Brownian motion in dynamically disordered media

The motion of Brownian test particles in a model random potential with time dependent correlations is investigated using four methods: renormalized perturbation, perturbation using Martin, Siggia, and Rose functional formalism (MSR), the Edwards variational method on the MSR functional, and renormalization group with the MSR function. The disorder averaged one-particle propagators determined by the renormalized perturbation expansion and MSR perturbation expansion are identical to the second and possibly higher order, and the two-particle propagators determined by these perturbation methods are identical at the first and possibly higher order. The one-particle propagator determined by the Edwards method is identical to the perturbation expansions at the first order, but the second-order analogue of the Edwards method has a more complex expression, which reduces to the second-order perturbation expression with additional higher-order terms. The diffusion constant and two-particle correlations are calculated from these propagators and are used to determine the effects of the random potential on the Brownian particles. Generally, the diffusion rate decreases with the disorder strength and increases with the temporal decay rate. The two competing mechanisms result in an enhancement of the diffusion constant for weak potentials with fast temporal fluctuations. The system exhibits two-particle correlations that are inherently non-Gaussian and indicate clustering behavior. The diffusion constant is also determined from a simple one-loop renormalization group calculation. In the static limit, the diffusion constant calculated by the renormalization group recovers the results of Deem and Chandler [M.W. Deem and D. Chandler, J. Stat. Phys. 76, 911 (1994)].