Grand canonical ensemble

Consider a system held at a temperature $T$ , whose energy $U$ and number of particles $N$ are allowed to vary. Imagine for example a system connected by a copper pipe to a large reservoir at fixed temperature.

Instead of fixing the number of particles, we fix the chemical potential $\mu$ . We define it analogously to temperature, such that

\frac{\partial S}{\partial N} = -\frac\mu T.

Let $U$ be the total combined energy of the system and reservoir, $N$ the combined number of particles.

We then consider the probability that the system is in a specific microstate $j$ . $P(E_j,N_j)$ depends on both the energy of the microstate and the number of particles. We argue

P_j(E_j,N_j) \propto \underbrace{\Gamma_S(j)}_{\text{degeneracy}} \cdot \Gamma_R(U-E_j,N-N_j).

Where $\Gamma_S$ is the multiplicity of the system and $\Gamma_R$ is the multiplicity of the reservoir. Since we are specifying a specific microstate $j$ , $\Gamma_S(j)$ is the degeneracy. We can write

\Gamma_R(U-E_j,N-N_j) = \exp\left(\frac 1k S_R(U-E_j,N-N_j)\right).

Then Taylor expand to find

\begin{align*} S_R(U-E_j,N-N_j) &\approx S_R(U,N) - E_j \frac{\partial S_R}{\partial U_R} - N_j \frac{\partial S_R}{\partial N_R} \\ &= S_R(U,N) - \frac{E_j}{T} + \frac{N_j \mu}{T}. \end{align*}

Finally we find

\begin{align*} P_j(E_j,N_j) &= \frac 1\xi e^{\beta E_j} e^{\beta \mu N_j} \\ \xi &= \sum_j e^{-\beta E_j} e^{\beta\mu N_j}. \end{align*}

Where $\beta=1/kT$ and $\xi$ is the Grand Partition Function, analogous to $Z$ in the canonical ensemble. Taking derivatives we find interesting relations to thermodynamic quantities.

\begin{align*} \frac{\partial}{\partial \beta} \log \xi &= \frac 1\xi \frac{\partial}{\partial \beta} \sum e^{\beta(\mu N_j-E_j)} = \mu \avg N - \avg U \\ \frac{\partial}{\partial \mu} \log \xi &= \frac1\xi \frac{\partial}{\partial \mu} \sum e^{\beta(\mu N_j-E_j)} = \beta \avg N \\ S &= -k \sum_j p_j \log p_j = - k \frac 1\xi \sum e^{\beta(\mu N_j-E_j)} \left[\beta \mu N_j - \beta E_j - \log \xi\right] \\ &= \frac 1T \frac 1\xi \sum (E_j - \mu N_j) e^{\beta (\mu N_j - E_j)} + k \log \xi \cdot \frac 1\xi \sum e^{\beta (\mu N_j-E_j)} \\ &= \frac{U - \mu N}{T} + k \log \xi. \end{align*}

We can also define a state function which we call the grand potential $\Omega$

\begin{align*} \Omega &= -kT \log \xi = U - TS - \mu N \\ \d\Omega &= -S \d T - p \d V - N \d\mu. \end{align*}