On single file and less dense processes

TL;DR

This paper investigates the diffusion behavior of hard rods in a one-dimensional system, deriving a universal scaling law that interpolates between single file and free diffusion, applicable to both normal and anomalous dynamics.

Contribution

It introduces a universal scaling law for particle displacement in 1D systems of hard rods, linking density laws to diffusion behavior, and extends understanding to anomalous diffusion scenarios.

Findings

Derives a universal scaling law for <|r|> based on density and system parameters.

Shows <|r|> scales as t^((1+a)/2) for normal diffusion.

Provides a framework for understanding anomalous diffusion with power-law waiting times.

Abstract

The diffusion process of N hard rods in a 1D interval of length L (--> inf) 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, <|r|>~ <|r|>_{free}/n^mu, where <|r|>_{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+a), where a is associated with the particles' density law of the system, rho~rho_0*L^(-a), 0<= a <=1. The scaling law for <|r|> leads to, <|r|>~rho_0^((a-1)/2) (<|r| >_{free})^((1+a)/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|>~t^((1+a)/2) for normal diffusion, with a Gaussian PDF in space for any value of a (deduced by a complementary analysis), and, <|r|>~t^((beta(1+a))/2), for anomalous diffusion in which the system's particles all have the same power-law waiting time PDF for individual events, psi~t^(-1-beta), 0<beta<1. Our analysis shows that the scaling <|r|>~t^(1/2) in a 'standard' single file is a direct result of the fixed particles' density condition imposed on the system, a=0.