3.3.11 Two-particle distribution function, bound states

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 conservation laws.

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].

### 3.3.12 Conclusions

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

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

Nonequilibrium Statistical Operator DOI: http://dx.doi.org/10.5772/intechopen.84707

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 the memory-function approach.

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 adopt the distribution of thermal equilibrium.
