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Identities for Correlation Functions in Classical Statistical Mechanics and the Problem of Crystal States

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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 \(\phi ^z(x_1,\dots ,x_k)\) be the cluster functions corresponding to the \(\rho ^z(x_1,\dots ,x_k)/z^k\) we prove identities of the type

$$\begin{aligned}&\phi ^{z_0+z'}(x_1,\dots ,x_k)\\&\quad =\sum _{n=0}^\infty {z'^n\over n!}\int dx_{k+1}\dots \int dx_{k+n}\,\phi ^{z_0}(x_1,\dots ,x_{k+n}) \end{aligned}$$

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.

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Correspondence to David Ruelle.

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Communicated by David Ruelle.

This note is dedicated to Joel Lebowitz on his 90-th birthday

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Ruelle, D. Identities for Correlation Functions in Classical Statistical Mechanics and the Problem of Crystal States. J Stat Phys 180, 1002–1009 (2020). https://doi.org/10.1007/s10955-020-02575-3

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