A generalized Fokker-Planck equation for particle transport in random media

A transport equation is derived for the probability density function (pdf) that gives the phase-space distribution of a particle moving in a random medium: the motion of the particle is determined by a general second-order stochastic differential equation that models transport in a medium exhibiting correlated fluctuations in both space and time. The derivation makes use of cumulant expansions and functional calculus. The most general form of the transport equation requires closure, and a simple closure approximation is discussed. No closure approximation is necessary when the stochastic component of the particle equation of motion is a correlated Gaussian process, and results for transport in an unbounded medium are derived for this exact case. These results are consistent with other studies and serve to validate the model.