Liouville's theorem

Liouville's theorem is the statement that phase space flows like an incompressible fluid. Imagine coloring a small blob of possible states -- a little region of positions and momenta -- and letting Hamilton's equations carry every point in it forward in time. The blob will stretch, shear, and fold into fantastically complicated shapes, but its total volume never changes. Density in phase space is conserved along the motion.

Precisely, let rho(q, p, t) be the density of representative points in phase space. Liouville's theorem says the total (convective) derivative vanishes: d rho / dt = partial rho / partial t + {rho, H} = 0, equivalently partial rho / partial t = - {rho, H}. This follows because the Hamiltonian flow is divergence-free in phase space: sum_i (partial q_dot_i / partial q_i + partial p_dot_i / partial p_i) = sum_i (partial^2 H /(partial q_i partial p_i) - partial^2 H /(partial p_i partial q_i)) = 0, the two mixed partials cancelling exactly -- this is where Hamilton's crucial minus sign earns its keep. The 2n-dimensional phase-space volume element dq dp is therefore invariant under time evolution and, more generally, under any canonical transformation.

This is the theorem that makes statistical mechanics possible. It justifies using phase-space volume as the natural measure for counting states, underlies the equal-a-priori-probability postulate of the microcanonical ensemble, and forbids trajectories from being squeezed toward attractors -- Hamiltonian systems cannot have the volume-shrinking attractors that dissipative systems do. It is also why you cannot beat the uncertainty principle classically: you can move phase-space area around, but never compress it.