Grand canonical ensemble

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

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

where:

• N is the number of particles
• $\mathrm{\Lambda }$ is the de Broglie thermal wavelength (which depends on the temperature)
• $\beta =\frac{1}{k_{B}T}$, 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)
• ${(R^{*})}^{3N}$ 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.:

$pV=k_{B}T\log Q_{\mu VT}$

See also

• Monte Carlo in the grand-canonical ensemble

References

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