Relative motion in stochastic velocity and force fields

Fundamental Fokker-Planck equations for the probability distribution function P(xi, t) for the relative distance xi of two test particles are derived exactly for Gaussian random velocity and force fields. The present formulation clarifies the validity of previous approximate theories on the relative diffusion in random fields. That is, almost all previous results are justified in higher dimensions. The case of one-dimensional velocity field is a remarkably exceptional one in which the relative diffusion shows a fractional power-law behavior, namely -- tsup(1/2) for long time t, because a long-time velocity correlation exists between two adjacent particles. (author)