Volume 83, Number 2, July 2008
Article Number 20004
Number of page(s) 6
Section General
Published online 10 July 2008
EPL, 83 (2008) 20004
DOI: 10.1209/0295-5075/83/20004

On single-file and less dense processes

O. Flomenbom1 and A. Taloni2, 3

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

received 5 January 2008; accepted in final form 3 June 2008; published July 2008
published online 10 July 2008

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, $\langle \vert r\vert\rangle \sim \langle \vert r\vert\rangle _{{\rm free}}/n^{\mu }$, where $\langle \vert r\vert\rangle _{{\rm free}}$ is the result for a free particle in the studied system and n is the number of particles in the covered length. The exponent $\mu $ 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 \vert r\vert\rangle $ leads to, $\langle \vert r\vert\rangle _{}\sim \rho _{0}^{(a-1)/2}(\langle \vert r\vert\rangle _{{\rm free}})^{(1+a)/2}$, 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.

02.50.Ey - Stochastic processes.
66.30.Pa - Diffusion in nanoscale solids.

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