3. Applications

The NSO method is a fundamental step in deriving equations of evolution to describe nonequilibrium phenomena. It can be shown that any currently used description can be deduced from this approach. We give three typical examples, the quantum master equations, see [9, 10], kinetic theory, see [11], and linear response theory, see [12]. In all of these applications, we have to define the set of relevant observables, and to eliminate the Lagrange parameters determined by the selfconsistency conditions. We shortly outline these applications, for a more exhaustive presentation see [3–5].

### 3.1 Quantum master equation

### 3.1.1 Open systems

The main issue is that any physical system cannot be completely separated from the surroundings, so that the isolated system is only a limiting case of the open system which is in contact with a bath. More general, we subdivide the degrees of freedom of the total system into the relevant degrees of freedom which describe the system S under consideration, and the irrelevant part describing the bath B. Examples are a harmonic oscillator coupled to a bath consisting of harmonic oscillators, such as an oscillating molecule interacting with phonons or photons, or radiation from a single atom embedded in the bath consisting of photons, see below.

The Hamiltonian H of the open system can be decomposed

$$H = H\_{\rm S} + H\_{\rm B} + H\_{\rm int.} \tag{31}$$

The system Hamiltonian acts only in the Hibert space of the system states leaving the bath states unchanged. It is expressed in terms of the system observables nonequilibrium state are the occupation numbers of the single-particle states, i.e.,

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.

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

The NSO method is a fundamental step in deriving equations of evolution to describe nonequilibrium phenomena. It can be shown that any currently used description can be deduced from this approach. We give three typical examples, the quantum master equations, see [9, 10], kinetic theory, see [11], and linear response theory, see [12]. In all of these applications, we have to define the set of relevant observables, and to eliminate the Lagrange parameters determined by the selfconsistency conditions. We shortly outline these applications, for a more exhaustive

The main issue is that any physical system cannot be completely separated from

H ¼ H<sup>S</sup> þ H<sup>B</sup> þ Hint: (31)

the surroundings, so that the isolated system is only a limiting case of the open system which is in contact with a bath. More general, we subdivide the degrees of freedom of the total system into the relevant degrees of freedom which describe the system S under consideration, and the irrelevant part describing the bath B. Examples are a harmonic oscillator coupled to a bath consisting of harmonic oscillators, such as an oscillating molecule interacting with phonons or photons, or radiation from a single atom embedded in the bath consisting of photons, see below.

The system Hamiltonian acts only in the Hibert space of the system states leaving the bath states unchanged. It is expressed in terms of the system observables

The Hamiltonian H of the open system can be decomposed

SNSOðÞ¼� t kBTrf g ρNSOð Þt ln ρNSOð Þt : (30)

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

the distribution function, see Section 3.2 for discussion.

describe the details of a process.

Non-Equilibrium Particle Dynamics

in the limit ϵ ! þ0.

3. Applications

presentation see [3–5].

3.1.1 Open systems

12

3.1 Quantum master equation

A possible definition of the entropy would be

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> <sup>¼</sup> 0.

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

$$\rho\_s(t) = \text{Tr}\_{\mathbb{B}} \rho(t) \tag{32}$$

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 of freedom,

$$
\rho\_{\rm rel}(\mathbf{t}) = \rho\_{\rm s}(\mathbf{t})\rho\_{\rm B}.\tag{33}
$$
