A Fokker–Planck equation for canonical non-Markovian systems: A local linearization approach

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Abstract
Correctly renormalized drift coefficients can be straightforwardly derived using the linear version of the generalized Langevin equation and linear reaction potentials (parabolas or inverted parabolas). Recent investigation via computer simulation of molecular dynamics and numerical solution of stochastic differential equations, shows that interesting cases exist where the nonlinear nature of the interaction between reacting system and ‘‘bath’’ and that of the reaction potential must be taken into account. Then it is shown that a Smoluchowski diffusion equation with correctly renormalized drift coefficients can be obtained by adopting a local linearization assumption, which, nevertheless allows the reaction coordinate to ‘‘feel’’ the influence of different transport properties in different regions of the reaction potential. Under the special condition where the system–bath interaction is assumed to be linear, this Smoluchowski equation is shown to coincide with that recently proposed by Okuyama and Oxtoby [J. Chem. Phys. 8 4, 5830 (1986)]. In the case where the renormalization corrections are neglected, this equation coincides with that proposed by our group [Fonseca, P. Grigolini, and D. Pareo, J. Chem. Phys. 8 3, 1039 (1985)].