===== Classical phase density ===
==== Set ====
| @#55CCEE: context     | @#55CCEE: $ \langle \mathcal M, H\rangle$ ... classical Hamiltonian system |
| @#FFBB00: definiendum | @#FFBB00: $ {\hat\rho} \in \mathrm{it} $ |
| @#55EE55: postulate   | @#55EE55: $\langle \mathcal M, H\rangle$ ... Hamiltonian system |
| @#DDDDDD: range       | @#DDDDDD: $ \Gamma_{\mathcal M} \equiv \mathcal M\times T\mathcal M $ |
| @#55EE55: postulate   | @#55EE55: $\hat\rho: \Gamma_{\mathcal M} \times \mathbb R \to \mathbb R_+ $ |
| @#DDDDDD: range       | @#DDDDDD: $\hat\rho:: \hat\rho({\bf q},{\bf p},t) $ |
| @#55EE55: postulate   | @#55EE55: $ \frac{\partial}{\partial t}{\hat\rho} = - \nabla ({\hat\rho} \cdot X_H )$ |

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

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=== Discussion ===
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:
=== Theorems ===
Liouville equation
^ $ \left(\frac{\mathrm \partial}{\mathrm \partial t}+X_H\cdot\nabla\right){\hat\rho}=0 $ ^
which can also be written as
^ $ \frac{\mathrm d}{\mathrm dt}{\hat\rho}(\pi(t),t)=0 $ ^
where $\pi$ is the solution of the [[Hamiltonian equations]].

=== Reference ===
Wikipedia: 
[[http://en.wikipedia.org/wiki/Continuity_equation|Continuity equation]],
[[http://en.wikipedia.org/wiki/Liouville%27s_theorem_%28Hamiltonian%29|Liouville equations]],
[[https://en.wikipedia.org/wiki/Cluster_expansion|Cluster expansion]]

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=== Refinement of ===
[[ODE system]]
=== Context ===
[[Classical Hamiltonian system]]
=== Related ===
[[Hamiltonian equations]]