Method of correlation functions

Abstract

In Chap. 1 we derived the distribution function for a closed system (micro-canonical ensemble), which is very rarely used in practical calculations but is of great fundamental importance, serving as a starting point for derivation of distribution functions of small (but macroscopic) subsystems corresponding to the canonical and grand canonical ensemble. But the latter are still difficult (in fact virtually impossible) to calculate for realistic interactions. In this chapter we study an effective approximate technique that makes it feasible to describe distribution functions of small groups containing n particles (n = 1, 2, 3,...), the so called n-particle distribution functions and n-particle correlation functions. With the help of this technique we will be able to construct the route from a statistical mechanical description to thermodynamics. For most problems the knowledge of just the first two distribution functions (i.e. n = 1 and n = 2) is sufficient; sometimes (albeit rarely) it is necessary to also know the ternary (n = 3) distribution function. In Chap. 8 we present an efficient approximate technique to derive n-particle distribution functions. Here we introduce definitions and obtain some important general results. Since the momentum and configurational parts of the partition function are separable, we focus on distribution functions in configuration space (the momentum part is given by the Maxwell distribution). We present derivations in the canonical ensemble; similar results can be obtained for the grand ensemble.