Statistical Evolution Equations

Abstract

So far, the problem of determination of statistical distribution functions, the problem of derivation of relevant relations (like phase transition conditions) as well as the problem of determination of the form of a distribution function by an elementary and macroscopic principle were in the centre of interest. Another essential aspect of every statistical theory of multi-component systems is the aspect of evolution of a physical system, i. e. the problem of time-space evolution. In order to describe all evolution possibilities of a physical system in a uniform way, one needs evolution equations, in which case it very often makes sense to use differential evolution equations or equations which base on small differences of physical quantities (such equations shall also be called differential evolution equations!). Such evolution equations shall be discussed in this chapter. It shall be started with an elementary evolution equation, and it shall be shown how to get differential evolution equations by a gradual determination of all physically relevant terms (such as fluctuation terms and drift terms). In this way it will be shown that one can get many different classes of differential evolution equations by using only one basic evolution equation. Thus, in this chapter universal dynamic aspects of statistical phenomena shall play a crucial role. The equations which shall now be considered are of the Fokker-Planck type, of a conjugate Fokker-Planck type, of a Schrödingcr type and of a conjugate complex Schrödinger type, with all equations emerging by a special choice of evolution equation parameters.