On the extension of the Kramers theory of chemical relaxation to the case of nonwhite noise

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Abstract
The first step of our approach consists of relating the generalized Brownian motion in a double-well potential to a suitable time-independent Fokker–Planck operator implying that an arbitrary large number of ‘‘virtual’’ variables be used. Then, to simplify the solution of this multidimensional Fokker–Planck equation, we develop a procedure of adiabatic elimination of the fastly relaxing variables. As a significant feature of this reduction scheme, we point out that no limitation on the number of the virtual variables is implied. The explicit form of the first correction term to the Smoluchowski equation is also shown to depend on whether or not the stochastic force is white. Via a comparison with the analytical results of Grote and Hynes’ theory [J. Chem. Phys. 73, 2715 (1980)] it is argued that the ‘‘exact’’ approach and the ‘‘reduction’’ procedure can be regarded as being complementary to one another.