On (in)elastic non-dissipative Lorentz gases and the (in)stability of classical pulsed and kicked rotors

TL;DR

This paper investigates the dynamics of particles in various Lorentz gas models, revealing phenomena like Fermi acceleration, diffusive behavior, and instability of pulsed and kicked rotors across different dimensions through numerical and theoretical analysis.

Contribution

It extends the random walk framework to multiple Lorentz gas models, demonstrating new growth behaviors and instability results for pulsed and kicked rotors in higher dimensions.

Findings

Fermi acceleration with kinetic energy growing as t^{2/5} in periodic scatterers

Diffusive motion with diffusion constant scaling as |p_0|^{5} in elastic Lorentz gases

Instability of pulsed and kicked rotors in dimensions d≥2 and all dimensions respectively

Abstract

We study numerically and theoretically the $d$-dimensional Hamiltonian motion of fast particles through a field of scatterers, modeled by bounded, localized, (time-dependent) potentials, that we refer to as (in)elastic non-dissipative Lorentz gases. We illustrate the wide applicability of a random walk picture previously developed for a field of scatterers with random spatial and/or time-dependence by applying it to four other models. First, for a periodic array of spherical scatterers in $d\geq2$, with a smooth (quasi)periodic time-dependence, we show Fermi acceleration: the ensemble averaged kinetic energy $\left<\|p(t)\|^2\right>$ grows as $t^{2/5}$. Nevertheless, the mean squared displacement $\left<\|q(t)\|^2\right>\sim t^2$ behaves ballistically. These are the same growth exponents as for random time-dependent scatterers. Second, we show that in the soft elastic and periodic Lorentz gas, where the particles' energy is conserved, the motion is diffusive, as in the standard hard Lorentz gas, but with a diffusion constant that grows as $\|p_0\|^{5}$, rather than only as $\|p_0\|$. Third, we note the above models can also be viewed as pulsed rotors: the latter are therefore unstable in dimension $d\geq 2$. Fourth, we consider kicked rotors, and prove them, for sufficiently strong kicks, to be unstable in all dimensions with $\left<\|p(t)\|^2\right>\sim t$ and $\left<\|q(t)\|^2\right>\sim t^3$. Finally, we analyze the singular case $d=1$, where $\left< \|p(t)\|^2\right>$ remains bounded in time for time-dependent non-random potentials whereas it grows at the same rate as above in the random case.