Lévy walks and generalized stochastic collision models

Abstract

A stochastic collision model is studied in which a test particle of a mass $M$ collides with bath particles of another mass $m.\mathrm{}$ If the distribution of time intervals between the collisions is long tailed, the relaxation of momentum of the test particle is algebraic. The diffusion is enhanced and a superdiffusion is characteristic of the test particle motion for long times. It is shown that for long times $〈x^2\mathrm{}\left(t\right)\mathrm{}〉$ is independent of the mass ratio $\epsilon =m/M.\mathrm{}$ The mass ratio is an important parameter controlling a transition time before which $〈x^2〉\sim t$ and after which diffusion is enhanced. Special attention is given to the Rayleigh limit where ε is small. It is shown that when $\epsilon =1\mathrm{}$ our results are identical to those obtained within the framework of the Lévy walk model.