The partition function of the grand canonical ensemble can be generally stated as $$ \mathcal{Z} = \sum_{r} e^{-\beta(E_{r} - N_r\mu)}\tag{1}$$ where $E_{r}$ is the energy of the micro-state $r$ of $N_r$ particles, $\beta$ the usual inverse temperature, and where $\mu$ is the chemical potential.
This is not the only way to state $\mathcal{Z}$.
Quoting Gould and Tobochnik,
[...] it is possible to distinguish the subset of all particles in a given single particle microstate from the particles in all other single particle microstates. For this reason we divide the system of interest into subsystems each of which is the set of all particles that are in a given single particle microstate. Because the number of particles in a given microstate varies, we need to use the grand canonical ensemble and assume that each subsystem is coupled to a heat bath and a particle reservoir independently of the other single particle microstates.
From this definition it arises that one can also compute
$$\mathcal{Z} = \prod_{k} Z_{k}\tag{2}$$
with $Z_k$ the partition function for a subsystem (one particle microstate).
And lastly, one can make the dependencies on the number of particle explicit by using the variation (Gould and Tobochnik sec. 6.11 ) $$ \mathcal{Z} = \sum_{N=1}^\infty e^{\beta\mu N} Z_c(N)\tag{3}$$
where $Z_c(N)$ is the canonical partition function of the system of $N$ particles.
Now my question is , shouldn't equation (3) be summed from $0$ to $\infty$ rather than 1 to $\infty$? This would yield $$ \mathcal{Z} = \sum_{N=0}^\infty e^{\beta\mu N} Z_c(N)=1+\sum_{N=1}^\infty e^{\beta\mu N} Z_c(N)\tag{*}$$
Stated more clearly: should we exclude the 0 particle state from the summation? The rational for excluding it is that
- the textbook says so,
- it will end up being a very important term in most situation, since everything else is a decreasing exponential.
- The interpretation of $Z_c(0)=1$ is rather ambiguous.
The reason for including it are:
- it makes sense physically (all particle in the reservoir), and
- it is essential for the coherence between eq. (2) and (3). i.e. in the fermion case, $Z_k = 1 +e^{-\beta(e_k-\mu)}$ and one has $\mathcal{Z}=1+....$