[QUE/SM-06005] SM-PROBLEM

Calculate the fluctuation in energy $(\overline{\Delta E})^2\,=\,\overline{E^2}\,-\,\overline{E}^2$ for the grand canonical ensemble. You may find the equations ( derive them!) useful: $$\overline{\Delta E})^2\,=\,-\left(\frac{\partial\overline{E}}{\partial\beta}\right)_{\mu,V}\,+\,\frac{\mu}{\beta}\left(\frac{\partial \overline{E}}{\partial \mu}\right)_{T,V} $$ $$ \left(\frac{\partial \overline{E}}{\partial T}\right)_{\mu,V}\,=\,\left(\frac{\partial \overline{E}}{\partial T}\right)_{\overline{N},V}\,+\,\left(\frac{\partial \overline{E}}{\partial\overline{N}}\right)_{T,V}\left(\frac{\partial \overline{N}}{\partial T}\right)_{\mu,V} $$ $$ \left(\frac{\partial \mu}{\partial T}\right)_{\overline{N},V} \,=\,-\frac{\left(\frac{\partial \overline{N}}{\partial T}\right)_{\mu,V}}{\left(\frac{\partial \overline{N}}{\partial \mu}\right)_{T,V}} $$ The Maxwell's relation $$ \left(\frac{\partial\mu}{\partial T}\right)_{\overline{N},V}\,=\,-\left(\frac{\partial S}{\partial \mu}\right)_{T,V} $$ Also the relation derived in class ( you can assume this ) $$ \overline{\Delta N^2}\,=\,\left(\frac{\partial\overline {N}}{\partial\mu}\right)_{T,V} $$