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

Let $z$ be the activity of point particles described by classical equilibrium statistical mechanics in ${\bf R}^ν$. The correlation functions $ρ^z(x_1,\dots,x_k)$ denote the probability densities of finding $k$ particles at $x_1,\dots,x_k$. Letting $ϕ^z(x_1,\dots,x_k)$ be the cluster functions corresponding to the $ρ^z(x_1,\dots,x_k)/z^k$ we prove identities of the type $ϕ^{z_0+z'}(x_1,\dots,x_k)$ $=\sum_{n=0}^\infty{z'^n\over n!}\int dx_{k+1}\dots\int dx_{k+n}\,ϕ^{z_0}(x_1,\dots,x_{k+n})$ 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 \- $ρ^z(x_1,\dots,x_k)$ are real analytic functions of $z$.

No reviews yet.

No comments yet.