Microcanonical entropy density

Extension: entropy density in the thermodynamic limit

Let $\Lambda\subset\mathbb{Z}^d$ (or a region in $\mathbb{R}^d$) be a finite volume with $|\Lambda|$ degrees of freedom and Hamiltonian $H_\Lambda$. Write the energy density as $u=E/|\Lambda|$.

A common microcanonical entropy is the finite-volume Boltzmann form (see Boltzmann microcanonical entropy ). The microcanonical entropy density is the thermodynamic-limit object

when the limit exists in an appropriate sense (often via upper/lower limits or regularizations), where $S_\Lambda(u)$ is a microcanonical entropy at energy density $u$.

Basic properties and interpretation

• Thermodynamic meaning: $s(u)$ is the entropy per degree of freedom at fixed energy density, matching thermodynamic entropy after identifying $u$ with the thermodynamic internal energy density.
• Concavity and stability: in many short-range systems, $s(u)$ is concave, reflecting stability ; nonconcavity is a hallmark of ensemble equivalence breakdown and can coexist with negative heat capacity (microcanonical) .
• Large deviations viewpoint: for suitable macroscopic observables, $s(u)$ appears as (minus) a rate function in a large deviation principle .

Duality with free energy (Legendre–Fenchel)

Let

be the log-partition density (see pressure (log-partition density) ), where $Z_\Lambda(\beta)$ is the canonical partition function at inverse temperature $\beta$.

Under standard hypotheses, $s$ and $\psi$ are related by convex duality:

This is a Legendre–Fenchel transform (see Fenchel conjugate and Legendre transform ) and is one of the equivalences packaged in Legendre duality (entropy/free energy) .

Prerequisites / cross-links

• canonical ensemble , statistical free energy
• second law (for the macroscopic role of entropy)