Grand partition function

The grand partition function $\Xi(\beta,V,\mu)$ is the normalization constant of the grand-canonical ensemble , appropriate when a system can exchange both energy and particles with reservoirs that fix inverse temperature $\beta$ and chemical potential $\mu$ (see chemical potential ).

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

Equivalently, in terms of canonical partition functions (see canonical partition function ) at fixed $N$,

Introducing the fugacity $z = e^{\beta\mu}$, this becomes $\Xi = \sum_{N\ge 0} z^N Z(\beta,V,N)$.

Role in the grand-canonical ensemble

The grand-canonical probability weight of a microstate $s$ with energy $E_s$ and particle number $N_s$ is proportional to

and division by $\Xi(\beta,V,\mu)$ ensures the probabilities sum to $1$.

This “energy minus chemical work” structure mirrors the thermodynamic combination $U-\mu N$ at fixed $(T,V,\mu)$, which is the natural setting for the grand potential .

Grand potential and pressure

The grand partition function generates the grand potential:

with $k_B$ the Boltzmann constant . For homogeneous systems in the thermodynamic limit , the pressure is obtained from $\Omega \approx -pV$, equivalently from the volume scaling of $\ln\Xi$:

This is one instance of pressure from partition functions .

Generating formulas for averages and fluctuations

As with other ensembles, $\ln \Xi$ generates equilibrium expectations (see observables from log partition functions and fluctuation formulas from log partition functions ).

Mean particle number

Particle-number fluctuations

Mean energy Using differentiation with respect to $\beta$ at fixed $\mu$,

These identities encode both equilibrium thermodynamics and linear response in the grand-canonical setting, and they are the basis for expressing susceptibilities such as the particle-number compressibility in terms of fluctuations.