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We solve a non-equilibrium statistical mechanics problem exactly, namely, thesingle-file dynamics of N hard-core interacting particles (the particles cannotpass each other) of size \Delta diffusing in a one dimensional system of finitelength L with reflecting boundaries at the ends. We obtain an exact expressionfor the conditional probability density function P_T(y_T,t|y_{T,0}) that atagged particle T (T=1,...,N) is at position y_T at time t given that it attime t=0 was at position y_{T,0}. Going beyond previous studies, we considerthe asymptotic limit of large N, maintaining L finite, using a non-standardasymptotic technique. We derive an exact expression for P_T(y_T,t|y_{T,0}) fora a tagged particle located roughly in the middle of the system, from which wefind that there are three time regimes of interest for finite-sized systems:(A) For times much smaller than the collision time t<< t_coll=1/(\rho^2D),where \rho=N/L is the particle concentration and D the diffusion constant foreach particle, the tagged particle undergoes normal diffusion; (B) for timesmuch larger than the collision time t>> t_coll but times smaller than theequilibrium time t<< t_eq=L^2/D we find a single-file regime whereP_T(y_T,t|y_{T,0}) is a Gaussian with a mean square displacement scaling ast^{1/2}; (C) For times longer than the equilibrium time $t>> t_eq,P_T(y_T,t|y_{T,0}) approaches a polynomial-type equilibrium probability densityfunction.

Diffusion of Finite-Sized Hard-Core Interacting Particles In a One-Dimensional Box - Tagged Particle Dynamics