Mixing in anharmonic potential well

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 ⟨time⟩−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.