INERTIAL EFFECTS IN THE SHORT-RANGE TOY MODEL

We examine the dynamics of the so-called Toy Model with an added inert ial 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 a nd there is no restoring harmonic term. When the dynamics are treated in the Hartree approximation (which becomes exact when the dimension o f the Space becomes infinite) the low-disorder (or high-temperature) r egime is diffusive and we examine the effect of the inertial term on t he asymptotic diffusion constant. The results of our calculations are compared with numerical simulations of the problem. We find that agree ment with the simulations in three dimensions is rather good. Above a critical value of the disorder the. variational calculation suggests t he existence of a dynamical transition with a non-zero anomaly. Howeve r, me find no numerical evidence for such a transition in finite dimen sions and suggest that it is a pathology of the large dimensional limi t.