Fokker–Planck equations for simple non-Markovian systems

Exact generalized Fokker–Planck equations are derived from the linear Mori–Kubo generalized Langevin equation for the case of Gaussian but non-Markovian noise. Fokker–Planck equations which generate the momentum and phase space probability distribution functions (pdf’s) for free Brownian particles and the phase space pdf for Brownian oscillators are presented. Also given is the generalized diffusion equation for the free Brownian particle pdf in the zero inertia limit. The generalized Fokker–Planck equations are similar in structure to the corresponding phenomenological equations. They, however, involve time-dependent friction and frequency functions rather than phenomenological constants. Explicit results for the frequency and friction functions are given for the Debye solid model. These functions enter as simple multiplicative factors rather than as retarded kernels. Further the phase space Fokker–Planck equations contain an extra diffusive term, a mixed phase space second partial derivative, not occurring in the phenomenological equations. For short times the generalized Fokker–Planck equations reduce to the appropriate Liouville equations. For systems with long time tail decay, e.g., the hydrodynamical Brownian particle and an oscillator in a harmonic lattice, the generalized equations do not asymptotically reduce to the phenomenological equations since these latter predict exponential decay. Moreover the exact generalized equations are not equivalent to the familiar approximate generalized Fokker–Planck equations with retarded kernels except when both types of equations reduce to the phenomenological form. The value of the approximate equations with retarded kernels as an improvement upon the phenomenological equations is thus subject to question.