The generalized Langevin equation: long-time behavior and diffusive transport in a periodic potential

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Résumé : In this presentation, we study the diffusive limit of solutions to the generalized Langevin equation (GLE) in a periodic potential. We begin by presenting sharp estimates on the rate of convergence to equilibrium for a simple Markovian approximation of the GLE. We then present asymptotic results for the effective diffusion coefficient in the small correlation time, overdamped and underdamped regimes. Finally, we employ a recently developed numerical method in order to calculate the effective diffusion coefficient for a wide range of friction coefficients, confirming our asymptotic results and corroborating the findings of earlier studies on the subject.