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Exercise. This is the solution to exercise 2.1.1 in the book.

Solution. In the context of the grand canonical ensemble with only one chemical species the functions ${Q}_{N}$ are probability distributions over the state of subsystems of the ensemble of volume $V$. Such subsystems are characterized by number of particles $N$ and total energy ${U}_{m}$. Thus ${Q}_{N}\left({U}_{m}\right)$ is the probability that a subsystem has $N$ particles and is in energy state ${U}_{m}$.

In the grand-canonical ensemble:

$${U}_{m}=TS-PV+N\mu $$The probability of a state is determined by its entropy (because entropy measures the number of possible configurations out of the total number of configurations available). In this case,

$${Q}_{N}\left({U}_{m}\right)={e}^{-S\u2215k}$$where $S$ is the entropy consistent with the number of particles and the energy of the system.

$${Q}_{N}\left({U}_{m}\right)={e}^{-\frac{PV}{kT}-\frac{{U}_{m}}{kT}+\frac{N\mu}{kT}}$$

The normalization condition in this case is:

$$\sum _{N=0}^{\infty}\sum _{m=0}^{\infty}{Q}_{N}\left({U}_{m}\right)=1$$