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grand_canonical_partition_function [2016/03/04 16:52]
nikolaj
grand_canonical_partition_function [2016/03/04 18:10] (current)
nikolaj
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 $= \prod_r \sum_{N=0}^{N^\text{max}} (e^{-\beta\ (\varepsilon_r-\mu)})^N $ $= \prod_r \sum_{N=0}^{N^\text{max}} (e^{-\beta\ (\varepsilon_r-\mu)})^N $
  
-$= \begin{cases} \prod_r \frac{1}{1-e^{-\beta\ (\varepsilon_r-\mu)}} & \mathrm{if}\ N^\text{max}=\infty \\\\ \prod_r(1+e^{-\beta\ (h_r-\mu)}) & \mathrm{if}\ N^\text{max}=1 \end{cases}$ ​+$= \begin{cases} \prod_r \frac{1}{1-e^{-\beta\ (\varepsilon_r-\mu)}} & \mathrm{if}\ N^\text{max}=\infty \\\\ \prod_r(1+e^{-\beta\ (\varepsilon_r-\mu)}) & \mathrm{if}\ N^\text{max}=1 \end{cases}$ ​
  
 where $\varepsilon_r$ are the energy eigenvalues we obtain where $\varepsilon_r$ are the energy eigenvalues we obtain
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 The $\propto$ expresses takes into account the possible multiplicity of states $r,​r',​\cdot$ with equal energy $\varepsilon_r,​\varepsilon_{r'​},​\dots$. The $\propto$ expresses takes into account the possible multiplicity of states $r,​r',​\cdot$ with equal energy $\varepsilon_r,​\varepsilon_{r'​},​\dots$.
  
-The above derivation presupposes that the different energy values are given by the discretely indexed expressions of the form $\sim N \varepsilon$. This is the historical beginning of quantum ​mechanics.  +The above derivation presupposes that the different energy values are given by the discretely indexed expressions of the form $\sim N \varepsilon$. This is the historical beginning of quantum ​mechaniscs
- +
-For spin 1 bosons, if we take a continuum limit (see discussing in [[Classical density of states|density of states]]) and write $\varepsilon_\omega\equiv \hbar \omega$, the [[Statistical internal energy|internal energy]] $U\equiv\langle H\rangle$ is +
- +
-^ $U=V\int_0^\infty \left(\frac{\hbar\omega^3}{\pi^2 c^3}\frac{1}{\mathrm e^{\beta\hbar\omega}-1}\right)\mathrm d\omega$ ^ +
- +
-The formlua for the integrand, the energy density, is known as //​Planck'​s law//.+
  
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