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

We review some properties of the stationary states of the Fokker-Planck equation for N interacting particles within a mean-field approximation, which yields a non-linear integrodifferential equation for the particle density. Analytical results show that for attractive long-range potentials the steady state is always a precipitate containing one. or several clusters of small size. For arbitrary potential, linear stability analysis allows the statement of the conditions under which the uniform equilibrium state is unstable against small perturbations and, via the Einstein relation, definition of a critical temperature T-c separating the two phases, uniform and precipitate. The corresponding phase diagram turns out to be strongly dependent on the pair-potential. In addition, numerical calculations reveal that the transition is hysteretic. We finally discuss the dynamics of relaxation for the uniform state suddenly cooled below T-c.