# Ensemble Inequivalence and Maxwell Construction in the Self-Gravitating Ring Model

## TL;DR

This paper investigates ensemble inequivalence in a long-range interacting model, demonstrating through numerical methods that traditional ensemble equivalence breaks down, and explores the applicability of the Maxwell construction in such systems.

## Contribution

It provides a detailed analysis of ensemble inequivalence in a self-gravitating ring model, combining numerical solutions and Monte Carlo simulations to challenge standard assumptions.

## Key findings

- Ensemble inequivalence occurs in the model due to long-range interactions.
- The Maxwell construction's applicability is limited and context-dependent.
- Numerical methods reveal non-trivial phase behavior in the model.

## Abstract

Equilibrium Statistical Mechanics is undoubtedly a cornerstone for the description of many particle systems. The common interpretation is based on ensemble theory as put forward by Gibbs, alongside the basic assumptions that different ensembles are equivalent, i.~e.\ the properties of the system can equally be obtained in any ensemble with the same results. However, the simplicity of the argument that provides such equivalence, mathematically grounded by the existence of Legendre transformation between the ensembles and the existence of its inverse, may break down for physical systems with long range interactions. In this paper we study the behavior of a simple toy model with a long range interaction and show from first principles, by solving numerically the mechanical equations of motion and Monte Carlo simulations, the inequivalence of ensembles, and discuss in what situations and how the Maxwell construction is applicable.