The Grand Canonical Ensemble

3.1 Definition and Motivation

In many physical situations, a system exchanges both energy and particles with a reservoir. The grand canonical ensemble describes such open systems. The macroscopic variables are the chemical potential $\mu$The volume $V$And the temperature $T$.

Definition. The grand partition function is

$\Xi = \sum_{N=0}^{\infty} \sum_{i} e^{-\beta(E_{i}^{(N)} - \mu N)}$

Where the outer sum is over all possible particle numbers $N$ and the inner sum is over all states with $N$ particles.

The probability that the system is in state $i$ with $N$ particles is

$P_{i,N} = \frac{e^{-\beta(E_{i}^{(N)} - \mu N)}}{\Xi}$

3.2 Thermodynamic Relations

Theorem 3.1. The grand potential $\Phi_G = -k_BT \ln \Xi$ satisfies

$\Phi_G = F - \mu N = -PV$

Proof. For a classical ideal gas, $\Xi = \sum_{N=0}^{\infty} e^{\beta \mu N} Z_N$ where $Z_N = z^N/N!$ is the canonical partition function. Therefore:

$\Xi = \sum_{N=0}^{\infty} \frac{(z e^{\beta \mu})^N}{N!} = \exp(z e^{\beta \mu})$

$\Phi_G = -k_BT \ln \Xi = -k_BT \cdot z e^{\beta \mu} = -PV$

The last equality follows from the ideal gas law $PV = Nk_BT$ with $N = z e^{\beta \mu}$. More generally, $\Phi_G = -PV$ holds for all systems. $\blacksquare$

Key relations from $\ln \Xi$:

$\langle N \rangle = \frac{1}{\beta}\frac{\partial \ln \Xi}{\partial \mu}\bigg|_{T,V}, \quad \langle E \rangle = -\frac{\partial \ln \Xi}{\partial \beta}\bigg|_{\mu,V} + \frac{\mu}{\beta}\frac{\partial \ln \Xi}{\partial \mu}\bigg|_{T,V}$

$S = k_B\left(\ln \Xi + \beta \langle E \rangle - \beta \mu \langle N \rangle\right)$

3.3 Number Fluctuations

Theorem 3.2. The particle number fluctuations in the grand canonical ensemble satisfy

$\langle N^2 \rangle - \langle N \rangle^2 = k_BT \left(\frac{\partial \langle N \rangle}{\partial \mu}\right)_{T,V}$

Proof. $\langle N^2 \rangle - \langle N \rangle^2 = \frac{1}{\beta^2}\frac{\partial^2 \ln \Xi}{\partial \mu^2} = \frac{1}{\beta}\frac{\partial}{\partial \mu}\left(\frac{1}{\beta}\frac{\partial \ln \Xi}{\partial \mu}\right) = \frac{1}{\beta}\frac{\partial \langle N \rangle}{\partial \mu}$. $\blacksquare$

For an ideal gas, $\langle N \rangle = z e^{\beta \mu}$So $\partial \langle N \rangle / \partial \mu = \beta \langle N \rangle$Giving relative fluctuations:

$\frac{\langle N^2 \rangle - \langle N \rangle^2}{\langle N \rangle^2} = \frac{1}{\langle N \rangle}$

This is Poisson …/4-statistics-and-probability/2_statistics: fluctuations scale as $1/\sqrt{N}$Negligible for macroscopic systems.

3.4 Worked Example: Ideal Gas in the Grand Canonical Ensemble

Problem. Compute $\Xi$, $\langle N \rangle$And $\langle E \rangle$ for a classical ideal gas in the grand canonical ensemble.