Random diffusion model

We study the random diffusion model. This is a continuum model for a conserved scalar density field varphi driven by diffusive dynamics. The interesting feature of the dynamics is that the bare diffusion coefficient D is density dependent. In the simplest case, D=D[over ]+D_{1}deltavarphi , where D[over ] 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 D1 . We develop perturbation theory to fourth order in D1 . 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 wave number 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 critical properties of the direct approach survive, going to higher order in perturbation theory.