SciPost Phys. 3, 018 (2017) ·
published 4 September 2017

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We analyze the classical problem of the stochastic dynamics of a particle
confined in a periodic potential, through the so called Il'in and Khasminskii
model, with a novel semianalytical approach. Our approach gives access to the
transient and the asymptotic dynamics in all damping regimes, which are
difficult to investigate in the usual Brownian model. We show that the
crossover from the overdamped to the underdamped regime is associated with the
loss of a typical time scale and of a typical length scale, as signaled by the
divergence of the probability distribution of a certain dynamical event. In the
underdamped regime, normal diffusion coexists with a non Gaussian displacement
probability distribution for a long transient, as recently observed in a
variety of different systems. We rationalize the microscopic physical processes
leading to the nonGaussian behavior, as well as the timescale to recover the
Gaussian statistics. The theoretical results are supported by numerical
calculations and are compared to those obtained for the Brownian model.