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

d Φ G = - S d T - N d μ - P d V

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,

Φ G = - k B T ln(Ξ) = - k B T Z 1 e βμ

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.

1 Landau free energy
2 References
3 See also
4 External links

[edit] Landau free energy

Some authors refer to the Landau free energy or Landau potential as:^[1]^[2]

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

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

[edit] References

^ 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 $\Omega = -PV \,\;$