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{{Short description|Thermodynamic potential}}
{{Statistical mechanics|cTopic=[[Thermodynamic potential|Potentials]]}}
The '''grand potential''' or '''Landau potential''' or '''Landau free energy''' is a quantity used in [[statistical mechanics]], especially for [[irreversible process]]es in [[Open system (systems theory)|open system]]s.
The grand potential is the characteristic state function for the [[grand canonical ensemble]].

==Definition==

The grand potential is defined by
<math display="block">
\Phi_\text{G} \stackrel{\mathrm{def}}{{}={}} U - T S - \mu N
</math>
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
<math display="block">\begin{align}
d\Phi_\text{G} & = dU - T\,dS - S\,dT - \mu d\,N - N\,d\mu \\
& = - P\,dV - S\,dT - N\,d\mu
\end{align}</math>
where ''P'' is [[pressure]] and ''V'' is [[volume]], using the [[fundamental thermodynamic relation]] (combined [[first law of thermodynamics|first]] and [[first law of thermodynamics|second]] [[thermodynamic laws]]);

<math display="block">dU = T\,dS - P\,dV + \mu\,dN</math>

When the system is in [[thermodynamic equilibrium]], Φ<sub>G</sub> is a minimum. This can be seen by considering that ''d''Φ<sub>G</sub> 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:<ref>{{cite book|author=Lee, J. Chang|year=2002|title=Thermal Physics - Entropy and Free Energies|chapter=5|publisher=New Jersey: World Scientific}}</ref><ref>Reference on "Landau potential" is found in the book: {{cite book|author=D. Goodstein|title=States of Matter|page=19}}</ref>

<math display="block">
\Omega \stackrel{\mathrm{def}}{{}={}} F - \mu N = U - T S - \mu N
</math>

named after Russian physicist [[Lev Landau]], which may be a synonym for the grand potential, depending on system stipulations. For homogeneous systems, one obtains <math> \Omega = -PV</math>.<ref>{{cite web|last1=McGovern|first1=Judith|title=The Grand Potential|url=http://theory.physics.manchester.ac.uk/~judith/stat_therm/node88.html|website=PHYS20352 Thermal and Statistical Physics|publisher=University of Manchester|access-date=5 December 2016}}</ref>

==Homogeneous systems (vs. inhomogeneous systems)==
In the case of a scale-invariant type of system (where a system of volume <math>\lambda V</math> has exactly the same set of microstates as <math>\lambda</math> systems of volume <math>V</math>), 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:

<math display="block">\left(\frac{\partial \langle P \rangle}{\partial V}\right)_{\mu,T} = 0,</math>

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

<math display="block">\left(\frac{\partial \langle N \rangle}{\partial V}\right)_{\mu,T} = \frac{N}{V}.</math>

In this case we simply have <math> \Phi_\text{G} = - \langle P\rangle V</math>, as well as the familiar relationship <math>G = \langle N \rangle \mu</math> for the [[Gibbs free energy]].
The value of <math> \Phi_\text{G} </math> 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 <math> \Phi_\text{G} = - \langle P\rangle V</math> 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.<ref>{{Cite journal | doi = 10.1063/1.1740312| last = Brachman | first = M. K.| title = Fermi Level, Chemical Potential, and Gibbs Free Energy | journal = The Journal of Chemical Physics | volume = 22 | issue = 6 | page = 1152| year = 1954 |bibcode = 1954JChPh..22.1152B }}</ref>
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]]), <math>\Phi_\text{G} \neq - \langle P\rangle V</math>.<ref>{{cite book |title=Thermodynamics of Small Systems |last1=Hill |first1=Terrell L. |year=2002 |publisher=Courier Dover Publications |isbn=9780486495095 }}</ref>

== See also ==
* [[Gibbs free energy|Gibbs energy]]
* [[Helmholtz free energy|Helmholtz energy]]

==References==
{{Reflist}}

== External links ==
* [https://web.archive.org/web/20060909024313/http://theory.ph.man.ac.uk/~judith/stat_therm/node88.html Grand Potential (Manchester University)]

[[Category:Thermodynamics]]
[[Category:Lev Landau]]

Grand potential - Revision history

imported>Hellacioussatyr at 16:28, 8 April 2025