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 ${\mathbf{R}}^{\nu }$. 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 ${\varphi }^z(x_1,\dots ,x_k)$ be the cluster functions corresponding to the ${\rho }^z(x_1,\dots ,x_k)/z^k$ we prove identities of the type $${\varphi }^{z_0+z^{\prime }}(x_1,\dots ,x_k)$$ $$=\sum _{n=0}^{\infty }\frac{z^{\prime n}}{n!}\int d{x}_{k+1}\cdots \int d{x}_{k+n}{\varphi }^{z_0}(x_1,\dots ,x_{k+n})$$ 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 $z$.

8 pages