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If $Z=Z(z,V,T)$ is the Grand canonical Partition function, $\beta =\frac{1}{k_BT}$,$z=e^{\beta \mu }$ is the fugacity and $\epsilon_{\vec p}$ is the energy of a single particle in pth momentum state, then how can we derive the relation for the average occupation number:$$<n_{\vec p}>= -\frac{1}{\beta}\frac{\partial lnZ(z,V,T) }{\partial \epsilon_{\vec p}}$$ Here, z and T are constant

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  • $\begingroup$ Have been searching it at Huang's book, but haven't found it yet. $\endgroup$ Jan 6, 2015 at 4:50

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The grand partition function is given by $$Z(z,V,T)=\sum_{N=0}^{\infty}[z^N\sum_{\{n_{\vec{p}}\}}e^{-\beta\sum_{\vec{p}}n_{\vec{p}}\epsilon_{\vec{p}}}]$$ where $N$ is the total number of particles and $\{n_{\vec{p}}\}$ means sum over the distributed occupation numbers that subject to the constraint: $$\sum_{\vec{p}}n_{\vec{p}}=N$$ Then as usual, the average occupation number for a state labeled by $\vec{p}$ would be given by: $$<n_{\vec{p}}>=\frac{\sum_{N=0}^{\infty}[z^N\sum_{\{n_{\vec{p}}\}}n_{\vec{p}}e^{-\beta\sum_{\vec{p}}n_{\vec{p}}\epsilon_{\vec{p}}}]}{\sum_{N=0}^{\infty}[z^N\sum_{\{n_{\vec{p}}\}}e^{-\beta\sum_{\vec{p}}n_{\vec{p}}\epsilon_{\vec{p}}}]}$$ which is exactly $$-\frac{1}{\beta}\frac{1}{Z}\frac{\partial Z}{\partial \epsilon_{\vec{p}}}=-\frac{1}{\beta}\frac{\partial \mathrm{ln}Z}{\partial \epsilon_{\vec{p}}}$$

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