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classical_phase_density [2015/08/18 20:27]
nikolaj
classical_phase_density [2015/08/18 20:29] (current)
nikolaj
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 ^ $ \frac{\mathrm d}{\mathrm dt}{\hat\rho}(\pi(t),​t)=0 $ ^ ^ $ \frac{\mathrm d}{\mathrm dt}{\hat\rho}(\pi(t),​t)=0 $ ^
 where $\pi$ is the solution of the [[Hamiltonian equations]]. where $\pi$ is the solution of the [[Hamiltonian equations]].
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-=== 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 [[https://​en.wikipedia.org/​wiki/​Cluster_expansion|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]]). 
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-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  
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-$\frac{V}{(\hbar c)^3}\int {\mathrm d}E\, f(\frac{V}{(\hbar c)^3}E)$. 
  
 === Reference === === Reference ===
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