On single-file and less dense processes

¹  Department of Chemistry, Massachusetts Institute of Technology - Cambridge, MA, 02139, USA
²  Department of Physics, Massachusetts Institute of Technology - Cambridge, MA, 02139, USA
³  Institute of Physics, Academia Sinica - 128 Academia Road, Taipei, 11529, Taiwan

Corresponding author: flomenbo@mit.edu

Received: 5 January 2008
Accepted: 3 June 2008

Abstract

The diffusion process of N hard rods in a 1D interval of length 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, , where is the result for a free particle in the studied system and n is the number of particles in the covered length. The exponent μ is given by, , where a is associated with the particles' density law of the system, , . The scaling law for leads to, , 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, for normal diffusion, with a Gaussian PDF in space for any value of a (deduced by a complementary analysis), and, , for anomalous diffusion in which the system's particles all have the same power-law waiting time PDF for individual events, , . Our analysis shows that the scaling in a “standard” single file is a direct result of the fixed particles' density condition imposed on the system, .