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

We study the nonequilibrium dynamics of a particle in a general N-dimensional random potential when N --> infinity. We demonstrate the existence of two asymptotic time regimes: (i) stationary dynamics and (ii) slow aging dynamics with violation of equilibrium theorems. We derive the equations obeyed by the slowly varying part of the two-time correlation and response functions and obtain an analytical solution of these equations. For short-range correlated potentials we find that (i) the scaling function is nonanalytic at similar times and this behavior crosses over to ultrametricity when the correlations become long range and (ii) aging dynamics persists in the limit of zero confining mass with universal features for widely separated times. We compare the numerical solution to the dynamical equations and generalize the dynamical equations to finite N by extending the variational method to the dynamics.