On single-file and less dense processes

The diffusion process of N hard rods in a 1D interval of length $L ( \to \infty ) $ 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, $\langle |r|\rangle \sim \langle |r|\rangle _{{\rm free}}/n^{\mu }$, where $\langle |r|\rangle _{{\rm free}}$ is the result for a free particle in the studied system and n is the number of particles in the covered length. The exponent μ is given by, $\mu =1/(1+a)$, where a is associated with the particles' density law of the system, $\rho \sim \rho _{0}L^{-a}$, $0\leqslant a\leqslant 1$. The scaling law for $\langle |r|\rangle $ leads to, $\langle |r|\rangle _{}\sim \rho _{0}^{(a-1)/2}(\langle |r|\rangle _{{\rm free}})^{(1+a)/2}$, 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, $\langle {r^2} \rangle \sim t^{{{1 + a} \over 2}} $ for normal diffusion, with a Gaussian PDF in space for any value of a (deduced by a complementary analysis), and, $\langle {r^2} \rangle \sim t^{{{\beta (1 + a)} \over 2}} $, for anomalous diffusion in which the system's particles all have the same power-law waiting time PDF for individual events, $\psi \sim t^{-1-\beta }$, $0 < \beta < 1$. Our analysis shows that the scaling $\langle r^{2}\rangle \sim t^{1/2}$ in a “standard” single file is a direct result of the fixed particles' density condition imposed on the system, $a=0$.