Grand canonical ensemble

Ensemble variables

• Chemical Potential, $\left.\mu \right.$

• Volume, $\left.V\right.$

• Temperature, $\left.T\right.$

Partition Function

Classical partition function (one-component system) in a three-dimensional space: $Q_{\mu VT}$

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

where:

• $\left.N\right.$ is the number of particles

• $\left.\Lambda \right.$ is the de Broglie thermal wavelength (which depends on the temperature)

• $\beta ={\frac {1}{k_{B}T}}$, with $k_{B}$ being the Boltzmann constant

• $\left.U\right.$ 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. $\int d(R^{*})^{3N}=1$

Helmholtz energy and partition function

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

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

where A is the Helmholtz energy function. Using the relation

$\left.U\right.=TS-PV+\mu N$

one arrives at

$\left.\Omega \right.=-PV$

i.e.:

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