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classical_phase_density [2015/08/16 18:10] nikolaj |
classical_phase_density [2015/08/18 20:29] nikolaj |
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| ==== Set ==== | ==== Set ==== | ||
| | @#55CCEE: context | @#55CCEE: $ \langle \mathcal M, H\rangle$ ... classical Hamiltonian system | | | @#55CCEE: context | @#55CCEE: $ \langle \mathcal M, H\rangle$ ... classical Hamiltonian system | | ||
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| | @#FFBB00: definiendum | @#FFBB00: $ {\hat\rho} \in \mathrm{it} $ | | | @#FFBB00: definiendum | @#FFBB00: $ {\hat\rho} \in \mathrm{it} $ | | ||
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| | @#55EE55: postulate | @#55EE55: $\langle \mathcal M, H\rangle$ ... Hamiltonian system | | | @#55EE55: postulate | @#55EE55: $\langle \mathcal M, H\rangle$ ... Hamiltonian system | | ||
| | @#DDDDDD: range | @#DDDDDD: $ \Gamma_{\mathcal M} \equiv \mathcal M\times T\mathcal M $ | | | @#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_+ $ | | | @#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) $ | | | @#DDDDDD: range | @#DDDDDD: $\hat\rho:: \hat\rho({\bf q},{\bf p},t) $ | | ||
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| - | Continuity equation: | ||
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| | @#55EE55: postulate | @#55EE55: $ \frac{\partial}{\partial t}{\hat\rho} = - \nabla ({\hat\rho} \cdot X_H )$ | | | @#55EE55: postulate | @#55EE55: $ \frac{\partial}{\partial t}{\hat\rho} = - \nabla ({\hat\rho} \cdot X_H )$ | | ||
| - | >todo: Total derivative | + | >todo: Total derivative for the 'Continuity equation' (last postulate) |
| >todo: Hamiltonian vector field | >todo: Hamiltonian vector field | ||
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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 === | ||
