Fluctuations and Complements

Abstract

Characterize the fluctuations of the energy in the grand canonical ensemble and prove that

$$\left\langle {{{\left( {\Delta E} \right)}^2}} \right\rangle = {\left\langle {{{\left( {\Delta E} \right)}^2}} \right\rangle _{can}} + \left\langle {{{\left( {\Delta N} \right)}^2}} \right\rangle \left( {\frac{{\partial U}}{{\partial N}}} \right)_{T,V}^2$$

where U=〈E〉 is the average energy, N the average number of particles, 〈(ΔN)²〉 its fluctuations, and 〈(ΔE)²〉 _can the energy fluctuations as obtained from the canonical ensemble.