Nonequilibrium diffusion induced by a spatio-temporal fluctuation

Spatio-temporal generalization of the Ornstein-Uhlenbeck process has been received considerable attention in the context of coagulations in a random flow. We shall explore the diffusion process with a symmetric spatio-temporal correlated noise in the presence of the external force for the underdamped case. In a nontrivial short correlation limit, the role of spatial correlation is explored and a Fokker-Planck equation is derived based on the stochastic Liouville equation. Our Fokker-Planck equation interpolates the recently proposed diffusion equation describing the generalized Ornstein-Uhlenbeck processes and the traditional Kramers equation. In the small limit of the characteristic value of the momentum given by the mass, spatial and temporal correlation lengths, the diffusion coefficient is proportional to the inverse of the momentum. On the other hand, for the large limit of the characteristic momentum constant, the usual Brownian motion without spatial randomness is reproduced. The analytic form of the steady state distribution is numerically verified with use of the stochastic simulation.