Show that the average quadratic deviation of energy in the grand canonical ensemble may be written as

Show that the average quadratic deviation of energy in the grand canonical ensemble may be written as

where U = (E) is the thermodynamic internal energy in terms of β, x, and V . Hence, show that we may also write

(you may use the Jacobian transformations of Appendix A.5). From this last expression, show that

where

is the average quadratic deviation of energy in the canonical ensemble. Finally, show that

since