Diffusion of finite-sized hard-core interacting particles in a one-dimensional box: Tagged particle dynamics

Abstract

We solve a nonequilibrium statistical-mechanics problem exactly, namely, the single-file dynamics of N hard-core interacting particles (the particles cannot pass each other) of size Delta diffusing in a one-dimensional system of finite length L with reflecting boundaries at the ends. We obtain an exact expression for the conditional probability density function rhoT(yT,t|yT,0) that a tagged particle T (T=1,...,N) is at position yT at time t given that it at time t=0 was at position yT,0. Using a Bethe ansatz we obtain the N -particle probability density function and, by integrating out the coordinates (and averaging over initial positions) of all particles but particle T , we arrive at an exact expression for rhoT(yT,t|yT,0) in terms of Jacobi polynomials or hypergeometric functions. Going beyond previous studies, we consider the asymptotic limit of large N , maintaining L finite, using a nonstandard asymptotic technique. We derive an exact expression for rhoT(yT,t|yT,0) for a tagged particle located roughly in the middle of the system, from which we find that there are three time regimes of interest for finite-sized systems: (A) for times much smaller than the collision time t<<taucoll=1/(rho2D) , where rho=N/L is the particle concentration and D is the diffusion constant for each particle, the tagged particle undergoes a normal diffusion; (B) for times much larger than the collision time t >>taucoll but times smaller than the equilibrium time t<<taueq=L2/D, we find a single-file regime where rhoT(yT,t|yT,0) is a Gaussian with a mean-square displacement scaling as t1/2; and (C) for times longer than the equilibrium time t>>taue , rhoT(yT,t|yT,0) approaches a polynomial-type equilibrium probability density function. Notably, only regimes (A) and (B) are found in the previously considered infinite systems.