Dynamics of heterogeneous hard spheres in a file

Normal dynamics in a quasi-one-dimensional channel of length L (-> infinity) 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 similar to D-gamma for small D, where 0 <= gamma < 1. The initial spheres' density rho is nonuniform and scales with the distance (from the origin) l as rho similar to l(-alpha), 0 <= alpha <= 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 < r(2)> similar to t((1-gamma)/(2c-gamma)), where c = 1/(1 + alpha). The PDF of the tagged particle is Gaussian in position. Generalizations of these results are considered.