Grand canonical ensemble

Ensemble variables

• Chemical Potential, $\mu$

• Volume, $V$

• Temperature, $T$

Partition Function

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

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

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 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.:

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