# Diffusion Phenomena in a Mixed Phase Space

## TL;DR

This paper analytically derives average particle velocities in strongly chaotic systems using Brownian dynamics and diffusion equations, validated on a simplified Fermi-Ulam model with mixed phase space.

## Contribution

It introduces a novel analytical method for calculating average velocities in chaotic systems with mixed phase space, extending to time-dependent billiards.

## Key findings

- Analytical velocities match numerical simulations.
- Method applicable to various time-dependent billiard systems.
- Provides a new approach for understanding diffusion in chaotic dynamics.

## Abstract

We show that, in strongly chaotic dynamical systems, the average particle velocity can be calculated analytically by consideration of Brownian dynamics in phase space, the method of images and use of the classical diffusion equation. The method is demonstrated on the simplified Fermi-Ulam accelerator model, which has a mixed phase space with chaotic seas, invariant tori and Kolmogorov-Arnold-Moser (KAM) islands. The calculated average velocities agree well with numerical simulations and with an earlier empirical theory. The procedure can readily be extended to other systems including time-dependent billiards.