Grand potential

The grand potential or Landau potential or Landau free energy is a quantity used in statistical mechanics, especially for irreversible processes in open systems. The grand potential is the characteristic state function for the grand canonical ensemble.

Definition

Grand potential is defined by

\Phi _{\rm {G}}\ {\stackrel {\mathrm {def} }{=}}\ U-TS-\mu N

where U is the internal 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

{\begin{aligned}d\Phi _{\rm {G}}&=dU-TdS-SdT-\mu dN-Nd\mu \\&=-PdV-SdT-Nd\mu \end{aligned}}

where P is pressure and V is volume, using the fundamental thermodynamic relation (combined first and second thermodynamic laws);

dU=TdS-PdV+\mu dN

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.

Landau free energy

Some authors refer to the grand potential as the Landau free energy or Landau potential and write its definition as:^[1]^[2]

\Omega \ {\stackrel {\mathrm {def} }{=}}\ F-\mu N=U-TS-\mu N

named after Russian physicist Lev Landau, which may be a synonym for the grand potential, depending on system stipulations. For homogeneous systems, one obtains $\Omega =-PV$ .^[3]

Homogeneous systems (vs. inhomogeneous systems)

In the case of a scale-invariant type of system (where a system of volume $\lambda V$ has exactly the same set of microstates as $\lambda$ systems of volume $V$ ), then when the system expands new particles and energy will flow in from the reservoir to fill the new volume with a homogeneous extension of the original system. The pressure, then, must be constant with respect to changes in volume:

\left({\frac {\partial \langle P\rangle }{\partial V}}\right)_{\mu ,T}=0,

and all extensive quantities (particle number, energy, entropy, potentials, ...) must grow linearly with volume, e.g.

\left({\frac {\partial \langle N\rangle }{\partial V}}\right)_{\mu ,T}={\frac {N}{V}}.

In this case we simply have $\Phi _{\rm {G}}=-\langle P\rangle V$ , as well as the familiar relationship $G=\langle N\rangle \mu$ for the Gibbs free energy. The value of $\Phi _{\rm {G}}$ can be understood as the work that can be extracted from the system by shrinking it down to nothing (putting all the particles and energy back into the reservoir). The fact that $\Phi _{\rm {G}}=-\langle P\rangle V$ is negative implies that the extraction of particles from the system to the reservoir requires energy input.

Such homogeneous scaling does not exist in many systems. For example, when analyzing the ensemble of electrons in a single molecule or even a piece of metal floating in space, doubling the volume of the space does double the number of electrons in the material.^[4] The problem here is that, although electrons and energy are exchanged with a reservoir, the material host is not allowed to change. Generally in small systems, or systems with long range interactions (those outside the thermodynamic limit), $\Phi _{G}\neq -\langle P\rangle V$ .^[5]

References

^ Lee, J. Chang (2002). "5". Thermal Physics - Entropy and Free Energies. New Jersey: World Scientific.
^ Reference on "Landau potential" is found in the book: D. Goodstein. States of Matter. p. 19.
^ McGovern, Judith. "The Grand Potential". PHYS20352 Thermal and Statistical Physics. University of Manchester. Retrieved 5 December 2016.
^ Brachman, M. K. (1954). "Fermi Level, Chemical Potential, and Gibbs Free Energy". The Journal of Chemical Physics. 22 (6): 1152. Bibcode:1954JChPh..22.1152B. doi:10.1063/1.1740312.
^ Hill, Terrell L. (2002). Thermodynamics of Small Systems. Courier Dover Publications. ISBN 9780486495095.