Grand potential

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:$dPhi\_\{G\}\; =\; -\; S\; dT\; -\; N\; dmu\; -\; P\; dV$Where P is pressure and V is volume.

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

For an ideal gas,:where Ξ is the grand partition function, k_B is Boltzmann constant, Z₁ is the partition function for 1 particle and β is equal to 1 / k_BT.

Landau free energy

Some authors refer to the "Landau free energy" or Landau potential as: [Lee, Joon Chang. (2002) book "Thermal Physics - Entropy and Free Energies" (ch. 5). New Jersey: World Scientific] [Reference on "Landau potential" is found in the book "States of Matter" by David Goodstein (page 19) as $Omega\; =\; F-\; mu\; N\; ,;$ where "F" is the Helmholtz free energy. For homogeneous systems, one obtains $Omega\; =\; -PV\; ,;$]

:$Omega\; stackrel\{mathrm\{def\{=\}\; F\; -\; mu\; N\; =\; U\; -\; T\; S\; -\; mu\; N$

named after Russian physicist Lev Landau, which may be a synonym for the grand potential, depending on system stipulations.

References

See also

* Gibbs energy
* Helmholtz energy

External links

* [http://theory.ph.man.ac.uk/~judith/stat_therm/node88.html Grand Potential (Manchester University)]