4.1 Information theory

structure of a linear system of coupled Boltzmann equations for the quasiparticles,

<sup>¼</sup> <sup>∑</sup>

0 <sup>ν</sup> þ β ∑ ν0

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

=π<sup>3</sup>=<sup>2</sup> is obtained [5, 15, 16].

Even more information is included if we also consider the nonequilibrium twoparticle distributions. As an example, we mention the Debye-Onsager relaxation effect, see [5, 14]. Another important case is the formation of bound states. It seems naturally to consider the bound states as new species and to include the occupation numbers (more precisely, the density matrix) of the bound particle states in the set of relevant observables [20, 21]. It needs a long memory time to produce bound states from free states dynamically in a low-density system, because bound states cannot be formed in binary collisions, a third particle is needed to fulfill the

The inclusion of initial correlation to improve the kinetic theory, in particular to fulfill the conservation of total energy, is an important step worked out during the last decades, see [22] where further references are given. Other approaches to

include correlations in the kinetic theory are given, e.g., in [23, 24].

conductivity [14] or the hopping conductivity [5, 12].

Transport coefficients are expressed in terms of correlation functions in equilibrium. The evaluation can be performed numerically (molecular-dynamic simulations), or using quantum statistical methods such as perturbation theory and the technique of Green functions. The generalized linear response theory has solved problems owing to the evaluation of correlation functions. Perturbation expansions are improved if higher orders are considered. The treatment of singular terms that appear in perturbation expansions is quite complex. Alternatively, the set of relevant observables can be extended. Examples are the virial expansion of the

It is not clear whether the rigorous evaluation of the correlation functions (i.e., the limit ϵ ! 0 only after full summation of the perturbation expansion) will give nontrivial results for the conductivity. For instance, arguments can be given that the exact evaluation of the force-force correlation function to calculate the

ν0 F<sup>ν</sup>0P<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

<sup>ν</sup>, (187)

F<sup>ν</sup><sup>0</sup> Δn<sup>ν</sup> ð Þ <sup>0</sup> jΔn<sup>ν</sup> : (188)

E � ð Þþ Pjn<sup>ν</sup> P; n\_ h i<sup>ν</sup> <sup>i</sup><sup>ϵ</sup>

f <sup>ν</sup>ðÞ¼ t Trf g ρð Þt n<sup>ν</sup> ¼ f

3.3.11 Two-particle distribution function, bound states

e m

see [11]

with P<sup>ν</sup><sup>0</sup>

averaged occupation numbers as

Non-Equilibrium Particle Dynamics

the result (186) with <sup>s</sup> <sup>¼</sup> 25<sup>=</sup><sup>2</sup>

conservation laws.

3.3.12 Conclusions

42

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 distribution which are not discussed here.

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 described by time-dependent correlation functions.

### 4.2 Hydrodynamics

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

not available, but hydrodynamic fluctuations and turbulent flow have been considered using the NSO method [4].
