Geometrical approach for the mean-field dynamics of a particle in a short range correlated random potential

We consider the zero temperature relaxational dynamics of a particle in a random potential with short range correlations. We first obtain a set of “two-times” mean-field equations (including the case of a finite, constant, driving force), and we present detailed results coming from a numerical integration of these equations. We restrict ourselves to the situation where the spatial correlations of the random potential decrease exponentially (otherwise our geometrical analysis fails). It is possible, in this case, to compute the spectrum of the Hessian of the energy landscape, and we subsequently propose a geometrical description of the “mean field aging” behavior. Our numerical results combined with further analytical arguments finally lead to the waiting-time dependence of the main characteristic time scales.