todo: Total derivative for the 'Continuity equation' (last postulate)
todo: Hamiltonian vector field

For all initial values $\pi(0)\in\Gamma_{\mathcal M}$, the solutions of the Hamiltonian equations of motion follow the Hamiltonian flow $X_H$. Because phase trajectories can't intersect (the Hamiltonian equations are first order in time), a given sub volume $\Sigma$ of $\Gamma_{\mathcal M}$ flows along $X_H$ with only smooth distortion of its boundary $\partial \Sigma$. Morally, the phase density counts the number of system points in any given subset of the phase volume: If we specify such a volume $\Sigma_{t_0}\subset \Gamma_{\mathcal M}$ where the index denotes some point in time, then $\int_{\Sigma_{t_0}}\hat\rho({\bf q},{\bf p},t_0)=\int_{\Sigma_{t_1}}\hat\rho({\bf q},{\bf p},t_1)$. The phase density doesn't literally count ensemble points, as there are be infinitely many. So $\hat\rho$ is assigned any initial value $\hat\rho({\bf q},{\bf p},0)$ which is soon factored out in a normalization, see Classical probability density function.

We denote the measure in $\Gamma_{\mathcal M}$ simply by $\mathrm d\Gamma$.

Using the Hamiltonian equations, we can pull out $X_H$ and get the Liouville equations:

Liouville equation

which can also be written as

where $\pi$ is the solution of the Hamiltonian equations.

Wikipedia: Continuity equation, Liouville equations, Cluster expansion

ODE system

Classical Hamiltonian system

Hamiltonian equations