Dissipative Liouville equation (${\partial }_{t}P+{\nabla }_z\cdot (bP)=0$)

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.

The divergence of the vector field generating a flow on phase space. Its time integral determines the Jacobian determinant of the characteristic flow map: $d\log |\det D{\Phi }_t|/dt=({\nabla }_z\cdot b)({\Phi }_t,t)$. A Hamiltonian vector field has zero phase-space divergence in canonical coordinates.