Grand Canonical Potential (Landau Free Energy)

Equivalent forms

\Omega = F - \mu N = U - TS - \mu N

\Omega = -PV

d\Omega = -S\,dT - P\,dV - N\,d\mu

The grand potential obeys Ω = −PV exactly — the entire equation of state is encoded in one number.

Symbol	Name	SI unit	Dimension	Range
$Ω$	Grand potentialoutput Thermodynamic potential for open systems at fixed T, V, μ	J	ML²T⁻²	-10000 – 0
$μ$	Chemical potential Energy cost per particle added from the reservoir	J	ML²T⁻²	-5 – 0
$T$	Temperature Temperature of the reservoir	K	Θ	10 – 10000

Unit systems

SI:: J
chemistry:: kJ/mol; $\mu in eV$ per particle
natural:: $\hbar = k_B = 1$

Where it holds

Valid for systems in thermal and chemical equilibrium with a reservoir. Extensive in V at fixed

T and \mu

.

Discovery

Josiah Willard Gibbs (ensemble) / Lev Landau (free energy form) · 1902

Gibbs introduced the grand canonical ensemble in 1902. Landau systematised its use as a free energy in the 1930s, applying it to phase transitions and quantum liquids.

Try this

What determines the equilibrium number of particles in an open system?

When a system can exchange both energy and particles with a reservoir, the equilibrium state minimises the grand potential Ω = F − μN. This governs adsorption, chemical reactions, semiconductor doping, and neutron star composition.

Research status: stable

Common misconceptions

\Omega = -PV

is exact (not an approximation) for systems large enough that P is well-defined. For finite systems edge corrections apply.

Derivation

Grand partition function

\Xi = \Sigma _{N=0}^{\infty } z^N Z_N

where

z = e^{\beta \mu }

is the fugacity and Z_N the N-particle canonical partition function.

\Omega = -k_BT \ln \Xi

.

For ideal quantum gases

\Xi = \Pi _i \Xi _i

:

\Xi _i = (1 \mp ze^{-\beta \varepsilon _i})^{\mp 1}

for bosons/fermions, reproducing BE/FD distributions.

Limiting cases

\mu \to -\infty

⟶ ⟨N⟩

\to 0

; system empties;

\Omega \to 0

Fixed N (canonical)⟶

\Omega \to F - \mu N

; Legendre transform recovers Helmholtz free energy

\Omega = -PV

always⟶ Equation of state

P = -\Omega /V

; all pressure laws derivable from grand potential

Canonical Partition Function→Helmholtz Free Energy→Gibbs Free Energy→Fermi-Dirac Distribution→