Inertial effects in the short-range Toy Model

¹ CNRS Laboratoire de Physique Théorique de l'ENS, 24 rue Lhomond, 75231 Paris CEDEX 05, France
² DAMTP, University of Cambridge, Silver Street, Cambridge, CB3 9EW, Cambridge, UK

Received: 10 December 1997
Accepted: 12 March 1998

Abstract

We examine the dynamics of the so-called Toy Model with an added inertial term. The problem is essentially the Kramers problem for a massive particle in a flow field given by the gradient of a quenched Gaussian random potential. The correlations of the potential are short range and there is no restoring harmonic term. When the dynamics are treated in the Hartree approximation (which becomes exact when the dimension of the space becomes infinite) the low-disorder (or high-temperature) regime is diffusive and we examine the effect of the inertial term on the asymptotic diffusion constant. The results of our calculations are compared with numerical simulations of the problem. We find that agreement with the simulations in three dimensions is rather good. Above a critical value of the disorder the variational calculation suggests the existence of a dynamical transition with a non-zero anomaly. However, we find no numerical evidence for such a transition in finite dimensions and suggest that it is a pathology of the large dimensional limit.