d ρ d t = ∂ ρ ∂ t + <munderover> ∑ i = 1 n </munderover> ( ∂ ρ ∂ q i q ˙ i + ∂ ρ ∂ p i p ˙ i )
Consider a Hamiltonian dynamical system with canonical coordinates and conjugate momenta. Then the phase space distribution determines the probability that the system will be found in the infinitesimal phase space volume.
Note the similarity with the quantum Liouville (von Neumann) equation where the Poisson brackets {.,.} are replaced by commutator brackets [.,.]
Classical Liouville Equation
partial differential equation for the that time rate of change of density of points in phase space