Phase space volume element

The natural measure for integration over classical phase space, preserved by Hamiltonian time evolution.

Phase space volume element

For a system with $n$ degrees of freedom and phase space $\Gamma = \mathbb{R}^{2n}$ , the phase space volume element is

d\Gamma = d^n q \, d^n p = dq_1 \cdots dq_n \, dp_1 \cdots dp_n.

This is the standard Lebesgue measure on $\mathbb{R}^{2n}$ , and more generally corresponds to the Liouville measure on a symplectic manifold .

Liouville’s theorem

A fundamental property is that Hamiltonian time evolution preserves the phase space volume: if $\Phi_t$ denotes the time- $t$ flow generated by a Hamiltonian $H$ , then

\Phi_t^* (d\Gamma) = d\Gamma.

This is the classical statement of Liouville’s theorem and underlies the consistency of equilibrium statistical mechanics.

In statistical mechanics

The microcanonical ensemble assigns equal probability to equal phase space volumes on the energy shell. The canonical and grand canonical measures are absolutely continuous with respect to $d\Gamma$ .