Grand potential

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Grand potential

The grand potential, also called the grand canonical potential or Landau free energy, is a thermodynamic quantity used for open systems that can exchange energy and particles with a reservoir at a fixed temperature T and chemical potential μ. It is the natural state function for the grand canonical ensemble.

Definition
Φ_G = U − T S − μ N
where U is internal energy, S is entropy, N is particle number, and μ is the chemical potential.

How it changes
dΦ_G = −P dV − S dT − N dμ
This comes from the fundamental relation dU = T dS − P dV + μ dN. You can also write
dΦ_G = dU − T dS − S dT − μ dN − N dμ.

Equilibrium meaning
When the system is in thermodynamic equilibrium at fixed volume V, temperature T, and chemical potential μ, Φ_G is at a minimum. If T and μ are constant and the volume is fixed, small changes do not increase Φ_G.

Relation to other energies
F = U − T S is the Helmholtz free energy, so Φ_G = F − μ N.
Some authors call Φ_G the Landau free energy or Landau potential.

Homogeneous (uniform) systems
For a uniform system, Φ_G = −P V, i.e., the grand potential equals minus pressure times volume. In this case, the Gibbs energy can be written as G = ⟨N⟩ μ.

Limitations
In small systems or systems with long-range interactions, Φ_G may not equal −P V.

Why it matters
The grand potential is useful for calculating average quantities like the mean particle number, energy, and entropy, and it can be obtained from derivatives with respect to μ, T, and V.