Legendre transform from Helmholtz free energy to grand potential

How the grand potential arises as the Legendre transform of the Helmholtz free energy with respect to particle number.

Legendre transform from Helmholtz free energy to grand potential

In equilibrium thermodynamics, different constraints lead naturally to different thermodynamic potentials. The grand potential is the potential adapted to fixed temperature $T$ , volume $V$ , and chemical potential $\mu$ , i.e. the setting of the grand canonical ensemble .

Definition (thermodynamic Legendre transform)

Let $F(T,V,N)$ be the Helmholtz free energy, i.e. the potential adapted to fixed $(T,V,N)$ (see Helmholtz free energy and thermodynamic system ).
The grand potential $\Omega(T,V,\mu)$ is defined as the Legendre transform of $F$ with respect to the particle number $N$ :

\Omega(T,V,\mu) \;=\; \inf_{N\in\mathbb{N}} \bigl(F(T,V,N) - \mu N\bigr).

At the minimizing $N=N^*(T,V,\mu)$ (when it exists), the stationarity condition is the familiar conjugacy relation

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

Here $\mu$ is the chemical potential , i.e. the intensive variable conjugate to $N$ .

In the thermodynamic limit (see thermodynamic limit ), one often treats $N$ as effectively continuous, and the infimum can be interpreted as a minimum over $N\ge 0$ .

Statistical-mechanical counterpart

In statistical mechanics, the same transform is encoded by switching from the canonical normalization factor canonical partition function to the grand partition function . Concretely, if the canonical free energy is defined by the construction free energy from a partition function , then the grand potential satisfies

\Omega(T,V,\mu) \;=\; -\frac{1}{\beta}\,\log \Xi(\beta,\mu,V),

where $\beta$ is the inverse temperature and $\Xi$ is constructed by the grand-canonical partition-function construction .

Physical interpretation

Passing from $F(T,V,N)$ to $\Omega(T,V,\mu)$ exchanges the constraint “fixed particle number” for “fixed chemical potential,” matching the physics of a system in particle exchange with a reservoir.
The term $-\mu N$ represents the energetic/entropic bookkeeping associated with exchanging particles with that reservoir.
In homogeneous equilibrium, $\Omega$ is extensive in $V$ and encodes the pressure $p$ via $\Omega(T,V,\mu) \;=\; -p(T,\mu)\,V$ (up to boundary corrections), connecting directly to the thermodynamic grand potential .