The Brownian Mean Field model

TL;DR

This paper analyzes the Brownian Mean Field model, a system of particles on a circle with cosine interactions, exploring its dynamics, thermodynamics, phase transitions, and fluctuations in both homogeneous and inhomogeneous phases.

Contribution

It provides a comprehensive description of the BMF model, including mean field approximation, fluctuation effects, and behavior near the critical point, extending understanding of mean field systems.

Findings

Characterization of the BMF model in mean field approximation

Analysis of fluctuations and stochastic evolution of magnetization

Behavior of the system near the critical point

Abstract

We discuss the dynamics and thermodynamics of the Brownian Mean Field (BMF) model which is a system of N Brownian particles moving on a circle and interacting via a cosine potential. It can be viewed as the canonical version of the Hamiltonian Mean Field (HMF) model. We first complete the description of this system in the mean field approximation. Then, we take fluctuations into account and study the stochastic evolution of the magnetization both in the homogeneous phase and in the inhomogeneous phase. We discuss its behavior close to the critical point.