Diffusion, super-diffusion and coalescence from a single step

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