For the grand canonical ensemble, obtain an expression for the ensemble average energy and ensemble average particle numbers.
In the grand canonical ensemble, a system is in contact with a reservoir that can exchange both energy and particles. The grand canonical partition function is defined as [ \mathcal{Z} = \sum_{N=0}^{\infty} \sum_{{i}} e^{-\beta (E_i - \mu N)}, ] where (E_i) is the energy of the microstate (i) with (N) particles, (\mu) is the chemical potential, (\beta = ____ ____ ___ ___ ____ _________ ________ ____ _________ ________ ______.
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