Negative heat capacity in the microcanonical ensemble

In the microcanonical ensemble, heat capacity can be negative when the entropy has locally convex curvature; this cannot occur in the canonical ensemble and signals possible ensemble inequivalence.

Negative heat capacity in the microcanonical ensemble

Prerequisites and notation

Microcanonical entropy density: microcanonical entropy density
Canonical ensemble comparison: canonical ensemble
Thermodynamic stability language: thermodynamic stability , heat capacity
Inequivalence context: ensemble inequivalence for long-range systems

Extension (and key example mechanism): negative microcanonical heat capacity

Microcanonical temperature and curvature criterion

Let $S(E)$ be the microcanonical entropy and write $S(E)=N s(u)$ with energy density $u=E/N$ . The microcanonical inverse temperature is

\beta(E)=\frac{\partial S}{\partial E}(E), \qquad T(E)=\frac{1}{\beta(E)}.

Differentiate $T$ with respect to $E$ :

\frac{dT}{dE} = -\frac{T^2}{N}\,s''(u).

Therefore the microcanonical heat capacity

C_{\mathrm{mic}}(E)=\left(\frac{dT}{dE}\right)^{-1}

has the sign

C_{\mathrm{mic}}(E) < 0 \quad\Longleftrightarrow\quad s''(u) > 0.

So negative heat capacity is equivalent to local convexity of the entropy density.

Why canonical heat capacity cannot be negative

In the canonical ensemble, energy fluctuations satisfy

\mathrm{Var}_\beta(E)=\frac{\partial^2}{\partial \beta^2}\log Z(\beta)\ge 0.

Using $U(\beta)=\langle E\rangle_\beta$ and $C_{\mathrm{can}}=\frac{dU}{dT}$ , one finds

C_{\mathrm{can}}(\beta)=\beta^2\,\mathrm{Var}_\beta(E)\ge 0.

Hence negative heat capacity is a specifically microcanonical (or finite/nonadditive) phenomenon and is incompatible with canonical stability.

Interpretation and where it appears

In additive short-range systems at equilibrium, $s(u)$ is typically concave in the thermodynamic limit, so $C_{\mathrm{mic}}\ge 0$ .
In nonadditive/long-range systems, $s(u)$ may be nonconcave, producing $C_{\mathrm{mic}}<0$ and indicating ensemble inequivalence .
In finite systems (e.g., small clusters) interface/surface contributions can also produce effective convex intruders in $S(E)$ over a range of energies.

Diagnostic use

Observed $C_{\mathrm{mic}}<0$ is a strong diagnostic that:

microcanonical and canonical descriptions may disagree over that energy range, and
a convex-analytic “envelope” construction is needed to relate entropy to canonical free energy (see Legendre duality ).