A Brownian particle in a microscopic periodic potential

Details A Brownian particle in a microscopic periodic potential A Brownian particle in a microscopic periodic potential

2011-10-04

arxiv.org/abs/1110.0688v1

Physics

Mathematical Physics

74 pages

Scientific paper

We study a model for a massive test particle in a microscopic periodic potential and interacting with a reservoir of light particles. In the regime considered, the fluctuations in the test particle's momentum due to collisions typically outweigh the shifts in momentum due to the periodic force, and so, the force is effectively a perturbative contribution. The mathematical starting point is an idealized reduced dynamics for the test particle given by a linear Boltzmann equation. In the limit that the mass ratio of a single reservoir particle to the test particle tends to zero, we show that under the standard normalizations for the test particle variables, there is convergence to the Ornstein-Uhlenbeck process. Our analysis is primarily directed towards bounding the perturbative effect of the periodic potential on the particle's momentum.

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