Liouville measure

Definition

Given a symplectic manifold $(M,\omega )$ of dimension $2n$, the Liouville measure is the top-degree volume form

which in local Darboux (canonical) coordinates $(q^1,\dots ,q^n,p_1,\dots ,p_n)$ becomes

Key Properties

• Coordinate-invariant: $\mathrm{\Omega }$ does not depend on the choice of canonical coordinates.
• Preserved by Hamiltonian flow: If ${\varphi }_t$ is the time-$t$ flow of any Hamiltonian $H$, then ${\varphi }_t^{\ast }\mathrm{\Omega }=\mathrm{\Omega }$. Liouville's theorem.
• σ-finite but non-normalizable: ${\int }_M\mathrm{\Omega }$ generally diverges on noncompact $M$.

Role in Statistical Mechanics

• “Counting” microstates
  For any measurable region $U\subset M$,$$\mathrm{\#}\{\text{microstates in }U\}\propto {\int }_U\mathrm{\Omega }\equiv {\int }_{U}d{q}^1\cdots d{q}^{n}d{p}_1\cdots d{p}_n.$$
• Reference volume for ensembles
  An ensemble is defined by a probability density $\rho (q,p)$ with
  ${\int }_M\rho \mathrm{\Omega }=1$.
  The ensemble measure is$$d{\mu }_{\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{e}\mathrm{m}\mathrm{b}\mathrm{l}\mathrm{e}}(q,p)=\rho (q,p)\mathrm{\Omega }.$$