# Probability for a phase space flow to return to its original state

A Hamiltonian system of $n$ interacting atoms, each of mass $M$, is confined within a cubical box of sides $V$. The average initial speed of each particle is $v$.

How do I estimate the timescale for the system to return close to its initial state so that each particle is within $L$ of its initial location and with a momentum vector $p$ satisfying $|p − p_\text{initial}| < Δp$?

I'm thinking about using Liouville's theorem, which states that "volume in phase space is conserved during the motion of a Hamiltonian system". I guess that I could find the ratio of the original phase space volume to the total phase space volume accessible... but then I'm not sure how to proceed onwards.

• – Bzazz May 24 '18 at 15:48