Identities for correlation functions in classical statistical mechanics and the problem of crystal states

Abstract

Let $z$ be the activity of point particles described by classical equilibrium statistical mechanics in ${\bf R}^\nu$. The correlation functions $\rho^z(x_1,\dots,x_k)$ denote the probability densities of finding $k$ particles at $x_1,\dots,x_k$. Letting $\phi^z(x_1,\dots,x_k)$ be the cluster functions corresponding to the $\rho^z(x_1,\dots,x_k)/z^k$ we prove identities of the type $$ \phi^{z_0+z'}(x_1,\dots,x_k) $$ $$ =\sum_{n=0}^\infty{z'^n\over n!}\int dx_{k+1}\dots\int dx_{k+n}\,\phi^{z_0}(x_1,\dots,x_{k+n}) $$ It is then non-rigorously argued that, assuming a suitable cluster property (decay of correlations) for a crystal state, the pressure and the translation invariant correlation functions \- $\rho^z(x_1,\dots,x_k)$ are real analytic functions of $z$.