Aging and relaxation dynamics in free-energy landscapes with multiple minima

We consider the stochastic dynamics of a system thermally relaxing in a free-energy landscape with multiple attractors, and show that lack of translational homogeneity in this landscape leads to aging effects, e.g. to the dependence of the susceptibilities on the time elapsed from a thermal quench to the imposition of the probing field. We then prove an inequality between response and correlation which generalizes the fluctuation dissipation theorem to a situation far from thermodynamical equilibrium. As an application and a check we specialize our formalism in a way which we suggest is appropriate for spin-glass systems: we assume a hierarchical organization of the landscape, and find aging behavior in the response curves in good agreement with relevant experimental data. We finally conclude with a summary and a brief discussion of different approaches to slow relaxation in complex systems.