Classical particles in the continuum subjected to high density boundary conditions

TL;DR

This paper proves that the thermodynamic limit of pressure in a classical particle system with superstable interactions remains unaffected by high-density boundary conditions, even when external particles' density increases with distance, under certain decay conditions of the pair potential.

Contribution

It establishes conditions under which the thermodynamic limit exists despite boundary conditions generated by external particles with increasing density.

Findings

Thermodynamic pressure limit is boundary-condition independent under specified decay.

Existence of the limit holds for Lennard-Jones type potentials with certain decay rates.

Boundary conditions with increasing external particle density do not affect the limit if decay conditions are met.

Abstract

We consider a continuous system of classical particles confined in a finite region $\Lambda$ of $\mathbb{R}^d$ interacting through a superstable and tempered pair potential in presence of non free boundary conditions. We prove that the thermodynamic limit of the pressure of the system at any fixed inverse temperature $\beta$ and any fixed fugacity $\lambda$ does not depend on boundary conditions produced by particles outside $\Lambda$ whose density may increase sub-linearly with the distance from the origin at a rate which depends on how fast the pair potential decays at large distances. In particular, if the pair potential $v(x-y)$ is of Lennard-Jones type, i.e. it decays as $C/\|x-y\|^{d+p}$ (with $p>0$) where $\|x-y\|$ is the Euclidean distance between $x$ and $y$, then the existence of the thermodynamic limit of the pressure is guaranteed in presence of boundary conditions generated by external particles which may be distributed with a density increasing with the distance $r$ from the origin as $\rho(1+ r^q)$, where $\rho$ is any positive constant (even arbitrarily larger than the density $\rho_0(\beta,\lambda)$ of the system evaluated with free boundary conditions) and $q\le {1\over 2}\min\{1, p\}$.