Grand potential

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The grand potential is a quantity used in statistical mechanics, especially for irreversible processes in open systems.

Grand potential is defined by

\Phi_{G} \ \stackrel{\mathrm{def}}{=}\ E - T S - \mu N

Where E is the energy, T is the temperature of the system, S is the entropy, μ is the chemical potential, and N is the number of particles in the system.

The change in the grand potential is given by

dΦG = − SdT − Ndμ − PdV

Where P is pressure and V is volume.

When the system is in thermodynamic equilibrium, ΦG is a minimum. This can be seen by considering that dΦG is zero if the volume is fixed and the temperature and chemical potential have stopped evolving.

For an ideal gas,

ΦG = − kBTln(Ξ) = − kBTZ1eβμ

where Ξ is the grand partition function, kB is boltzmann constant, Z1 is the partition function for 1 particle and β is equal to 1 / kBT.


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