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 occurr...