Diffusive behavior for randomly kicked Newtonian particles in a spatially periodic medium

TL;DR

This paper proves a central limit theorem for the momentum of a randomly kicked Newtonian particle in a periodic medium, showing superdiffusive behavior of the position with a $t^{3/2}$ scaling.

Contribution

It establishes a new CLT for the momentum and superdiffusive scaling for position in a frictionless, periodically forced particle system.

Findings

Momentum converges to a Brownian motion after rescaling.

Position exhibits superdiffusive scaling with time $t^{3/2}$.

High energy states dominate the particle's long-term behavior.

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.