Liquid-vapor transition in a model of a continuum particle system with finite-range modified Kac pair potential

TL;DR

The paper proves the existence of a liquid-vapor phase transition for a continuum particle system in dimension $d>1$ with a finite-range modified Kac potential of range $γ^{-1}$, for small $γ>0$. The authors introduce a boxed particle/box-model that reduces the continuum problem to a lattice-like spin system, enabling the use of reflection positivity and the Dobrushin–Shlosman criterion to establish a first-order transition. They show that, for non-convex mean-field free energy $φ_λ$, there exist two coexisting translation-invariant Gibbs measures at a critical chemical potential $λ_c(γ)$, with densities approaching the mean-field values as $γ\downarrow0$, thus connecting finite-range results to the classical LP limit. The work provides a rigorous framework for LVT in finite-range continuum systems without non-physical multi-body terms, bridging rigorous MF theory with realistic interactions and offering a method to quantify coexistence and density jumps in the liquid and vapor phases.

Abstract

We prove the existence of a phase transition in dimension $d>1$ in a continuum particle system interacting with a pair potential containing a modified attractive Kac potential of range $γ^{-1}$, with $γ>0$. This transition is "close", for small positive $γ$, to the one proved previously by Lebowitz and Penrose in the van der Waals limit $γ\downarrow0$. It is of the type of the liquid-vapor transition observed when a fluid, like water, heated at constant pressure, boils at a given temperature. Previous results on phase transitions in continuum systems with stable potentials required the use of unphysical four-body interactions or special symmetries between the liquid and vapor. The pair interaction we consider is obtained by partitioning space into cubes of volume $γ^{-d}$, and letting the Kac part of the pair potential be uniform in each cube and act only between adjacent cubes. The "short-range" part of the pair potential is quite general (in particular, it may or may not include a hard core), but restricted to act only between particles in the same cube. Our setup, the "boxed particle model", is a special case of a general "spin" system, for which we establish a first-order phase transition using reflection positivity and the Dobrushin--Shlosman criterion.

Figures (2)

• Figure 1: A schematic phase diagram of a fluid in the temperature-pressure plane $(T,P)$. There exists a critical point $(T_{c},P_{c})$ below which an LVT occurs, and the density (at which the free energy is minimized) jumps discontinuously when crossing the liquid-vapor transition line (thick black line). The other lines correspond to fluid-solid transitions.
• Figure 2:

Theorems & Definitions (67)

• Remark 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Remark 1.7
• Theorem 2.1: Dobrushin--Shlosman criterion
• Lemma 2.2
• proof