SQUARE45

Let

\mathcal{S}

be the system, and

\mathcal{R}

be the reservoir. The system is characterized by a Hamiltonian

\mathcal{H}(\mathbf{x}, N)

and a fixed chemical potential

\mu

and temperature

T

. The Grand Canonical Partition Function

\mathcal{Z}

is defined over the state space

\Omega_{\text{states}}

and particle number

N

: \n\n$$\mathcal{Z}(\mu, T, V) = \sum_{N=0}^{\infty} \frac{z^N}{N!} \text{Tr}\\text{e}\left(e^{-\beta \mathcal{H}_N}\right) = \text{Tr}\\text{e}\left(e^{-\beta (\mathcal{H} - \mu\hat{N})\right)$$\n\nwhere $\beta = 1/(k_B T)$, $z = e^{\beta \mu}$ is the fugacity, $\mathcal{H}$ is the system Hamiltonian operator, and $\hat{N}$ is the particle number operator. The Grand Potential $\Omega$ is then derived via the thermodynamic relation:\n\n$$\Omega(\mu, T, V) = -k_B T \ln \mathcal{Z}(\mu, T, V)$$\n\nFurthermore, the expectation value of any observable $\mathcal{O}$ is given by:\n\n$$\langle \mathcal{O} \rangle = \frac{1}{\mathcal{Z}} \text{Tr}\\text{e}\left(\mathcal{O} e^{-\beta (\mathcal{H} - \mu\hat{N})}\right)$$

Grand Canonical Ensemble

The statement of the theorem