Grand canonical ensemble

Gibbs equilibrium distribution at fixed temperature and chemical potential: energy and particle number both fluctuate, normalized by the grand partition function.

Grand canonical ensemble

The grand canonical ensemble models a system that can exchange both energy and particles with a reservoir, so temperature and chemical potential are fixed while energy and particle number fluctuate.

It is the natural ensemble for open systems, lattice gases, and quantum gases, and it provides a direct route to the thermodynamic grand potential and pressure.

Fix an inverse temperature inverse temperature $\beta>0$ and a chemical potential $\mu$ . For each particle number $N$ , let $\Gamma_N$ be the $N$ -particle phase space (or Hilbert space sector), and let $H_N$ be the corresponding Hamiltonian .

The grand canonical weight of a microstate $x\in\Gamma_N$ is proportional to $e^{-\beta(H_N(x)-\mu N)}$ . The normalized probability density is

\rho_{\beta,\mu}(N,x) \;=\; \frac{e^{-\beta(H_N(x)-\mu N)}}{\Xi(\beta,\mu)},

where the normalizing constant

\Xi(\beta,\mu) \;=\; \sum_{N\ge 0}\int_{\Gamma_N} e^{-\beta(H_N(x)-\mu N)}\, d\Gamma_N

is the grand partition function (also called the grand canonical partition function).

(Quantum version: $\Xi(\beta,\mu)=\mathrm{Tr}\,\exp\{-\beta(\hat H-\mu \hat N)\}$ .)

Ensemble averages

For an observable $A$ that may depend on $N$ and $x$ , the ensemble average is

\langle A\rangle_{\beta,\mu} \;=\; \sum_{N\ge 0}\int_{\Gamma_N} A(N,x)\,\rho_{\beta,\mu}(N,x)\, d\Gamma_N.

Derivative formulas (mean values and fluctuations)

Thermodynamic observables are generated by derivatives of $\ln\Xi$ (see observables from log partition functions ):

Mean particle number:
$\langle N\rangle_{\beta,\mu} \;=\; \frac{1}{\beta}\,\frac{\partial}{\partial\mu}\ln \Xi(\beta,\mu).$
Particle-number fluctuations:
$\mathrm{Var}_{\beta,\mu}(N) \;=\; \frac{1}{\beta^2}\,\frac{\partial^2}{\partial\mu^2}\ln \Xi(\beta,\mu),$
which connects directly to response coefficients such as compressibility (a special case of susceptibility–fluctuation relations ).
Mean energy:
$\left\langle E\right\rangle_{\beta,\mu} \;=\; -\frac{\partial}{\partial\beta}\ln \Xi(\beta,\mu) \;+\; \mu\,\langle N\rangle_{\beta,\mu}.$

More generally, mixed derivatives of $\ln\Xi$ generate covariances, fitting into fluctuation formulas from log partition functions .

Grand potential and pressure

The thermodynamic grand potential is obtained from $\Xi$ by

\Omega(\beta,\mu) \;=\; -\frac{1}{\beta}\ln \Xi(\beta,\mu),

matching the thermodynamic grand potential and arising as a Legendre transform of free energy in equilibrium thermodynamics.

For a homogeneous system of volume $V$ , one has $\Omega = -pV$ , so the pressure can be read from the partition function as in pressure from the partition function :

p \;=\; \frac{1}{\beta V}\,\ln \Xi(\beta,\mu).

Relation to other ensembles

If particle-number fluctuations are suppressed (e.g., by conditioning on a fixed $N$ ), the grand canonical ensemble reduces to the canonical ensemble . In the thermodynamic limit , predictions for many bulk observables often coincide across these ensembles under suitable conditions (ensemble equivalence).