Showing papers by "David Ruelle published in 2020"
TL;DR: In this article, it was shown that the correlation functions of point particles described by classical equilibrium statistical mechanics are real analytic functions of the point particle activity of a point particle in a crystal state, assuming a suitable cluster property (decay of correlations).
Abstract: Let z be the activity of point particles described by classical equilibrium statistical mechanics in
$$\mathbf{R}^
u $$
. 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.
TL;DR: In this article, it is shown that the correlation functions of point particles described by classical equilibrium statistical mechanics are real analytic functions of the point particles in a crystal state. But it is not shown that these correlation functions are analytic functions for all point particles.
Abstract: Let $z$ be the activity of point particles described by classical equilibrium statistical mechanics in ${\bf R}^
u$. 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 $z$.