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classical_hamiltonian_system [2015/08/16 15:29]
nikolaj
classical_hamiltonian_system [2015/08/16 15:30] (current)
nikolaj
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 === Discussion === === Discussion ===
 The Hamiltonian function is related to the Lagrangian function via Legendre transformation. The Hamiltonian function is related to the Lagrangian function via Legendre transformation.
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-== Volume in statistical physics == 
-It's worth noting that $\hbar$ translates energy to frequency (or time) and $c$ further translates time to length. This way one 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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-$\frac{V}{(\hbar c)^3}\int {\mathrm d}E\, f(\frac{V}{(\hbar c)^3}E)$. 
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-Such a volume $V$ may actually also not just be a parameter of the theory, but fixed by the space ${\mathcal M}$ - e.g. the integral over the space, if such a number exists. ​ 
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-This e.g. happens in [[Classical density of states]] and is how $V$ enters $U=\langle{H}\rangle$ and eventually $p := -\frac{\partial U}{\partial V} = \frac{N}{V}\cdot k_B T$. 
  
 === Reference === === Reference ===
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