Nonequilibrium diffusion induced by a spatio-temporal fluctuation

Abstract

Spatio-temporal generalization of the Ornstein–Uhlenbeck process has been receivedconsiderable attention in the context of coagulations in a random flow. We shall explore thediffusion process with a symmetric spatio-temporal correlated noise in the presence of theexternal force for the underdamped case. In a nontrivial short correlation limit, the role ofspatial correlation is explored and a Fokker–Planck equation is derived based on thestochastic Liouville equation. Our Fokker–Planck equation interpolates the recentlyproposed diffusion equation describing the generalized Ornstein–Uhlenbeck processes andthe traditional Kramers equation. In the small limit of the characteristic value ofthe momentum given by the mass, spatial and temporal correlation lengths, thediffusion coefficient is proportional to the inverse of the momentum. On the otherhand, for the large limit of the characteristic momentum constant, the usualBrownian motion without spatial randomness is reproduced. The analytic form ofthe steady state distribution is numerically verified with use of the stochasticsimulation.