Diffusion, super-diffusion and coalescence from a single step

Abstract

From the exact single step evolution equation of the two-point correlationfunction of a particle distribution subjected to a stochastic displacement fieldu(x), we derive different dynamical regimes whenu(x) is iterated to build a stochastic velocity field. First we show that spatially uncorrelated fieldsu(x) lead to both standard and anomalous diffusion equations. When the fieldu(x) is spatially correlated each particle performs a simple free Brownian motion, but the trajectories ofdifferent particles result to be mutually correlated. The two-point statistical properties of the fieldu(x) induce two-point spatial correlations in the particle distribution satisfying a simple butnon-trivial diffusion-like equation. These displacement–displacement correlationslead the system to three possible regimes: coalescence, simple clustering and acombination of the two. The existence of these different regimes is shown, in theone-dimensional system, through computer simulations and a simple theoretical argument.