Grand canonical ensemble

The grand-canonical ensemble is for "open" systems, where the number of particles, $N$, can change. It can be viewed as an ensemble of canonical ensembles; there being a canonical ensemble for each value of $N$, and the (weighted) sum over $N$ of these canonical ensembles constitutes the grand canonical ensemble. The weighting factor is $\exp [\beta \mu ]$ and is known as the fugacity. The grand-canonical ensemble is particularly well suited to simulation studies of adsorption.

Ensemble variables

• chemical potential, $\mu$
• volume, $V$
• temperature, $T$

Grand canonical partition function

The grand canonical partition function for a one-component system in a three-dimensional space is given by:

${\mathrm{\Xi }}_{\mu VT}=\sum _{N=0}^{\mathrm{\infty }}\exp [\beta \mu N]Q_{NVT}$

where $Q_{NVT}$ represents the canonical ensemble partition function. For example, for a classical system one has

${\mathrm{\Xi }}_{\mu VT}=\sum _{N=0}^{\mathrm{\infty }}\exp [\beta \mu N]\frac{V^N}{N!{\mathrm{\Lambda }}^{3N}}\int d(R^{*}{)}^{3N}\exp [-\beta U(V,(R^{*}{)}^{3N})]$

where:

• $N$ is the number of particles
• $\mathrm{\Lambda }$ is the de Broglie thermal wavelength (which depends on the temperature)
• $\beta$ is the inverse temperature
• $U$ is the potential energy, which depends on the coordinates of the particles (and on the interaction model)
• ${(R^{*})}^{3N}$ represent the $3N$ position coordinates of the particles (reduced with the system size): i.e. $\int d(R^{*}{)}^{3N}=1$

Helmholtz energy and partition function

The corresponding thermodynamic potential, the grand potential, $\mathrm{\Omega }$, for the aforementioned grand canonical partition function is:

$\mathrm{\Omega }=A-\mu N$,

where A is the Helmholtz energy function. Using the relation

$U=TS-pV+\mu N$

one arrives at

$\mathrm{\Omega }=-pV$

i.e.:

$pV=k_{B}T\ln {\mathrm{\Xi }}_{\mu VT}$

See also

• Grand canonical Monte Carlo
• Mass-stat

References

Related reading

• Richard C. Tolman "On the Establishment of Grand Canonical Distributions", Physical Review 57 pp. 1160-1168 (1940)