Diffusion of interacting particles in one dimension

Abstract

We study the diffusion of N particles on an infinite line. The particles obey the standard diffusion equation and interact by a hard-core interaction. The problem has an exact solution, from which we derive the single-particle and two-particle probability distributions for arbitrary initial conditions, as expansions in powers of t^-½, where t denotes time. Explicit expressions are given for the moments of the displacement for each of the particles and correlations of displacements between any pair of particles. The mth moment grows as t^m/2 in the leading order. Correlations in the system are quite strong. Two of the interesting features are as follows. (1) Correlation between the displacements of the central particle and that of any other particle decays with the label distance exponentially, but with a correlation length of the order of N. (2) Correlations of a particle near one edge with those on the other edge are much larger than with those near the center. This implies that the size of the assembly expands quite symmetrically with time as t^½.