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 $\exp \left[\beta \mu \right]$ and is known as the fugacity. The grand-canonical ensemble is particularly well suited to simulation studies of adsorption.

Ensemble variables

• chemical potential, $\left.\mu \right.$
• volume, $\left.V\right.$
• temperature, $\left.T\right.$

Grand canonical partition function

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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Xi _{\mu VT}=\sum _{N=0}^{\infty }\exp \left[\beta \mu N\right]Q_{NVT}}

i.e. for a classical system one has

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Xi _{\mu VT}=\sum _{N=0}^{\infty }\exp \left[\beta \mu N\right]{\frac {V^{N}}{N!\Lambda ^{3N}}}\int d(R^{*})^{3N}\exp \left[-\beta U\left(V,(R^{*})^{3N}\right)\right]}

where:

• $N$ is the number of particles
• Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.\Lambda \right.} is the de Broglie thermal wavelength (which depends on the temperature)
• $\beta ={\frac {1}{k_{B}T}}$, with $k_{B}$ being the Boltzmann constant
• $U$ is the potential energy, which depends on the coordinates of the particles (and on the interaction model)
• Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left(R^{*}\right)^{3N}} represent the $3N$ position coordinates of the particles (reduced with the system size): i.e. Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int d(R^{*})^{3N}=1}

Helmholtz energy and partition function

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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Omega =\left.A-\mu N\right.} ,

where A is the Helmholtz energy function. Using the relation

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.U\right.=TS-pV+\mu N}

one arrives at

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.\Omega \right.=-pV}

i.e.:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.pV=k_{B}T\ln \Xi _{\mu VT}\right.}

See also

• Monte Carlo in the grand-canonical ensemble

References

Related reading

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