On single-file and less dense processes

The diffusion process of N hard rods in a 1D interval of length L(->infinity) is studied using scaling arguments and an asymptotic analysis of the exact N-particle probability density function (PDF). In the class of such systems, the universal scaling law of the tagged particle's mean absolute displacement reads, <vertical bar r vertical bar >approximate to <vertical bar r vertical bar > free/n(mu), where <vertical bar r vertical bar > free is the result for a free particle in the studied system and n is the number of particles in the covered length. The exponent mu is given by, mu=1/(1+alpha), where alpha is associated with the particles' density law of the system, rho approximate to rho 0L(-alpha), 0 <=alpha <= 1. The scaling law for <vertical bar r vertical bar > leads to, <vertical bar r vertical bar >approximate to rho 0((alpha-1)/2)(<vertical bar r vertical bar > free)((1+alpha)/2), an equation that predicts a smooth interpolation between single-file diffusion and free-particle diffusion depending on the particles' density law, and holds for any underlying dynamics. In particular, < r(2)>approximate to t(1+alpha/2) for normal diffusion, with a Gaussian PDF in space for any value of alpha (deduced by a complementary analysis), and, < r2 >approximate to t(beta(1+alpha)/2) , for anomalous diffusion in which the system's particles all have the same power-law waiting time PDF for individual events, psi approximate to t(-1-beta), 0 <beta < 1. Our analysis shows that the scaling < r(2)>approximate to t(1/2) in a "standard" single file is a direct result of the fixed particles' density condition imposed on the system, alpha=0. Copyright (C) EPLA, 2008.