Statistical mechanics
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Thegrand potential orLandau potential orLandau free energy is a quantity used instatistical mechanics, especially forirreversible processes inopen systems.The grand potential is the characteristic state function for thegrand canonical ensemble.

The grand potential is defined by

${\displaystyle {\mathrm{\Phi }}_{\mathrm{G}}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}U-TS-\mu N}$

whereU is theinternal energy,T is thetemperature of the system,S is theentropy, μ is thechemical potential, andN is the number of particles in the system.

The change in the grand potential is given by

whereP ispressure andV isvolume, using thefundamental thermodynamic relation (combinedfirst andsecondthermodynamic laws);

${\displaystyle dU=TdS-PdV+\mu dN}$

When the system is inthermodynamic 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.

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

${\displaystyle \mathrm{\Omega }\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}F-\mu N=U-TS-\mu N}$

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

In the case of a scale-invariant type of system (where a system of volume${\displaystyle \lambda V}$ has exactly the same set of microstates as${\displaystyle \lambda }$ systems of volume${\displaystyle V}$), 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:

${\displaystyle {\left(\frac{\mathrm{\partial }⟨P⟩}{\mathrm{\partial }V}\right)}_{\mu ,T}=0,}$

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

${\displaystyle {\left(\frac{\mathrm{\partial }⟨N⟩}{\mathrm{\partial }V}\right)}_{\mu ,T}=\frac{N}{V}.}$

In this case we simply have${\displaystyle {\mathrm{\Phi }}_{\mathrm{G}}=-⟨P⟩V}$, as well as the familiar relationship${\displaystyle G=⟨N⟩\mu }$ for theGibbs free energy.The value of${\displaystyle {\mathrm{\Phi }}_{\mathrm{G}}}$ 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${\displaystyle {\mathrm{\Phi }}_{\mathrm{G}}=-⟨P⟩V}$ 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 thethermodynamic limit),${\displaystyle {\mathrm{\Phi }}_G\ne -⟨P⟩V}$.^[5]

• Gibbs energy
• Helmholtz energy

• Grand Potential (Manchester University)

• Thermodynamics
• Lev Landau

• Articles with short description
• Short description matches Wikidata