Random Diffusion Model

Details Random Diffusion Model Random Diffusion Model

2007-10-02

arxiv.org/abs/0710.0641v1

Physics

Condensed Matter

Soft Condensed Matter

71 pages, 27 figures

Scientific paper

10.1103/PhysRevE.78.031123

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.

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