Variational layer expansion for kinetic processes

Often the analysis of the Fokker-Planck (FP) operator near the saddle point is sufficient to characterize the activated processes. However, there are also situations where the kinetic processes are controlled by the dynamics far away from the saddle points. Correspondingly, the knowledge of FP kinetic modes in all the phase space is required in order to describe accurately the activated processes. To this aim we propose a variational method for approximating the site-localizing functions that ate defined as linear combinations of the FP slow eigenfunctions and describe the stable-state populations. The starting point is the layer expansion method that has been developed by Matkowsky and Schuss [Slam J. Appl. Math. 33, 365 (1977); 36, 604 (1979); 40, 242 (1981)], which we apply to the covariant form of the FP equation. Error-function profiles across the separatrix are derived in this way for the site-localizing functions. The same kind of profile is found in the numerical solutions of a bistable two-dimensional Smoluchowski equation, but about a line (the so-called stochastic separatrix) that is, in general, different from the deterministic separatrix. Thus the layer expansion has to be generalized by considering the separatrix as a parametric function to be optimized according to a variational criterion for the decay rates. After discretization along the separatrix of the integral relation for the rate, the variational problem is solved numerically, with satisfactory agreement with the exact numerical results.