Self-diffusion for particles with stochastic collisions in one dimension

Color diffusion in a classical fluid composed of two species differing only by color is intimately connected with the asymptotic behavior of trajectories of test particles in the equilibrium system. We investigate here such behavior in a one-dimensional system of "hard" points with density p and velocities ±1. Colliding particles reflect each other with probability p and pass through each other with probability 1 -p. We show that for p > 0 the appropriately scaled trajectories of n particles converge to p^-1b (t)+ (1-p)(ρp)^-1b_j(t), j = 1, ..., n. The b(t),b_j(t) are standard, independent Brownian motions. The common presence of b(t) means that motions are not independent and hence that the macroscopic state of the colored system is not in local equilibrium with respect to color.