Correlation functions and the uniqueness of the state in classical statistical mechanics

Abstract

A general criterion is derived which assures the uniqueness of the state of a classical statistical mechanical system in terms of a given system of correlation functions. The criterion is$\sum\limits_k {(m_{k + j}^A )} ^{ - 1/k} = \infty$ for allj and all bounded setsA, where

$$m_k^A = (k!)^{ - 1} \int\limits_A \cdots \int\limits_A {\varrho _k } (x_1 , \ldots ,x_k )dx_1 \ldots dx_1 .$$