A WKB treatment of diffusion in a multidimensional bistable potential

Abstract

We extend to the case of a finite set of stochastic variables whose distributionP obeys a nonlinear Fokker-Planck equation our previous treatment of diffusion in a bistable potentialU, in the limit of small, constant diffusion coefficient. This is done with the help of an extended WKB approximation due to Gervais and Sakita. The treatment is valid if there exists a well-defined most probable path connecting the minima ofU, and if the valley ofU along that path has a slowly varying width, and weak curvature and twisting. We find that: (i) the final approach to equilibrium is governed by Eyring's generalization of the Kramers high-viscosity rate, which we rederive; (ii) for intermediate times, if the initial distribution is concentrated in the region of instability (close vicinity of the saddle point ofU),P has, along the most probable path, the behavior described by Suzuki's scaling statement for a one-dimensional system. In a second part of this time domain,P enters the diffusive regions around the minima ofU and relaxes toward local longitudinal equilibrium on a time comparable with Suzuki's time scale. The time for relaxation toward transverse local equilibrium may, depending on the initial conditions, compete with these longitudinal times.