Legendre duality between free energy and entropy

Statement (entropy–free energy duality)

Let $s(u)$ be the microcanonical entropy density as a function of energy density $u$ (see Boltzmann (microcanonical) entropy ), and let

be the canonical pressure/free-energy potential (from canonical partition function ), where $\beta$ is inverse temperature (see temperature ).

Under standard short-range assumptions ensuring existence of these limits and a large-deviation principle for the energy, the canonical potential is the Legendre–Fenchel transform of $s$:

Moreover, $s$ is the concave Legendre–Fenchel biconjugate of itself, and in particular

with $s^{**}=s$ whenever $s$ is concave and upper semicontinuous (see Fenchel conjugate and Fenchel–Moreau theorem ).

Key hypotheses

• Existence of thermodynamic limits for $s(u)$ and $\psi(\beta)$ (typically short-range, stable interactions).
• A large-deviation principle for the energy under the canonical ensemble (often via Varadhan’s lemma ).
• Regularity ensuring $Z_N(\beta)$ grows exponentially in $N$ and $s(u)$ is well-defined on its domain.

Conclusions

• Forward transform: $\psi(\beta)$ is obtained from $s(u)$ by a Legendre–Fenchel transform (a convex-analysis version of the Legendre transform ).
• Inverse transform (when concave): if $s$ is concave (as expected from thermodynamic stability), then $s(u)=\inf_\beta(\psi(\beta)+\beta u)$.
• Thermodynamic identification: in physical units, the Helmholtz free energy $F(T)$ (see Helmholtz free energy ) is related to $\psi(\beta)$ by scaling, and the duality expresses $F$ as the appropriate Legendre transform of entropy/internal energy.

Cross-links to definitions

• Microcanonical side: microcanonical measure , Boltzmann entropy .
• Canonical side: canonical ensemble , partition function , statistical free energy .
• Convex analysis: Legendre transform , Fenchel conjugate , Fenchel–Moreau theorem .

Proof idea / significance

The canonical partition function is a Laplace transform of the microcanonical density of states. In the thermodynamic limit, Laplace-type asymptotics (a “Laplace principle”) turn $\frac{1}{N}\log Z_N(\beta)$ into a supremum of $s(u)-\beta u$, yielding the transform. The inverse transform follows from general biconjugation results (Fenchel–Moreau) once the correct convex/concave conventions are applied.

This duality is the formal backbone behind passing between energy-controlled and temperature-controlled descriptions of equilibrium.