Liouville volume preservation

Take the divergence of the phase-space velocity field by differentiating each position rate with respect to its position and each momentum rate with respect to its momentum. Hamilton's equations turn these terms into mixed second derivatives of H with opposite signs. For a twice continuously differentiable Hamiltonian, the derivatives commute and the sum vanishes. The argument works even when H depends explicitly on time: energy can change while canonical phase volume remains preserved.

The volume element here is the canonical product of all position and momentum differentials. In noncanonical variables, its expression may include a coordinate-dependent measure factor. A compressible-looking coordinate plot can therefore be misleading if the correct measure is omitted. Conversely, an arbitrary divergence-free vector field in a higher-dimensional space need not be Hamiltonian. Liouville preservation is a consequence of the canonical structure, but preservation of volume alone is not its full definition.

Let ρ be an ensemble density relative to canonical volume. Conservation of ensemble probability gives a continuity equation. Since the Hamiltonian velocity field is divergence-free, that equation states that the fine-grained density is constant along each trajectory. A fixed laboratory point can still see density increase or decrease as different parts of the ensemble pass through it. Material constancy and constancy at a fixed coordinate are different derivatives, just as in ordinary fluid transport.

For an autonomous Hamiltonian, any sufficiently regular density that is a function of H alone is stationary, because its bracket with H vanishes. That provides a class of equilibrium distributions but not a proof that every initial ensemble approaches one. Fine filaments can develop indefinitely under reversible motion. Coarse-graining averages over those filaments and may produce an apparently smoother distribution, introducing an observational procedure beyond the exact Liouville equation.

A free particle shears phase space: momentum stays fixed while position shifts by momentum times elapsed time divided by mass. A rectangular patch becomes a parallelogram with the same area. An oscillator instead rotates an appropriately scaled phase plane. Both motions preserve canonical volume, despite very different shapes and transport. The free shear is a useful warning against picturing incompressibility as rigid motion or as conservation of every distance between nearby trajectories.

Recurrence needs additional hypotheses, notably a finite invariant measure for the relevant region and measure-preserving evolution. Even then the statement concerns almost-everywhere return arbitrarily close under appropriate conditions, not an exact short period for every point. Mixing and ergodicity are separate dynamical properties and do not follow merely from zero divergence. Friction in the ordinary oscillator phase plane gives the contrasting case: its flow contracts area, signaling that the closed canonical two-variable description has been abandoned.