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:

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

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

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

where:

• $N$ is the number of particles
• $\left.\Lambda \right.$ 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)
• $\left(R^{*}\right)^{3N}$ represent the $3N$ position coordinates of the particles (reduced with the system size): i.e. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega = \left. A - \mu N \right. } ,

where A is the Helmholtz energy function. Using the relation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left.U\right.=TS -pV + \mu N}

one arrives at

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. \Omega \right.= -pV}

i.e.:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. p V = k_B T \ln \Xi_{\mu V T } \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)