Self-diffusion for particles with stochastic collisions in one dimension

Abstract

Color diffusion in a classical fluid composed of two species differingonly 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 densityp and velocities ±1. Colliding particles reflect each other with probabilityp and pass through each other with probability 1 −p. We show that forp > 0 the appropriately scaled trajectories ofn particles converge to p⁻¹b(t)+ (1-p)(ρp)⁻¹b_j(t),j = 1, ...,n. Theb(t),b _j (t) are standard, independent Brownian motions. The common presence ofb(t) means that motions are not independent and hence that the macroscopic state of the colored system isnot in local equilibrium with respect to color.