Diffusion of finitesized hardcore interacting particles in a onedimensional box: Tagged particle dynamics
Author

L. Lizana

Tobias Ambjörnsson
Summary, in English
We solve a nonequilibrium statisticalmechanics problem exactly, namely, the singlefile dynamics of N hardcore interacting particles (the particles cannot pass each other) of size Delta diffusing in a onedimensional system of finite length L with reflecting boundaries at the ends. We obtain an exact expression for the conditional probability density function rho T(yT,t vertical bar 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 Nparticle 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 rho T(yT,t vertical bar 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 rho T(yT,t vertical bar 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 finitesized systems: (A) for times much smaller than the collision time t <tau(coll)=1/(rho D2), 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 tau(coll) but times smaller than the equilibrium time t <tau(eq)=L2/D, we find a singlefile regime where rho T(yT,t vertical bar yT,0) is a Gaussian with a meansquare displacement scaling as t(1/2); and (C) for times longer than the equilibrium time t tau(eq), rho T(yT,t vertical bar yT,0) approaches a polynomialtype equilibrium probability density function. Notably, only regimes (A) and (B) are found in the previously considered infinite systems.