ρð Þt , and we need an additional principle to find the correct one in between many possible distributions which all fulfill the conditions (16). We ask for the most probable distribution at time t, where the information entropy has a maximum value (see Section 4)

and Tr ρrel f g ð Þt ¼ 1. Once more, we use Lagrange multipliers λnð Þt to account for the self-consistency conditions (18). Since the averages are, in general, time dependent, the corresponding Lagrange multipliers are now time-dependent functions as

where the Lagrange multipliers λnð Þt (thermodynamic parameters) are determined by the self-consistency conditions (18). Φð Þt is the Massieux-Planck function, needed for normalization purposes and playing the role of a thermodynamic potential. Generalizing the equilibrium case, Eq. (12), we can consider the relevant

Relations similar to the relations known from equilibrium thermodynamics can

as known from the thermodynamics of irreversible processes. In contrast to Eq. (15), this expression can have a positive value so that Srelð Þt can increase with time. The relevant statistical operator ρrelð Þt is not the wanted nonequilibrium statistical operator ρð Þt because it does not obey the Liouville-von Neumann equation. Also, Srelð Þt is not the thermodynamic entropy because it is based on the arbitrary choice of the set Bf g<sup>n</sup> of relevant observables, and not all possible variables are correctly reproduced. As example, we consider below the famous Boltzmann

<sup>λ</sup>nð Þ<sup>t</sup> <sup>B</sup>\_ <sup>n</sup>

, <sup>Φ</sup>ðÞ¼ <sup>t</sup> ln Tr e� <sup>∑</sup>

SrelðÞ¼� t k<sup>B</sup> Tr ρrelð Þt ln ρrel f g ð Þt : (20)

: (16)

δ Tr ρrelð Þt ln ρrel ½ f g ð Þt � ¼ 0 (17)

Tr <sup>ρ</sup>rel f g ð Þ<sup>t</sup> <sup>B</sup><sup>n</sup> <sup>¼</sup> h i <sup>B</sup><sup>n</sup> <sup>t</sup> (18)

n λnð Þt B<sup>n</sup> 

<sup>t</sup> (21)

, (19)

have to be reproduced by the statistical operator ρð Þt , i.e.,

processes.

2.1.1 The relevant statistical operator

Non-Equilibrium Particle Dynamics

with the self-consistency conditions

well. We find the generalized Gibbs distribution

�Φð Þ�t ∑ n λnð Þt B<sup>n</sup>

be derived. In particular, the production of entropy results as

<sup>∂</sup>Srelð Þ<sup>t</sup> <sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>∑</sup> n

ρrelðÞ¼ t e

entropy in nonequilibrium

8

entropy that is based on the single-particle distribution function, but does not take higher order correlation functions into account.
