Classical phase density

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Set

context	$ \langle \mathcal M, H\rangle$ … classical Hamiltonian system
definiendum	$ {\hat\rho} \in \mathrm{it} $
postulate	$\langle \mathcal M, H\rangle$ … Hamiltonian system
range	$ \Gamma_{\mathcal M} \equiv \mathcal M\times T\mathcal M $
postulate	$\hat\rho: \Gamma_{\mathcal M} \times \mathbb R \to \mathbb R_+ $
range	$\hat\rho:: \hat\rho({\bf q},{\bf p},t) $
postulate	$ \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

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.

Volume in statistical physics

A characteristic volume $V$ may be given by an integral over the spatial part of ${\mathcal M}$. This is e.g. how $V$ arises in the statistical mechanics derivation in the classical setting of the ideal gas law $p := -\frac{\partial}{\partial V}\langle{H}\rangle = \frac{N}{V}\cdot k_B T$. See also Cluster expansion. Introducing the density $n=\frac{N}{V}$, this holds true for infinite volumes. In the derivation via quantum gases in an infinite volume, a volume parameter is introduced in when the momenta are quantized (see Classical density of states).

A remark on the latter case: Note that the physical constants $\hbar$ and $c$ can be used to translate energy to frequency (or time) and further translates time to length. Using this, we can write down models involving a volume parameter $V$, defining a characteristic energy $\frac{(\hbar c)^3}{V}$. This may then e.g. be embedded via (unitless!) expressions as complicated as

$\frac{V}{(\hbar c)^3}\int {\mathrm d}E\, f(\frac{V}{(\hbar c)^3}E)$.