Grand Canonical Ensemble §

The weight is derived by cutting a small system out of a large isolated (microcanonical) reservoir. The reservoir has $N_{\text{tot}}$ particles and energy $E_{\text{tot}}$; giving the system $(E,N)$ leaves the reservoir $(E_{\text{tot}}-E, N_{\text{tot}}-N)$. The probability is proportional to the reservoir’s entropy, expanded to first order:

Using $(\partial S/\partial E)_{N}=\beta$ and $(\partial S/\partial N)_{E}=\beta\mu$ for the reservoir and dropping the constant $S_{\text{res}}(E_{\text{tot}},N_{\text{tot}})$ gives exactly $e^{\beta(\mu N - E)}$. The chemical potential is defined by the reservoir’s entropy slope:

so it is the free-energy cost of adding one particle. This is the same construction that turns the microcanonical ensemble into the canonical one, with particle exchange added on top. See MicrocanonicalEnsemble and CanonicalEnsemble.

Because $\ln \Xi$ is the log-sum-exp of all weights, it is the single object that plays the role of the canonical partition function. The factorization is the key: an ideal gas of $M$ independent levels has $\Xi$ as a product of $M$ one-level factors, so the mean occupation of a level is independent of all the others. This is why the grand ensemble turns a many-body sum into a sum of one-particle terms.

The derivative identities that follow are the workhorses of the subject:

The last line says the variance of the particle number is the second derivative of $\ln \Xi$ — the same pattern as the canonical ensemble, where energy variance is the second $\beta$-derivative of $\ln Z$.

The grand potential is the free energy of a system allowed to trade particles: $\Omega = F - \mu N$. Minimizing $\Omega$ over $N$ at fixed $T,\mu,V$ gives the most probable particle number, and in the thermodynamic limit the fluctuations vanish. See Thermodynamic Potentials and Equilibrium.

The relation $\operatorname{Var}(N) = k_B T\, \partial_\mu \langle N\rangle$ says that how much $N$ jitters is the same number as how much $N$ shifts when you nudge $\mu$. It is the grand-canonical fluctuation formula; see Statistical Foundations of Thermodynamics.

Note the three statistics give different fluctuations for the same mean number: fermions are anti-bunched ($f(1-f) < f$), bosons are bunched ($f(1+f) > f$), and classical particles are Poisson ($\operatorname{Var}=\langle N\rangle$). This is the seed of phenomena like fermion pressure and Bose condensation. For the thermodynamic limit of these fluctuations and the onset of collective behavior, see PhaseTransitions.

In the thermodynamic limit ($N \to \infty$), the grand canonical and canonical ensembles give identical thermodynamic quantities. The relative particle-number fluctuation scales as $\sqrt{\operatorname{Var}(N)}/\langle N\rangle \sim 1/\sqrt{N}$, so for macroscopic systems the two ensembles are indistinguishable in practice. This is why one can safely use the grand canonical ensemble (which factorizes over levels) even when the total particle number is effectively fixed: the constraint $N = \text{const}$ and the constraint $\mu = \text{const}$ lead to the same macroscopic predictions. The equivalence breaks down for small systems (nanoparticles, single molecules) or near critical points where fluctuations are enhanced.

The difference is the exclusion rule on a single level. A fermion level can hold 0 or 1 particle, so $\Xi_i = e^{-\beta\varepsilon} + e^{\beta(\mu-\varepsilon)}$ and $f = \Xi_i' / \Xi_i = 1/(e^x+1)$. A boson level can hold $0,1,2,\dots$ particles, so $\Xi_i$ is a geometric series that converges only when $\mu < \varepsilon$ — this is why the boson chemical potential is bounded above by the ground-state energy. A classical particle has no restriction, and the dilute limit $f\ll 1$ makes both quantum formulas collapse to $e^{-x}$.

The interactive simulation below plots the three distributions versus $x = (\varepsilon-\mu)/k_B T$ at fixed $T$. The Fermi curve is a smooth step at $x=0$ (sharp as $T\to 0$), the boson curve blows up as $x\to 0^+$ (a precursor of Bose condensation), and the classical curve is the common exponential tail.

The occupation of a state is the bridge to PhaseTransitions: when a boson level’s occupation diverges as $\mu \to \varepsilon_0$, the excess particles pile into the ground state — Bose–Einstein condensation.

For a fermion, $f(\varepsilon) = 1/(e^{(\varepsilon-\mu)/T}+1)$.

• Level 0: $f_0 = 1/(e^{0}+1) = 0.5$.
• Level 1: $f_1 = 1/(e^{2}+1) = 0.1192$.

Mean number: $\langle N \rangle = f_0 + f_1 = 0.6192$.

Variance (fermion levels are independent): $\operatorname{Var}(N) = f_0(1-f_0) + f_1(1-f_1) = 0.25 + 0.1050 = 0.3550$.

Check with fluctuation–dissipation: $k_B T\,\partial_\mu\langle N\rangle = 0.3550 = \operatorname{Var}(N)$. ✓

Grand partition function: $\ln\Xi = \ln(1+e^0) + \ln(1+e^{-2}) = \ln 2 + \ln(1+e^{-2}) = 0.8201$, so $\Omega = -T\ln\Xi = -0.8201$.

The mean energy is $\langle E\rangle = 0\cdot f_0 + 2\cdot f_1 = 0.2384$.

Note that $f_i \le 1$ for fermions (exclusion), and the variance is smaller than the classical/Poisson value $\langle N\rangle = 0.6192$ — a signature of anti-bunching.