Diffusion of finite-sized hard-core interacting particles in a one-dimensional box: Tagged particle dynamics
Author(s)
Lizana, L.; Ambjornsson, T.
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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 Δ 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 ρT(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 ρT(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 ρT(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«τcoll=1/(ϱ2D), where ϱ=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«τcoll but times smaller than the equilibrium time t«τeq=L2/D, we find a single-file regime where ρT(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«τeq, ρT(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.
Date issued
2009-11Department
Massachusetts Institute of Technology. Department of ChemistryJournal
Physical Review E
Publisher
American Physical Society
Citation
Lizana, L., and T. Ambjörnsson. “Diffusion of Finite-sized Hard-core Interacting Particles in a One-dimensional Box: Tagged Particle Dynamics.” Physical Review E 80.5 (2009) : 051103. © 2009 The American Physical Society
Version: Final published version
ISSN
1539-3755
1550-2376