Classical Particles in the Continuum Subjected to High Density Boundary Conditions

Abstract

We consider a continuous system of classical particles confined in a finite region \(\Lambda \) of \(\mathbb {R}^d\) interacting through a superstable and regular pair potential and subjected to non-free and non-periodic 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 the field produced by particles outside \(\Lambda \) whose local density may increase sub-linearly with the distance from the origin at a rate that depends on how fast the pair potential decays at large distances. In particular, if the pair potential v(r) decays as \(C/r^{d+p}\) (with \(p>0\)) when the distance r approaches infinity, then the existence of the thermodynamic limit of the pressure is guaranteed in presence of boundary conditions produced by external particles which may be distributed with a local density increasing with the distance r from the origin as \(\rho (1+ r^q)\), where \(q\le {1\over 2}\min \{1, p\}\) and \(\rho \) is any positive constant (even arbitrarily larger than the mean density \(\rho _0(\beta ,\lambda )\) the system has when submitted to free boundary conditions).