# Grand canonical ensemble

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

## Ensemble variables

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

## Grand canonical partition function

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

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

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

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

where:

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

## Helmholtz energy and partition function

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

${\displaystyle \Omega =\left.A-\mu N\right.}$,

where A is the Helmholtz energy function. Using the relation

${\displaystyle \left.U\right.=TS-pV+\mu N}$

one arrives at

${\displaystyle \left.\Omega \right.=-pV}$

i.e.:

${\displaystyle \left.pV=k_{B}T\ln \Xi _{\mu VT}\right.}$