Particle systems with weakly attractive interaction

TL;DR

This paper develops a mathematical framework for classical particle systems with weakly attractive, stable interactions, establishing inequalities and constructing invariant measures applicable to various conditions.

Contribution

It introduces a lattice approximation method to prove FKG inequalities and constructs infinite volume measures for attractive particle systems with broad applicability.

Findings

FKG inequalities hold for such particle systems

Infinite volume measures are constructed and shown to be invariant

Method applies to arbitrary activity, temperature, and long-range interactions

Abstract

Systems of classical continuous particles in the grand canonical ensemble interacting through purely attractive, yet stable, interactions are defined. By a lattice approximation, FKG ferromagnetic inequalities are shown to hold for such particle systems. Using these inequalities, a construction of the infinite volume measures is given by a monotonicity and upper bound argument. Invariance under Euclidean transformations is proven for the infinite volume measures. The construction works for arbitrary activity and temperature and for integrable long range interactions. Also, inhomogeneous systems of particles with different "charge" can be treated.