On the Scale-Invariant Distribution of the Diffusion Coefficient for Classical Particles Diffusing in Disordered Media.-

TL;DR

This paper investigates the distribution of diffusion coefficients in disordered media, revealing a scale-invariant inverse Gaussian form in certain regimes and identifying critical behaviors and limitations of the model.

Contribution

It applies a renormalization group approach to characterize the distribution P(D) of diffusion coefficients, identifying stable and unstable regimes and analyzing differences between annealed and quenched disorder.

Findings

Inverse Gaussian distribution is stable under rescaling in the annealed case.

Pure diffusion occurs at small disorder with P(D) approaching a delta function.

Strong disorder leads to infinite cumulants and ill-defined diffusion processes.

Abstract

The scaling form of the whole distribution P(D) of the random diffusion coefficient D(x) in a model of classically diffusing particles is investigated. The renormalization group approach above the lower critical dimension d=0 is applied to the distribution P(D) using the n-replica approach. In the annealed approximation (n=1), the inverse gaussian distribution is found to be the stable one under rescaling. This identification is made based on symmetry arguments and subtle relations between this model and that of fluc- tuating interfaces studied by Wallace and Zia. The renormalization-group flow for the ratios between consecutive cumulants shows a regime of pure diffusion for small disorder, in which P(D) goes to delta(D-<D>), and a regime of strong disorder where the cumulants grow infinitely large and the diffusion process is ill defined. The boundary between these two regimes is associated with an unstable fixed-point and a subdiffusive behavior: <x**2>=Ct**(1-d/2). For the quenched case (n goes to 0) we find that unphysical operators are generated raisng doubts on the renormalizability of this model. Implications to other random systems near their lower critical dimension are discussed.