3.3.10 Single-particle distribution function and the general form of the linearized Boltzmann equation

Kinetic equations are obtained if the occupation numbers n<sup>ν</sup> of single-(quasi-) particle states ∣νi is taken as the set of relevant observables Bf g<sup>n</sup> . The single-particle state ν is described by a complete set of quantum numbers, e.g., the momentum, the spin and the species in the case of a homogeneous multi-component plasma. In thermal equilibrium, the averaged occupation numbers of the quasiparticle states are given by the Fermi or Bose distribution function, nh i<sup>ν</sup> eq ¼ f 0 <sup>ν</sup> ¼ Tr ρeqn<sup>ν</sup> n o. These equilibrium occupation numbers are changed under the influence of the external field. We consider the deviation Δn<sup>ν</sup> ¼ n<sup>ν</sup> � f 0 <sup>ν</sup> as relevant observables. They describe the fluctuations of the occupation numbers. The response equations, which eliminate the corresponding response parameters Fν, have the

structure of a linear system of coupled Boltzmann equations for the quasiparticles, see [11]

$$\frac{\mathcal{E}}{m}\mathbf{E} \cdot \left[ (\mathbf{P}|\mathbf{n}\_{\nu}) + \langle \mathbf{P}; \dot{\mathbf{n}}\_{\nu} \rangle\_{i\epsilon} \right] = \sum\_{\nu'} F\_{\nu'} P\_{\nu'\nu} \tag{187}$$

resistance leads to a vanishing result, and the correlation function of stochastic forces must be considered, in analogy to the corresponding term in the Langevin equation [6, 25]. A related projection operator technique was used by Mori [26] for

There are close relation to other approaches, such as kinetic theory or quantum master equations, where the response function of the bath is considered. Irreversibility is not inherent in the equilibrium correlation functions, but in the assumption that a nonequilibrium state is considered as a fluctuation in equilibrium with a prescribed value of the relevant quantity. Other degrees of freedom are forced to

The method of nonequilibrium statistical operator (NSO) to describe irreversible processes is based on a very general concept of entropy, the Shannon information entropy (10). This concept is not restricted to dynamical properties like energy, particle numbers, momentum, etc., occurring in physics, but may be applied also to other properties occurring, e.g., in economics, financial market, and game theory. The generalized Gibbs distributions (13) and (19) are obtained if the averages of a

given set of observables are known. Other statistical ensembles may be

constructed, where the values of some observables have a given distribution. For instance, the canonical ensemble follows if the particle numbers are fixed, and the microcanonical ensemble has in addition a fixed energy in the interval ΔE around E, see [1, 2]. There exist alternative concepts of entropy to valuate a probability

In physics, we have a dynamical evolution that forms the equilibrium distribution for ergodic systems, and any initial distribution that is compatible with the values of the conserved quantities can be used to produce the correct equilibrium distribution. The main problem is the microscopic approach to evaluate the dynamical averages, which can be done using quantum statistical methods such as Green function theory or path integral calculations, or, alternatively, numerical simulations of the microscopic equations of motion such as molecular dynamics. In more general, complex systems, we do not know the exact dynamics of the time evolution. However, we can observe time-dependent correlation functions which reflect the time evolution, and properties such as the fluctuation-dissipation theorem are not related to a specific dynamical model for the complex system. The most interesting issue of the NSO method is the selection of the set of relevant observables to describe a nonequilibrium process. The better the choice of the set of relevant observables is, for which a dynamical model for the time evolution can be found, the less influence is produced by the irrelevant observables which may be

An important application is the description of hydrodynamic processes and its relation to kinetic theory. The NSO method allows to treat this problem, selecting the single-particle distribution as well as the hydrodynamic variables as set of relevant observables. This approach has been worked out in [23]. A more general presentation is found in [4], and transport processes in multi-component fluids and superfluid systems are investigated. Until now, a rigorous theory of turbulence is

the memory-function approach.

Nonequilibrium Statistical Operator

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

4. Concluding remarks

4.1 Information theory

adopt the distribution of thermal equilibrium.

distribution which are not discussed here.

described by time-dependent correlation functions.

4.2 Hydrodynamics

43

with P<sup>ν</sup><sup>0</sup> <sup>ν</sup> ¼ n\_ <sup>ν</sup> ð Þþ <sup>0</sup> jΔn<sup>ν</sup> n\_ <sup>ν</sup><sup>0</sup> ; n\_ h i<sup>ν</sup> <sup>i</sup><sup>ϵ</sup>. The response parameters F<sup>ν</sup> are related to the averaged occupation numbers as

$$f\_{\nu}(t) = \text{Tr}\left\{\rho(t)\mathbf{n}\_{\nu}\right\} = f\_{\nu}^{0} + \beta \sum\_{\nu'} F\_{\nu'}(\Delta \mathbf{n}\_{\nu'}|\Delta \mathbf{n}\_{\nu}).\tag{188}$$

The general form of the linear Boltzmann equation (187) can be compared with the expression obtained from kinetic theory. The left-hand side can be interpreted as the drift term, where self-energy effects are included in the correlation function P; n\_ h i<sup>ν</sup> <sup>i</sup><sup>ϵ</sup>. Because the operators n<sup>ν</sup> are commuting, from the Kubo identity follows n\_ <sup>ν</sup> ð Þ¼ <sup>0</sup> jn<sup>ν</sup> ð Þ 1=ℏβ n<sup>ν</sup> h i ½ � <sup>0</sup> ; n<sup>ν</sup> ¼ 0. In the general form, the collision operator is expressed in terms of equilibrium correlation functions of fluctuations that can be evaluated by different many-body techniques. In particular, for the Lorentz model the result (186) with <sup>s</sup> <sup>¼</sup> 25<sup>=</sup><sup>2</sup> =π<sup>3</sup>=<sup>2</sup> is obtained [5, 15, 16].
