Abstract
We prove a central limit theorem for the momentum distribution of a particle undergoing an unbiased spatially periodic random forcing at exponentially distributed times without friction. The start is a linear Boltzmann equation for the phase space density, where the average energy of the particle grows linearly in time. Rescaling time, the momentum converges to a Brownian motion, and the position is its time-integral showing superdiffusive scaling with time t 3/2. The analysis has two parts: (1) to show that the particle spends most of its time at high energy, where the spatial environment is practically invisible; (2) to treat the low energy incursions where the motion is dominated by the deterministic force, with potential drift but where symmetry arguments cancel the ballistic behavior.
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Communicated by G. Gallavotti
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Clark, J., Maes, C. Diffusive Behavior for Randomly Kicked Newtonian Particles in a Spatially Periodic Medium. Commun. Math. Phys. 301, 229–283 (2011). https://doi.org/10.1007/s00220-010-1149-x
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DOI: https://doi.org/10.1007/s00220-010-1149-x