The Generalized Stochastic Smoluchowski Equation

We study the dynamics of a system of overdamped Brownian particles governed by the generalized stochastic Smoluchowski equation associated with a generalized form of entropy and involving a long-range potential of interaction [P.H. Chavanis, Entropy 17, 3205 (2015)]. We first neglect fluctuations and provide a macroscopic description of the system based on the deterministic mean field Smoluchowski equation. We then take fluctuations into account and provide a mesoscopic description of the system based on the stochastic mean field Smoluchowski equation. We establish the main properties of this equation and derive the Kramers escape rate formula, giving the lifetime of a metastable state, from the theory of instantons. We relate the properties of the generalized stochastic Smoluchowski equation to a principle of maximum dissipation of free energy. We also discuss the connection with the dynamical density functional theory of simple liquids.