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We analyze the classical problem of the stochastic dynamics of a particleconfined in a periodic potential, through the so called Il'in and Khasminskiimodel, with a novel semi-analytical approach. Our approach gives access to thetransient and the asymptotic dynamics in all damping regimes, which aredifficult to investigate in the usual Brownian model. We show that thecrossover from the overdamped to the underdamped regime is associated with theloss of a typical time scale and of a typical length scale, as signaled by thedivergence of the probability distribution of a certain dynamical event. In theunderdamped regime, normal diffusion coexists with a non Gaussian displacementprobability distribution for a long transient, as recently observed in avariety of different systems. We rationalize the microscopic physical processesleading to the non-Gaussian behavior, as well as the timescale to recover theGaussian statistics. The theoretical results are supported by numericalcalculations and are compared to those obtained for the Brownian model.

Unifying description of the damping regimes of a stochastic particle in a periodic potential