Stochastic modeling of generalized Fokker–Planck equations. I.

Abstract

A relatively simple method is developed whereby the many-body features of a typical generalized Fokker–Planck equation (GFPE) for a diffusing molecule are first replaced by stochastic bath variables that are assumed to be Markovian. Then the combined molecular and bath variables are characterized as a multidimensional Markov process obeying a stochastic–Liouville equation, which is, in general, incomplete, because it ignores the back reaction of the molecule on the bath variables. In the final step, the equation is completed by subjecting it to the appropriate constraints required for detailed balance. In this form the augmented Fokker–Planck equation (AFPE) properly describes relaxation to thermal equilibrium, and, for the appropriate limiting conditions, it reduces to the classical Fokker–Planck equation. This procedure for stochastic modeling of GFPE is both an improvement on and a generalization of a method previously outlined by Hwang, Mason, Hwang, and Freed (HMHF). Detailed illustrations of AFPE’s are presented for the simple case of a planar rotator subjected to fluctuating torques, and these models are extended to the case of three-dimensional rotational diffusion. Examples include fluctuating torque models related to that used by HMHF. It is shown that only if the fluctuating torque is independent of the orientation of the molecule (more precisely of any fluctuating equilibrium orientation of the molecule), does the model become equivalent to the usual generalized Langevin equation. Otherwise, more general nonlinear models are obtained, which, however, are easily handled by the present methods. Models related to the slowly relaxing local structure (SRLS) model of Polnaszek and Freed are also developed. They are shown to be a consequence of requiring relaxation to the instantaneous value of the fluctuating potential associated with the torque, whereas the fluctuating torque models are a consequence of requiring relaxation to a uniform orientational distribution. They differ further in that the SRLS models are ’’nonfrictional’’. For these reasons we characterize the fluctuating torques as being ’’collision-induced’’ and the SRLS as being ’’structure-induced’’.