Eur. Phys. J. B (2014) 87: 120
https://doi.org/10.1140/epjb/e2014-40586-6

Regular Article

The Brownian mean field model

Laboratoire de Physique Théorique (IRSAMC), CNRS and UPS, Université de Toulouse, 31062 Toulouse, France

^a e-mail: chavanis@irsamc.ups-tlse.fr

Received: 19 December 2013
Received in final form: 26 March 2014
Published online: 28 May 2014

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. The BMF model displays a second order phase transition from a homogeneous phase to an inhomogeneous phase below a critical temperature T_c = 1 / 2. We first complete the description of this model in the mean field approximation valid for N → +∞. In the strong friction limit, the evolution of the density towards the mean field Boltzmann distribution is governed by the mean field Smoluchowski equation. For T<T_c, this equation describes a process of self-organization from a non-magnetized (homogeneous) phase to a magnetized (inhomogeneous) phase. We obtain an analytical expression for the temporal evolution of the magnetization close to T_c. Then, we take fluctuations (finite N effects) into account. The evolution of the density is governed by the stochastic Smoluchowski equation. From this equation, we derive a stochastic equation for the magnetization and study its properties both in the homogenous and inhomogeneous phase. We show that the fluctuations diverge at the critical point so that the mean field approximation ceases to be valid. Actually, the limits N → +∞ and T → T_c do not commute. The validity of the mean field approximation requires N(T − T_c) → +∞ so that N must be larger and larger as T approaches T_c. We show that the direction of the magnetization changes rapidly close to T_c while its amplitude takes a long time to relax. We also indicate that, for systems with long-range interactions, the lifetime of metastable states scales as e^N except close to a critical point. The BMF model shares many analogies with other systems of Brownian particles with long-range interactions such as self-gravitating Brownian particles, the Keller-Segel model describing the chemotaxis of bacterial populations, the Kuramoto model describing the collective synchronization of coupled oscillators, the Desai-Zwanzig model, and the models describing the collective motion of social organisms such as bird flocks or fish schools.