TFAE: Legendre Duality Between Entropy and Free Energy

Work in the thermodynamic limit for a sequence of finite systems (classical or lattice) where both a microcanonical entropy density and a canonical log-partition density exist:

• Microcanonical entropy density $s(u)$ (see microcanonical entropy density ).
• Canonical pressure / log-partition density $p(\beta)$ (see pressure (log-partition) density ) for inverse temperature $\beta$ (see canonical ensemble ).

The following are equivalent ways to state “entropy–free energy Legendre duality holds without loss,” i.e. the canonical and microcanonical descriptions match at the level of equilibrium thermodynamics.

1. Legendre–Fenchel transform from entropy to pressure.
   The canonical pressure is the Legendre–Fenchel transform of $s$:

   $$p(\beta) = \sup_{u}\,\{ s(u) - \beta u \}.$$

2. Biconjugacy from pressure back to entropy (no concavity defect).
   The entropy is recovered exactly from $p$ by the dual transform:

   $$s(u) = \inf_{\beta}\,\{ p(\beta) + \beta u \}.$$

   Equivalently, $s$ equals its concave upper-semicontinuous envelope (it has no “nonconcave” regions).

3. Unique saddle point / matching equations of state.
   For each $\beta$ in the region of interest, the supremum in $p(\beta)=\sup_{u}\{s(u)-\beta u\}$ is attained at a unique $u_\beta$, and the duality relation holds:

   $$\beta \in \partial s(u_\beta),$$

   where $\partial s$ denotes the subgradient (singleton if $s$ is differentiable).
   In differentiable regions this becomes $\beta = s'(u_\beta)$.

4. Differentiability of pressure and a unique energy density.
   The pressure $p(\beta)$ is differentiable at $\beta$ and

   $$-p'(\beta) = u_\beta,$$

   meaning the canonical ensemble has a unique thermodynamic energy density at $\beta$.
   Non-differentiability of $p$ corresponds to a nontrivial subgradient and (typically) coexistence.

5. Ensemble equivalence at the level of equilibrium macrostates.
   The set of equilibrium macrostates selected by the microcanonical ensemble at energy $u_\beta$ coincides with the set selected by the canonical ensemble at inverse temperature $\beta$ (see ensemble averages ).
   Failure of this equivalence is captured by ensemble equivalence breakdown and often appears in long-range systems .

6. Large-deviation characterization of energy fluctuations.
   Under the canonical ensemble at $\beta$, the empirical energy density satisfies a large deviation principle with good rate function

   $$I_\beta(u) = \beta u - s(u) + p(\beta),$$

   and $I_\beta$ has a unique minimizer at $u=u_\beta$.
   This is the probabilistic expression of “canonical energy concentrates at the Legendre-dual point.”

Prerequisites and context: Fenchel (convex conjugate) , Legendre transform , microcanonical entropy density , pressure (log-partition) density , canonical ensemble , canonical partition function .