4.3 The limit ϵ ! 0

It is the source term of the extended von Neumann equation (27) that introduces irreversible behavior. Different choices for the set of relevant observables are elected for different applications, in particular quantum master equations, kinetic theory, and linear response theory. It is claimed that this choice of the set of relevant observables is only a technical issue and has no influence on the result, only if the limit ε ! 0 is correctly performed in the final result.

Hamiltonian dynamics is realized by the self-consistency conditions for <sup>ψ</sup>^<sup>t</sup>

system is considered as a stochastic process, see also [6].

Institute of Physics, University of Rostock, Rostock, Germany

\*Address all correspondence to: gerd.roepke@uni-rostock.de

provided the original work is properly cited.

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

An interesting example is the electrical conductivity. In the stationary case which is homogeneous in time, the system remains near thermodynamic equilibrium as long as the electrical field is weak so that the produced heat can be exported. We have to consider an open system. If the conductor is embedded in vacuum, heat export is given by radiation. Bremsstrahlung is emitted during the collision of charged particles. Emission of photons can be considered as a measuring process to localize the charged particle during the collision process. The time evolution of the

see Eq. (28).

Nonequilibrium Statistical Operator

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

Author details

Gerd Röpke

45

relð Þ r; t ,

However, calculations are not performed this way. For instance, the limit ε ! 0 is performed already in a finite order of perturbation theory. The self-consistency conditions (18) guarantee that a finite source term will not influence the Hamiltonian dynamics of the relevant observables. A closer investigation of a finite source term and its influence on the nonequilibrium evolution would be of interest.

### 4.4 Heat production and entropy

A serious problem is that irreversibility is connected with the production of entropy [6]. For instance, in the case of electrical conductivity, heat is produced. In principle, we have to consider an open system coupled to a bath that absorbs the produced heat. In the Zubarev NSO method considered here, it is the right-hand of the extended von Neumann equation (27) that contains the source term. We impose the stationary conditions so that ρrel, in particular T, are not explicitly depending on time. Then, the source term acts like an additional process describing the coupling to a bath without specifying the microscopic process. The parameter ϵ now has the meaning of a relaxation time, and is no longer arbitrarily small but is of the order E<sup>2</sup> .

From a systematic microscopic point of view, one can introduce a process into the system Hamiltonian which describes the cooling of the system via the coupling to a bath, as known from the quantum master equations for open systems. Phonons related to the motion of ions can be absorbed by the bath, but one can calculate the electrical conductivity also for (infinitely) heavy ions so that the scattering of the electrons, accelerated by the field, is elastic. Collisions of electrons with the bath may help, but an interesting process to reduce the energy is radiation. Electrons which are accelerated during the collisions emit bremsstrahlung. This heat transfers the gain of energy of electrons, which are moving in the external field, to the surroundings.

### 4.5 Open systems: coupling to the radiation field

A general approach to scattering theory was given by Gell-Mann and Goldberger [27] (see also [1, 2]) to incorporate the boundary condition into the Schrödinger equation. The equation of motion in the potential Vð Þr reads

$$\frac{\partial}{\partial t}\boldsymbol{\mu}\_{\boldsymbol{\epsilon}}(\boldsymbol{r},t) + \frac{\mathrm{i}}{\hbar}\mathrm{H}\boldsymbol{\mu}\_{\boldsymbol{\epsilon}}(\boldsymbol{r},t) = -\boldsymbol{\varepsilon} \left[\boldsymbol{\mu}\_{\boldsymbol{\epsilon}}(\boldsymbol{r},t) - \boldsymbol{\mu}\_{\boldsymbol{\epsilon}\operatorname{rel}}^{\hat{\mathbf{t}}}(\boldsymbol{r},t)\right].\tag{189}$$

With H ¼ H0 þ V, the relevant state is an eigenstate ∣pi of H0 which changes its value at the scattering time ^t where the asymptotic state ∣p<sup>0</sup> i is formed. As known from the Langevin equation, one can consider ψϵð Þ¼ <sup>r</sup>; <sup>t</sup> <sup>ϱ</sup><sup>1</sup>=<sup>2</sup> exp ið Þ <sup>S</sup>=<sup>ℏ</sup> as a stochastic process [5] related to a stochastic potential Vð Þ r; t ; Eq. (189) appears as an average. The relaxation term is related to the fluctuations of Vð Þ r; t . The average
