Mean-field treatment of the many-body Fokker–Planck equation

Abstract

We review some properties of the stationary states of the Fokker - Planck\nequation for N interacting particles within a mean field approximation, which\nyields a non-linear integrodifferential equation for the particle density.\nAnalytical results show that for attractive long range potentials the steady\nstate is always a precipitate containing one cluster of small size. For\narbitrary potential, linear stability analysis allows to state the conditions\nunder which the uniform equilibrium state is unstable against small\nperturbations and, via the Einstein relation, to define a critical temperature\nTc separating two phases, uniform and precipitate. The corresponding phase\ndiagram turns out to be strongly dependent on the pair-potential. In addition,\nnumerical calculations reveal that the transition is hysteretic. We finally\ndiscuss the dynamics of relaxation for the uniform state suddenly cooled below\nTc.\n