Non-markovian diffusion and Fokker-Planck equations for brownian oscillators

A generalized diffusion equation is derived from the Mori-Kubo generalized Langevin for a brownian oscillator subject to gaussian random but in general non-markovian noise. This equation involves a time-dependent diffusion function rather than a phenomenological diffusion constant. For long times the diffusion function approaches a constant for overdamped markovian oscillators; only in the limit of extreme overdamping is the phenomenological theory recovered. A previously derived generalized phase space Fokker-Planck equation for the brownian oscillator is shown to have incorrect short-time behaviour. The difficulty is traced to a transient systematic component of the Mori random force which is non-vanishing for classical lattices at 0 K. Fokker-Planck and diffusion equations for the brownian oscillator are derived from a generalized Langevin representation equivalent to, but distinct from, that of Mori and Kubo. The random force in this representation lacks the systematic transient component. The Fokker-Planck and diffusion equations obtained from this alternative Langevin representation are thus correct at all times.