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grand_canonical_partition_function [2016/03/04 16:52] nikolaj |
grand_canonical_partition_function [2016/03/04 18:10] 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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