The Grand Canonical Ensemble

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

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

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 $\ln \Xi$:

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

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

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.

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

• Confusing the grand canonical ensemble with the canonical ensemble: In the canonical ensemble, $N$ is fixed and $T$ is specified. In the grand canonical ensemble, $\mu$ is specified and $N$ fluctuates. Using the wrong ensemble for a problem (e.g., fixing $N$ when the system exchanges particles with a reservoir) leads to incorrect results.
• Forgetting that $\Xi$ is a sum over both $N$ and states: The grand partition function sums over all particle numbers and all microstates for each $N$. It is not simply a product of single-particle partition functions unless particles are non-interacting.
• Assuming fluctuations are always negligible: While relative fluctuations scale as $1/\sqrt{N}$, in small systems (nanoparticles, quantum dots, biological macromolecules) $N$ can be small enough that fluctuations become significant and the canonical and grand canonical ensembles give different predictions.
• Misapplying $\Phi_G = -PV$: This relation holds for homogeneous systems in thermodynamic equilibrium. For non-equilibrium or inhomogeneous systems (e.g., systems with interfaces or external fields), the grand potential includes additional terms.

Problem. A surface has $M$ independent adsorption sites, each of which can be either empty or occupied by at most one gas molecule with energy $-\varepsilon$. Derive the average coverage $\theta = \langle N \rangle / M$ in equilibrium with a gas reservoir at chemical potential $\mu$.

Solution. Each site is a two-level system: empty with energy 0, occupied with energy $-\varepsilon$. The single-site grand partition function is:

$\xi = 1 + e^{\beta(\mu + \varepsilon)}$

Since sites are independent, $\Xi = \xi^M = (1 + e^{\beta(\mu + \varepsilon)})^M$.

$\langle N \rangle = \frac{1}{\beta}\frac{\partial \ln \Xi}{\partial \mu} = M \frac{e^{\beta(\mu + \varepsilon)}}{1 + e^{\beta(\mu + \varepsilon)}}$

$\theta = \frac{\langle N \rangle}{M} = \frac{e^{\beta(\mu + \varepsilon)}}{1 + e^{\beta(\mu + \varepsilon)}} = \frac{1}{1 + e^{-\beta(\mu + \varepsilon)}}$

This is the Langmuir adsorption isotherm. Since the gas reservoir is ideal, $\mu = k_BT \ln(P/P_0)$, giving $\theta = KP/(1 + KP)$ where $K = e^{\beta\varepsilon}/P_0$, recovering the standard Langmuir form.

For non-interacting quantum gases, the grand partition function factorises over single-particle states:

$\Xi = \prod_i \Xi_i, \quad \Xi_i = \begin{cases} 1 + e^{-\beta(\varepsilon_i - \mu)} & \text{(fermions)} \\ \frac{1}{1 - e^{-\beta(\varepsilon_i - \mu)}} & \text{(bosons)} \end{cases}$

The average occupation number follows directly:

$\langle n_i \rangle = -\frac{1}{\beta}\frac{\partial \ln \Xi_i}{\partial \varepsilon_i} = \frac{1}{e^{\beta(\varepsilon_i - \mu)} \pm 1}$

where $+$ is for fermions (Fermi-Dirac) and $-$ is for bosons (Bose-Einstein). This unified derivation from the grand canonical ensemble illustrates its power: both quantum statistics emerge naturally from the same formalism, with the only difference being whether each single-particle state can be occupied at most once (fermions) or any number of times (bosons).

• The grand canonical ensemble extends the canonical ensemble by allowing particle number fluctuations, making it suitable for open systems in contact with both a heat reservoir and a particle reservoir.
• $\Phi_G = -PV$ connects microscopic statistics to macroscopic thermodynamics: The grand potential directly gives the equation of state, linking the partition function to pressure and volume.
• Fluctuations scale as $1/\sqrt{N}$: For macroscopic systems ($N \sim 10^{23}$), relative particle number fluctuations are negligible ($\sim 10^{-12}$), justifying the use of the canonical ensemble for most practical purposes.
• The fugacity $z = e^{\beta\mu}$ parameterises particle number: The grand partition function is a power series in $z$, where each coefficient encodes the thermodynamics of the $N$-particle sector.
• Ideal gas statistics emerge naturally: The grand canonical treatment of the ideal gas reproduces the canonical results ($\langle E \rangle = \frac{3}{2}Nk_BT$, $PV = Nk_BT$) without the need to compute $N$-particle partition functions.

• Adsorption and surface science: The grand canonical ensemble describes gas molecules adsorbing on a surface, where the number of adsorbed particles fluctuates as the system exchanges molecules with the gas phase.
• Semiconductor physics: Carrier concentrations in semiconductors are calculated using grand canonical methods, where electrons and holes are exchanged with reservoirs at fixed chemical potential.
• Nuclear physics: The statistical model of nuclear reactions uses the grand canonical ensemble to describe particle production in high-energy collisions, where the number of produced pions, kaons, etc. fluctuates.
• Chemical equilibrium: Reactions in solution are naturally described in the grand canonical ensemble, where the chemical potentials of reactants and products are fixed by the reservoir.
• Monte Carlo simulations: Grand canonical Monte Carlo (GCMC) simulations insert and delete particles to sample the grand canonical distribution, used extensively in studies of porous materials and fluid adsorption.