Random Diffusion Model

TL;DR

This paper analyzes a continuum model of density-dependent diffusion, developing perturbation theory to understand its dynamics, including ergodic transitions and critical behavior, with implications for metastability and instability.

Contribution

It introduces a perturbation approach to the random diffusion model, revealing the nature of ergodic transitions and critical phenomena beyond one-loop order.

Findings

Mode coupling theory predicts an ergodic-nonergodic transition at one-loop order.

Higher-order perturbation shows the transition is not supported, indicating a more complex critical behavior.

A characteristic kinetic length grows as a power law in time near critical coupling.

Abstract

We study here the random diffusion model. This is a continuum model for a conserved scalar density field $\phi$ driven by diffusive dynamics. The interesting feature of the dynamics is that the {\it bare} diffusion coefficient $D$ is density dependent. In the simplest case $D=\bar{D}+D_{1}\delta \phi $ where $\bar{D}$ is the constant average diffusion constant. In the case where the driving effective Hamiltonian is quadratic the model can be treated using perturbation theory in terms of the single nonlinear coupling $D_{1}$. We develop perturbation theory to fourth order in $D_{1}$. The are two ways of analyzing this perturbation theory. In one approach, developed by Kawasaki, at one-loop order one finds mode coupling theory with an ergodic-nonergodic transition. An alternative more direct interpretation at one-loop order leads to a slowing down as the nonlinear coupling increases. Eventually one hits a critical coupling where the time decay becomes algebraic. Near this critical coupling a weak peak develops at a wavenumber well above the peak at $q=0$ associated with the conservation law. The width of this peak in Fourier space decreases with time and can be identified with a characteristic kinetic length which grows with a power law in time. For stronger coupling the system becomes metastable and then unstable. At two-loop order it is shown that the ergodic-nonergodic transition is not supported. It is demonstrated that the {\it critical} properties of the direct approach survive going to higher order in perturbation theory.