Dynamical correlation patterns: A new representation of the Liouville equation

Abstract

The reduced distribution functions characterizing a many-particle system are grouped together into a set called “distribution vector”. The BBGKY hierarchy then appears as a single matrix equation, algebraically equivalent to the Liouville equation. Each reduced distribution function is further decomposed into a “dynamical” cluster representation. Contrary to the usual representation, based on the functional form of the various terms (factorization), the new “correlation patterns” are defined by using a purely dynamical criterion: separate equations of evolution are written for the correlation patterns. These equations can again be viewed as components of the single (matrix) Liouville equation, in the correlation-patterns representation. This representation has been devised in order to be combined with the modern, very powerful methods of non-equilibrium statistical mechanics. It provides a concrete realization of the formerly developed abstract theory. There is no difficulty with the thermodynamic limit, which can be taken from the very beginning. The method is particularly well adapted to the study of inhomogeneous systems, in which case it provides a simple and clear-cut separation of inhomogeneity and correlation effects. The relation between usual (“static”) and dynamic correlation patterns is studied. Finally, a diagram technique, suitable for practical calculations, is constructed.