[NOTES/SM-06003] Grand Canonical Ensemble

For an isolated system in equilibrium the assumption of ``equal a priori probabilities applies'' and all the micro states have a the same probabilities. This changes for an open system which exchanges energy and particles with the environment and is equilibrium with the environment. In this case the probabilities of different micro states depend on the energy $E$ and the total number of particles $N$ in the micro state. This probability is given by the Gibbs distribution. The probability that the system is in a micro state with $N$ particles and energy $E_k$ is given by
\begin{equation}
P(E_k,N) = \Zca^{-1} e^{\beta(\mu N -E_k)}\label{EQ01}
\end{equation}
where $\Zca$ is the grand canonical partition function and is determined from the requirement that the sum of all probabilities be equal to one.
\begin{equation} \Zca = \sum_\text{micr} e^{\beta(\mu N -E_k)} \label{EQ02}. \end{equation}
where $\sum_\text{micr}$ denoted sum over all micro states and is given by
\begin{equation} \sum_{micr} \to \sum_{N=0}^\infty\sum_{E} \Omega(E,N)\end{equation}
A  \(\sum_{micr}\) corresponds to a sum over all  micro states, there is one term for each micro state.
The sum in the right hand side is over all  levels having energy $E$  and number of particles $N$; in general there will be many such micro states, $\Omega(E,N)$denotes the number of such micro states. The right hand side has exactly one term for each
set of values of \(E,N\).