Mixing in an anharmonic potential well

TL;DR

This paper proves phase-space mixing for integrable systems in anharmonic potential wells, showing convergence rates depend on non-harmonicity and regularity conditions, with implications for understanding long-term behavior of such systems.

Contribution

It establishes phase-space mixing and convergence rates for solutions to Liouville's equation in anharmonic potentials, including cases where non-harmonicity conditions fail.

Findings

Weak convergence with rate ~1/time under non-harmonicity

Slower mixing rate when non-harmonicity fails at certain energies

Faster convergence for higher regularity functions

Abstract

We prove phase-space mixing for solutions to Liouville's equation for integrable systems. Under a natural non-harmonicity condition, we obtain weak convergence of the distribution function with rate $\langle \mathrm{time} \rangle^{-1}$. In one dimension, we also study the case where this condition fails at a certain energy, showing that mixing still holds but with a slower rate. When the condition holds and functions have higher regularity, the rate can be faster.