Dynamics of heterogeneous hard spheres in a file.

Normal dynamics in a quasi-one-dimensional channel of length L (→∞) of N hard spheres are analyzed. The spheres are heterogeneous: each has a diffusion coefficient D that is drawn from a probability density function (PDF), W∼D(-γ) for small D, where 0≤γ<1. The initial spheres' density ρ is nonuniform and scales with the distance (from the origin) l as ρ∼l(-α), 0≤α≤1. An approximation for the N-particle PDF for this problem is derived. From this solution, scaling law analysis and numerical simulations, we show here that the mean square displacement for a particle in such a system obeys ∼t((1-γ)/(2c-γ)), where c=1/(1+α). The PDF of the tagged particle is gaussian in position. Generalizations of these results are considered.