2.3.2 Selection of the set of relevant observables

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 a Markov process.

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 reproduce already the correctly known averages Bh i<sup>n</sup> <sup>t</sup> , see Eq. (18), the less the 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 sufficient. We discuss this issue in Section 3.

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 order perturbation expansions to obtain acceptable results.

### 2.3.3 Entropy of the nonequilibrium state

An intricate problem is the definition of entropy for the nonequilibrium state. In nonequilibrium, entropy is produced, as investigated in the phenomenological approach to the thermodynamics of irreversible processes, considering currents induced by the generalized forces.

Such a behavior occurs for the relevant entropy defined by the relevant distribution (20),

$$S\_{\rm rel}(t) = -k\_{\rm B} \text{Tr} \{ \rho\_{\rm rel}(t) \ln \rho\_{\rm rel}(t) \}. \tag{29}$$

A famous example that shows the increase of the relevant entropy with time is the Boltzmann H theorem where the relevant observables to define the

nonequilibrium state are the occupation numbers of the single-particle states, i.e., the distribution function, see Section 3.2 for discussion.

Note that the increase of entropy cannot be solved this way. It is related to socalled coarse graining. The information about the state is reduced because the degrees of freedom to describe the system are reduced. This may be an averaging in phase space over small cells. The loss of information then gives the increase of entropy. This procedure is artificial, anthropomorphic, depending on our way to describe the details of a process.

The method of nonequilibrium statistical operator ρNSOð Þt allows to extend the set of relevant observables arbitrarily so that the choice of the set of relevant observables seems to be irrelevant. All missing correlations are produced dynamically. We can start with any set of relevant operators, but have to wait for a sufficient long time to get the correct statistical operator, or to go to very small ϵ. A possible definition of the entropy would be

$$\mathcal{S}\_{\rm NSO}(t) = -k\_{\rm B} \text{Tr} \{ \rho\_{\rm NSO}(t) \ln \rho\_{\rm NSO}(t) \}. \tag{30}$$

Aν. The bath Hamiltonian acts only in the Hilbert space of the bath states leaving the system states unchanged. It is expressed in terms of the bath observables Bμ. Both sets of operators are assumed to be hermitean and independent so that Aν; B<sup>μ</sup>

We project out the relevant part of the nonequilibrium statistical operator ρð Þt

where the trace over the bath can be performed after the eigenstates of the bath are introduced. The operator TrB means the trace over the quantum states of the heat bath. If we have no further information, we construct the relevant statistical operator taking the equilibrium distribution ρ<sup>B</sup> ¼ ρeq (13) for the irrelevant degrees

Starting with the extended Liouville-von Neumann equation (27), we perform

1 iℏ

> ∂ ∂t ρsð Þt

<sup>i</sup><sup>ℏ</sup> Hs; <sup>ρ</sup><sup>s</sup> ½ � ð Þ<sup>t</sup> <sup>ρ</sup><sup>B</sup> � <sup>ρ</sup><sup>B</sup>

Hs; ρ<sup>s</sup> ½ �¼ ð Þt

cyclic invariance of the trace TrB. To obtain a closed equation for ρsð Þt , the full nonequilibrium statistical operator ρð Þt occurring on the right-hand side has to be

> ∂ ∂t ρðÞ�t

inserting the time evolution for ρð Þt (8) and ρsð Þt (34) given above:

1

We eliminate ρðÞ¼ t ΔρðÞþt ρsð Þt ρ<sup>B</sup> and collect all terms with Δρð Þt on the left-hand side. We can assume that h i Hint <sup>B</sup> ¼ TrB Hintρ<sup>B</sup> ð Þ¼ 0 because the heat bath do not exert external forces on the system (if not, replace Hs by Hs þ h i Hint <sup>B</sup> and Hint by Hint � h i Hint B) so that also TrB Hint; ρ<sup>B</sup> ð Þ ½ � ρsðÞ¼ t 0 and the last term

The deviation Δρð Þt vanishes when Hint ! 0. In lowest order with respect to

For this, we calculate the time evolution of the irrelevant part of the statistical

of freedom,

eliminated.

∂ ∂t þ ε ΔρðÞ¼ <sup>t</sup>

We obtain

�ρ<sup>B</sup> 1

∂ ∂t þ ε ΔρðÞ�<sup>t</sup>

13

1

3.1.2 Born-Markov approximation

Nonequilibrium Statistical Operator

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

operator ΔρðÞ¼ t ρðÞ�t ρrelð Þt ,

the trace TrB over the variables of the bath (see Eq. (32)),

1 iℏ

ρsð Þ�t

∂ ∂t

<sup>i</sup><sup>ℏ</sup> ½ �� <sup>H</sup>; <sup>ρ</sup>ð Þ<sup>t</sup>

<sup>i</sup><sup>ℏ</sup> ½ð Þ Hs <sup>þ</sup> <sup>H</sup>int <sup>þ</sup> <sup>H</sup><sup>B</sup> ; <sup>Δ</sup>ρð Þ<sup>t</sup> � þ <sup>ρ</sup><sup>B</sup>

1

<sup>i</sup><sup>ℏ</sup> TrBf g <sup>H</sup>intρ<sup>B</sup> <sup>ρ</sup>sðÞþ<sup>t</sup> <sup>ρ</sup>Bρsð Þ<sup>t</sup> <sup>1</sup>

1

Hint, the solution is found as

<sup>i</sup><sup>ℏ</sup> HB; ρsð Þt ρ<sup>B</sup> ½ � disappears since HB; ρ<sup>B</sup> ½ �¼ 0.

ΔρðÞ¼ t

∂ ∂t

since the remaining terms disappear and <sup>1</sup>

ρsðÞ¼ t TrBρð Þt (32)

ρrelðÞ¼ t ρsð Þt ρB: (33)

TrB Hint ½ � ; ρð Þt (34)

<sup>i</sup> <sup>ℏ</sup> TrBðHBρðÞ�t ρð Þt HBÞ ¼ 0 because of

<sup>ρ</sup><sup>B</sup> (35)

<sup>i</sup><sup>ℏ</sup> TrB <sup>H</sup>int ½ � ; <sup>ρ</sup>ð Þ<sup>t</sup> : (36)

1

<sup>i</sup><sup>ℏ</sup> <sup>H</sup>int; <sup>ρ</sup>sð Þ<sup>t</sup> <sup>ρ</sup><sup>B</sup> ½ �:

(37)

1

<sup>i</sup><sup>ℏ</sup> TrB <sup>H</sup>int ½ �¼ ; <sup>Δ</sup>ρð Þ<sup>t</sup>

<sup>i</sup><sup>ℏ</sup> TrBf g ρBHint vanishes. Also, the term

1

<sup>¼</sup> 0.

The destruction of the reversibility of the von Neumann equation (27) is connected with the source term on the right-hand side that produces the mixing by averaging over the past in Eq. (25). This source term is responsible for the entropy production. At present, there is no proof that the entropy SNSOð Þt will increase also in the limit ϵ ! þ0.
