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

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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.