The generalized Smoluchowski equation and non-Markovian dynamics

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Abstract
Starting from the generalized Langevin equation, we derive a generalized Smoluchowski equation. This equation provides a general framework for problems of non-Markovian barrier crossing dynamics. We consider the dynamics of a Brownian particle in a harmonic well and on an inverted harmonic potential, and rederive the Grote–Hynes formula for barrier crossing rate constants in the high viscosity limit. In so doing, we answer some questions about possible limits to the validity of that result at high viscosities.