Constructing the grand canonical partition function

The grand canonical ensemble is used when a system can exchange particles with a reservoir, so the particle number fluctuates while temperature $T$, volume $V$, and chemical potential $\mu$ are fixed (see grand canonical ensemble and chemical potential ). Its normalization constant is the grand partition function.

Definition (classical/abstract formulation)

Let $Z_N(\beta,V)$ be the canonical partition function at fixed $N$. The grand partition function is

whenever the series converges.

It is common to introduce the fugacity $z = e^{\beta\mu}$, so the same definition reads

The corresponding grand canonical probability measure on the disjoint union of $N$-particle phase spaces is obtained by weighting each $N$ sector by $e^{\beta\mu N}$ and normalizing by $\Xi$. Expectations with respect to this measure are again ensemble averages , now including fluctuations of $N$.

Quantum version (trace with number operator)

For a quantum system with Hamiltonian $\hat H$ and particle number operator $\hat N$,

This makes explicit that the reservoir control parameter $\mu$ couples to $\hat N$ in the same way that external fields couple linearly to their conjugate observables.

Link to thermodynamic potentials

The grand partition function encodes the grand potential via the construction grand potential from a partition function :

This statistical-mechanical $\Omega$ corresponds to the thermodynamic grand potential and is related to the Helmholtz free energy by the Legendre construction Legendre transform from $F$ to $\Omega$ .

Physical interpretation

• The factor $e^{\beta\mu N}$ rewards or suppresses sectors with different particle number, depending on $\mu$; it implements exchange with a particle reservoir at chemical potential $\mu$.
• The sum over $N$ turns fixed-$N$ canonical physics into variable-$N$ grand canonical physics, and $\log \Xi$ acts as the generating function for particle-number statistics (see observables from derivatives of log-partition functions ).
• In homogeneous equilibrium, $\Omega$ is typically proportional to $-V$, so $\Xi$ encodes the pressure through $\Omega = -pV$ (up to boundary corrections).