Diffusion , super - diffusion and coalescence from 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 velocity field. First we show that spatially uncorrelated fields u(x) lead to both standard and anomalous diffusion equation. 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 twopoint 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, in the one-dimensional system, is shown through computer simulations and a simple theoretical argument. PACS numbers: 05.40.-a, 05.40.Fb, 02.50.Ey Submitted to: Journal of Statistical Mechanics: Theory and Experiments Diffusion, super-diffusion and coalescence from single step 2