2.3.1 The extended Liouville-von Neumann equation

The nonequilibrium statistical operator ρϵð Þt , Eq. (25), obeys the extended von Neumann equation

$$\frac{\partial \rho\_{\epsilon}(t)}{\partial t} + \frac{\mathbf{i}}{\hbar} [\mathbf{H}^{\ell}, \rho\_{\epsilon}(t)] = -\epsilon (\rho\_{\epsilon}(t) - \rho\_{\text{rel}}(t)). \tag{27}$$

Similar problems are known for systems (magnetism) where the ground state

5. Any real system is in contact with the surroundings. The intrinsic dynamics described by the Hamiltonian H<sup>t</sup> is modified due to the coupling of the open system to the bath. Within the quantum master equation approach, we can approximate the influence term describing the coupling to the bath by a relaxation term such as the source term. At present, we consider the source term as a purely mathematical tool to select the retarded solution of the von Neumann equation, and physical results are obtained only after performing the

The Zubarev method to solve the initial value problem for the Liouville-von Neumann equation is based on the selection of the set Bf g<sup>n</sup> of relevant observables which characterize the nonequilibrium state. The corresponding relevant statistical operator ρrelð Þt is some approximation to ρð Þt . According to the Bogoliubov principle of weakening of initial correlations, the missing correlations to get ρð Þt are produced dynamically. This process, the dynamical formation of the missing correlations, needs some relaxation time τ. If we would take instead of ρrelð Þt the exact (but unknown) solution ρð Þt , the relaxation time τ is zero. The Liouville-von Neumann equation, which is a first-order differential equation with respect to time, describes

There is no rigorous prescription how to select the set of relevant observables f g B<sup>n</sup> . The more relevant observables are selected so that their averages with ρrelð Þt

effort to produce the missing correlations dynamically, and the less relaxation time τ is needed. Taking into account that usually perturbation theory is used to treat the dynamical time evolution (23), a lower order of perturbation theory is then suffi-

In conclusion, the selection of the set of relevant observables is arbitrary, as a minimum the constants of motion C<sup>n</sup> have to be included because their relaxation time is infinite, their averages cannot be produced dynamically. The resulting ρNSOð Þt (26) should not depend on the (arbitrary) choice of relevant observables f g B<sup>n</sup> if the limit ε ! 0 is correctly performed. However, usually perturbation theory is applied, so that the result will depend on the selection of the set of relevant observables. The inclusion of long-living correlations into Bf g<sup>n</sup> allows to use lower

An intricate problem is the definition of entropy for the nonequilibrium state. In

A famous example that shows the increase of the relevant entropy with time is

SrelðÞ¼� t kBTr ρrelð Þt ln ρrel f g ð Þt : (29)

nonequilibrium, entropy is produced, as investigated in the phenomenological approach to the thermodynamics of irreversible processes, considering currents

Such a behavior occurs for the relevant entropy defined by the relevant

the Boltzmann H theorem where the relevant observables to define the

, see Eq. (18), the less the

has a lower symmetry than the Hamiltonian.

2.3.2 Selection of the set of relevant observables

reproduce already the correctly known averages Bh i<sup>n</sup> <sup>t</sup>

order perturbation expansions to obtain acceptable results.

cient. We discuss this issue in Section 3.

2.3.3 Entropy of the nonequilibrium state

induced by the generalized forces.

distribution (20),

11

limit ϵ ! 0.

Nonequilibrium Statistical Operator

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

a Markov process.

as can be seen after simple derivation with respect to time. In contrast to the von Neumann equation (8), a source term arises on the right-hand side that becomes infinitesimal small in the limit ϵ ! þ0. This source term breaks the time inversion symmetry, so that for any finite value of ϵ, the solution ρϵð Þt describes in general an irreversible evolution with time.

The source term can be interpreted in the following way:


$$\frac{\partial}{\partial t} \langle \mathbf{B}\_{\hbar} \rangle^{t} = \text{Tr} \left\{ \frac{\partial \rho\_{c}(t)}{\partial t} \mathbf{B}\_{\hbar} \right\} = -\text{Tr} \left\{ \frac{\mathbf{i}}{\hbar} [\mathbf{H}^{t}, \rho\_{c}(t)] \mathbf{B}\_{\hbar} \right\} = \left\langle \frac{\mathbf{i}}{\hbar} [\mathbf{H}^{t}, \mathbf{B}\_{\hbar}] \right\rangle^{t} = \left\langle \dot{\mathbf{B}}\_{\hbar} \right\rangle^{t}. \tag{28}$$

The source term cancels because of the self-consistency conditions (18). Thus, the time evolution of the relevant observables satisfies the dynamical equations of motion according to the Hamiltonian H<sup>t</sup> .


nonequilibrium state are used to construct the relevant statistical operator, the less

t<sup>0</sup> ! �∞ is less active to produce the remaining missing correlating. The past that is of relevance, given by the relaxation time τ, becomes shorter, if the relevant (longliving) correlations are already correctly implemented. The limit ε ! þ0 has to be

The nonequilibrium statistical operator ρϵð Þt , Eq. (25), obeys the extended von

as can be seen after simple derivation with respect to time. In contrast to the von Neumann equation (8), a source term arises on the right-hand side that becomes infinitesimal small in the limit ϵ ! þ0. This source term breaks the time inversion symmetry, so that for any finite value of ϵ, the solution ρϵð Þt describes in general an

1. The source term implements the "initial condition" in the equation of motion as expressed by ρrelð Þt . Formally, the source term looks like a relaxation process. In addition to the internal dynamics, the system evolves toward the

2. The construction of the source term is such that the time evolution of the relevant variables is not affected by the source term (we use the invariance of

> ; ρϵ ½ � ð Þt B<sup>n</sup>

The source term cancels because of the self-consistency conditions (18). Thus, the time evolution of the relevant observables satisfies the dynamical equations of

3. The value of ϵ has to be small enough, ϵ ≪ 1=τ, so that all relaxation processes to establish the correct correlations, i.e., the correct distribution of the irrelevant observables, can be performed. However, ℏϵ has to be large compared to the energy difference of neighbored energy eigenstates of the system so that mixing is possible. For a system of many particles, the density of energy eigenvalues is high so that we can assume a quasi-continuum. This is necessary to allow for dissipation. The van Hove limit means that the limit

4.Differential equations can have degenerated solutions. For instance, we know the retarded and advanced solution of the wave equation that describes the emission of electromagnetic radiation. An infinitesimal small perturbation can destroy this degeneracy and select out a special solution, here the retarded one.

.

<sup>¼</sup> <sup>i</sup>

<sup>ℏ</sup> <sup>H</sup><sup>t</sup> ½ � ; <sup>B</sup><sup>n</sup> <sup>t</sup>

<sup>¼</sup> <sup>B</sup>\_ <sup>n</sup> <sup>t</sup>

: (28)

<sup>ℏ</sup> <sup>H</sup><sup>t</sup>

ϵ ! þ0 has to be performed after the thermodynamic limit.

; ρϵ ½ �¼� ð Þt ϵ ρϵðÞ�t ρrel ð Þ ð Þt : (27)

dynamical formation of the correct correlations in ρð Þt is needed. The limit

performed after the thermodynamic limit, see below.

2.3 Discussion of the Zubarev NSO approach

Non-Equilibrium Particle Dynamics

Neumann equation

irreversible evolution with time.

relevant distribution.

h i <sup>B</sup><sup>n</sup> <sup>t</sup> <sup>¼</sup> Tr <sup>∂</sup>ρϵð Þ<sup>t</sup>

∂t Bn 

motion according to the Hamiltonian H<sup>t</sup>

∂ ∂t

10

2.3.1 The extended Liouville-von Neumann equation

<sup>∂</sup>ρϵð Þ<sup>t</sup> ∂t þ i ℏ Ht

The source term can be interpreted in the following way:

the trace with respect to cyclic permutations),

¼ �Tr <sup>i</sup>

Similar problems are known for systems (magnetism) where the ground state has a lower symmetry than the Hamiltonian.

5. Any real system is in contact with the surroundings. The intrinsic dynamics described by the Hamiltonian H<sup>t</sup> is modified due to the coupling of the open system to the bath. Within the quantum master equation approach, we can approximate the influence term describing the coupling to the bath by a relaxation term such as the source term. At present, we consider the source term as a purely mathematical tool to select the retarded solution of the von Neumann equation, and physical results are obtained only after performing the limit ϵ ! 0.
