Large time nonequilibrium dynamics of a particle in a random potential.

We study the o -equilibrium dynamics of a particle in a general N dimensional random potential whenN !1. We demonstrate the existence of two asymptotic time regimes: i. stationary dynamics, ii. slow aging dynamics with violation of equilibrium theorems. We derive the equations obeyed by the slowly varying part of the two-times correlation and response functions and obtain an analytical solution of these equations. For short-range correlated potentials we nd that: i. the scaling function is non analytic at similar times and this behaviour crosses over to ultrametricity when the correlations become long range, ii. aging dynamics persists in the limit of zero con ning mass with universal features for widely separated times. We compare with the numerical solution to the dynamical equations and generalize the dynamical equations to nite N by extending the variational method to the dynamics. 2