# Differences

This shows you the differences between two versions of the page.

 classical_phase_density [2015/08/18 20:27]nikolaj classical_phase_density [2015/08/18 20:29] (current)nikolaj Both sides previous revision Previous revision 2015/08/18 20:29 nikolaj 2015/08/18 20:27 nikolaj 2015/08/16 18:10 nikolaj 2015/08/16 16:09 nikolaj 2015/08/16 16:01 nikolaj 2015/08/16 15:58 nikolaj 2015/08/16 15:57 nikolaj 2015/08/16 15:33 nikolaj 2014/03/21 11:11 external edit2013/12/21 00:36 nikolaj 2013/11/06 19:07 nikolaj 2013/11/06 19:07 nikolaj 2013/11/05 19:03 nikolaj 2013/11/05 18:47 nikolaj 2013/11/05 18:44 nikolaj old revision restored (2013/11/04 19:20) 2015/08/18 20:29 nikolaj 2015/08/18 20:27 nikolaj 2015/08/16 18:10 nikolaj 2015/08/16 16:09 nikolaj 2015/08/16 16:01 nikolaj 2015/08/16 15:58 nikolaj 2015/08/16 15:57 nikolaj 2015/08/16 15:33 nikolaj 2014/03/21 11:11 external edit2013/12/21 00:36 nikolaj 2013/11/06 19:07 nikolaj 2013/11/06 19:07 nikolaj 2013/11/05 19:03 nikolaj 2013/11/05 18:47 nikolaj 2013/11/05 18:44 nikolaj old revision restored (2013/11/04 19:20) Line 25: Line 25: ^ $\frac{\mathrm d}{\mathrm dt}{\hat\rho}(\pi(t),​t)=0$ ^ ^ $\frac{\mathrm d}{\mathrm dt}{\hat\rho}(\pi(t),​t)=0$ ^ where $\pi$ is the solution of the [[Hamiltonian equations]]. where $\pi$ is the solution of the [[Hamiltonian equations]]. - - === 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]]). - - 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 - - $\frac{V}{(\hbar c)^3}\int {\mathrm d}E\, f(\frac{V}{(\hbar c)^3}E)$. === Reference === === Reference ===