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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 cluster of small size. For
arbitrary potential, linear stability analysis allows to state the conditions
under which the uniform equilibrium state is unstable against small
perturbations and, via the Einstein relation, to define a critical temperature
Tc separating 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
Tc.Comment: 13 pages, 8 figure