 Dissipative Liouville equation

Conservation of probability under a deterministic characteristic flow map with velocity $b(z,t)$ gives the displayed continuity equation on phase space. Along a characteristic, $dP/dt=-P{\nabla }_z\cdot b$. Unlike the Hamiltonian Liouville equation, it allows contraction of phase-space volume and increasing density along trajectories. It still preserves total probability when boundary fluxes vanish.