Kinetics of activated processes from nonstationary solutions of the Fokker–Planck equation for a bistable potential

Abstract

The kinetics of thermally activated processes are studied by the nonstationary solutions of the Fokker–Planck equation, or Kramers’ equation, for a particle moving in a bistable potential and coupled to a heat bath. An alternate direction implicit method is formulated and used to determine the time evolution of the probability density function and probability density current in the phase space for a large range of the strength of coupling to the heat bath. In addition to the rate constant in a first-order rate equation, corresponding to the lowest real eigenvalue in an eigenfunction expansion, transient processes corresponding to higher and complex eigenvalues are also studied. The results for strong coupling to the heat bath are compared to those obtained from the Smoluchowski equation. The present results are also used to evaluate the accuracy of previous treatments.

Combined Hermite spectral-finite difference method for the Fokker-Planck equation

“…Burschka and Titulater [9,10] calculated probability densities for the equation (1.3) with the absorbing boundary conditions (1.4)-(1.5). Several numerical methods, such as finite difference methods [12], Galerkin method [33] and mixed Hermite spectral-finite difference method [14,40] have also been developed to solve the Fokker-Planck problems. Tang et al [40] developed a mixed Hermite spectral-finite difference method, i.e., the Hermite spectral approximation in the velocity direction and finite-difference in the x-direction, for solving the Fokker-Planck equation with finite boundaries in space.…”

Spectral and Pseudospectral Methods of Solution of the Fokker-Planck and Schrödinger Equations

“…A simple model for the isomerization kinetics used by many researchers (Larson and Kostin 1978;Bernstein and Brown 1984;Voigtlaender and Risken 1985;Blackmore and Shizgal 1985a, b;Cartling 1987;Drozdov 1999;Drozdov and Tucker 2001;Felderhof 2008) is defined with the drift and diffusion coefficients given by…”

Spectral and pseudospectral methods for unbounded domains

“…Cartling [101] 曾应用有限差分法计算问题 (4.7), Moore 和 Flaberty [102] 则应用了 Galerkin 方法. 但 不管真解多么光滑, 上述方法数值解的精度是有限的.…”