Diffusion, super-diffusion and coalescence from single step

TL;DR

This paper derives a unified framework for understanding particle diffusion, super-diffusion, and coalescence from a single stochastic displacement step, revealing how spatial correlations influence different dynamical regimes.

Contribution

It introduces a theoretical approach linking single-step stochastic displacements to various diffusion regimes and coalescence, supported by simulations and analytical arguments.

Findings

Uncorrelated fields lead to standard and anomalous diffusion.

Spatial correlations induce non-trivial diffusion-like equations.

Different regimes of coalescence and clustering are identified.

Abstract

From the exact single step evolution equation of the two-point correlation function of a particle distribution subjected to a stochastic displacement field $\bu(\bx)$, we derive different dynamical regimes when $\bu(\bx)$ is iterated to build a velocity field. First we show that spatially uncorrelated fields $\bu(\bx)$ lead to both standard and anomalous diffusion equation. When the field $\bu(\bx)$ 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 $\bu(\bx)$ 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, in the one-dimensional system, is shown through computer simulations and a simple theoretical argument.