The Grand Canonical Ensemble

The canonical ensemble fixes the particle number and lets the energy fluctuate. Many systems fix neither. A region of a gas open to the rest of the container, a metal surface adsorbing molecules from a vapor, a single quantum mode that any number of bosons may occupy — each exchanges particles as freely as energy with its surroundings. The grand canonical ensemble describes a system in contact with a reservoir that fixes the temperature and the chemical potential while both the energy and the particle number of the system fluctuate. Its central object is the grand partition function, from which the whole thermodynamics follows exactly as the canonical thermodynamics follows from $Z$.

A system open to a particle reservoir

Consider a small system $\mathcal{S}$ in contact with a large reservoir $\mathcal{R}$, the two together forming an isolated compound with fixed total energy $E_{\mathrm{tot}}$ and fixed total particle number $N_{\mathrm{tot}}$. The wall between them passes both energy and particles. A microstate of $\mathcal{S}$ is now labeled by two quantities: its energy $E_s$ and its particle number $N_s$. When $\mathcal{S}$ is in a definite microstate $s$, the reservoir holds the remainder $E_{\mathrm{tot}}-E_s$ and $N_{\mathrm{tot}}-N_s$, and it does so in ${\Omega }_{\mathcal{R}}(E_{\mathrm{tot}}-E_s,N_{\mathrm{tot}}-N_s)$ ways.

By the postulate of equal a priori probabilities applied to the isolated compound, the probability of the system microstate $s$ is proportional to the number of reservoir microstates consistent with it,

The system energies and particle numbers are tiny against the reservoir totals, so expand the reservoir entropy $S_{\mathcal{R}}=k_B\ln {\Omega }_{\mathcal{R}}$ to first order in both $E_s$ and $N_s$:

These two reservoir derivatives coincide with the statistical definitions of temperature and chemical potential established for the microcanonical ensemble,¹

Both are properties of the reservoir alone, hence constants that the reservoir imposes on the system. Writing $\beta =1/k_{B}T$, the reservoir count reduces to a weight on the system microstate,

The Gibbs factor

The exponential weight is the two-variable analogue of the Boltzmann factor.

The Gibbs factor differs from the Boltzmann factor $e^{-\beta E_s}$ by the extra weight $e^{\beta \mu N_s}$. Raising $\mu$ multiplies the weight of every particle-carrying state, favoring microstates with more particles; lowering it favors emptier states. The chemical potential is the reservoir's control knob on the system's population, precisely as the temperature is its knob on the energy. For a fixed particle number the Gibbs factor collapses to the Boltzmann factor, so the canonical ensemble is the $N_s$-fixed section of the grand ensemble.

The grand partition function

Normalizing the Gibbs factor requires summing it over every microstate the system can occupy — over all particle numbers, and for each particle number over all microstates of that many particles.

The double sum factorizes the counting cleanly. The inner sum over the $N$-particle microstates is the canonical partition function $Z(N,V,T)$ studied in the previous module,² so the grand partition function is a $\mu$-weighted sum of canonical partition functions,

This makes $\Xi$ a generating function: it packages the canonical partition functions of every particle number into one object, with $e^{\beta \mu }$ marking the particle number as a bookkeeping variable. Fixing $\mu$ selects a distribution over $N$ rather than a single value, which is the whole point of opening the system to particle exchange.

Fugacity

The weight $e^{\beta \mu }$ appears once per particle, so it is convenient to name it.

Written this way $\Xi$ is a polynomial (or power series) in $z$ whose coefficient of $z^N$ is the $N$-particle canonical partition function. The fugacity runs from $0$ (the empty limit $\mu \to -\infty$, where only the $N=0$ term survives) upward; $z=1$ marks $\mu =0$. For the classical gases of the next lesson $z$ stays small, and the low-order terms of the series dominate; for the quantum gases of the following module $z$ is the natural expansion variable in which the Bose and Fermi occupation functions are written. The fugacity is the same control knob as $\mu$, rescaled to enter as a simple multiplicative activity per particle.

The grand potential

The bridge to thermodynamics is the logarithm of the grand partition function, exactly as $F=-k_{B}T\ln Z$ bridges the canonical ensemble.

To connect $\Phi$ to the potentials already in hand, start from the Helmholtz free energy $F=U-TS$ and subtract the particle term. The grand potential is the double Legendre transform of the energy that trades $S$ for $T$ and $N$ for $\mu$,

Its differential follows from the fundamental relation $\mathbf{d}U=T\mathbf{d}S-P\mathbf{d}V+\mu \mathbf{d}N$. Substituting $U=\Phi +TS+\mu N$ and cancelling gives

The natural variables are read off the differential: $\Phi$ is a function of $T$, $V$, and $\mu$, and its three first derivatives return the entropy, pressure, and particle number,

The grand potential is $-PV$

The three natural variables of $\Phi$ are $T$, $V$, and $\mu$, of which only $V$ is extensive. Doubling the system at fixed $T$ and $\mu$ doubles $\Phi$ and doubles $V$, so $\Phi$ must be proportional to $V$ with a coefficient built from the intensive variables alone.

The relation $PV=k_{B}T\ln \Xi$ is the grand-ensemble counterpart of $F=-k_{B}T\ln Z$ and is often the fastest route to an equation of state: compute $\ln \Xi$, read off the pressure directly, and obtain the density from $N=-\partial \Phi /\partial \mu$.

Mean energy and particle number

The averages come from derivatives of $\ln \Xi$, mirroring the canonical construction. Treat $\ln \Xi$ as a function of $\beta$ and the product $\beta \mu$; each derivative pulls down the corresponding quantity from the Gibbs exponent. The mean particle number is the derivative with respect to $\mu$ at fixed temperature,

This agrees with $N=-\partial \Phi /\partial \mu$ once $\Phi =-k_{B}T\ln \Xi$ is inserted. The mean energy is obtained by differentiating in $\beta$ at fixed fugacity, which isolates the energy exponent,

The combination $⟨E⟩-\mu ⟨N⟩$ is the quantity conjugate to $\beta$ in the Gibbs exponent, so it emerges most naturally from the $\beta$-derivative; the separate mean energy then follows by adding $\mu ⟨N⟩$. The entropy closes the set through $\Phi =⟨E⟩-TS-\mu ⟨N⟩$, giving

When the grand ensemble is the right choice

The three ensembles predict the same thermodynamics for a macroscopic system, so the choice among them is a matter of which calculation is easiest. The grand canonical ensemble is the convenient one whenever the constraint of fixed particle number obstructs the sum.

• Quantum ideal gases. Counting configurations of indistinguishable bosons or fermions at fixed total $N$ ties the single-particle occupation numbers together through ${\sum }_i{n}_i=N$. Releasing $N$ decouples the modes: the grand partition function factorizes over single-particle states, and each mode is summed independently. This is the derivation of the Bose-Einstein and Fermi-Dirac distributions in the next module.
• Adsorption and chemical equilibrium. Surfaces, binding sites, and reacting species exchange particles with a bath at fixed $\mu$; the equilibrium occupation follows directly from the Gibbs factor.
• Open subvolumes and fluctuations. A region of a fluid with imaginary walls exchanges particles with the surrounding fluid; the grand ensemble is the natural setting for density fluctuations, taken up in the next lesson.

Summary

• A system exchanging energy and particles with a reservoir at fixed $T$ and $\mu$ has microstate probability set by the Gibbs factor$e^{-\beta (E_s-\mu N_s)}$, obtained by expanding the reservoir entropy to first order in $E_s$ and $N_s$ with $\partial S_{\mathcal{R}}/\partial E=1/T$ and $\partial S_{\mathcal{R}}/\partial N=-\mu /T$.
• The grand partition function$\Xi ={\sum }_N{e}^{\beta \mu N}Z(N,V,T)={\sum }_N{z}^{N}Z(N,V,T)$ normalizes the Gibbs factor; it is a generating function in the fugacity $z=e^{\beta \mu }$ whose $z^N$ coefficient is the canonical partition function of $N$ particles.
• The grand potential $\Phi =-k_{B}T\ln \Xi =F-\mu N$ has differential $\mathbf{d}\Phi =-S\mathbf{d}T-P\mathbf{d}V-N\mathbf{d}\mu$, so $S$, $P$, and $N$ follow by differentiation. Because $V$ is its only extensive variable, $\Phi =-PV$, giving $PV=k_{B}T\ln \Xi$.
• Averages come from derivatives of $\ln \Xi$: $⟨N⟩=z\partial \ln \Xi /\partial z$ and $⟨E⟩-\mu ⟨N⟩=-\partial \ln \Xi /\partial \beta$. The grand ensemble is the tool of choice when fixed $N$ obstructs the sum, above all for the quantum ideal gases.