Ensemble Inequivalence and Microcanonical Saddles

Canonical, grand-canonical, and microcanonical ensembles impose different gravitational boundary conditions and can select different AdS saddles. Negative heat capacity excludes a saddle from canonical local equilibrium but can be compatible with a microcanonical entropy maximum. This is ensemble inequivalence among states of a fixed theory—not the ensemble averaging of theories encountered in gravitational path integrals.

Required background. Euclidean Actions, Boundary Terms, and Free-Energy Comparisons fixes boundary terms and potentials. AdS Black Branes and Holographic Thermodynamics supplies the equation of state. Thermodynamic Limits, Phases, and Ensemble Equivalence supplies the general convexity criteria.

Helpful background. Large Deviations and Phase Coexistence explains Legendre–Fenchel rather than naive Legendre transforms.

First application. Compare a small AdS black hole at fixed temperature and fixed energy and explain why one ensemble may exclude or stabilize the saddle.

Potentials and constrained saddles

For an entropy $S(E,Q,J)$,

$$\beta=\frac{\partial S}{\partial E}, \qquad -\beta\mu=\frac{\partial S}{\partial Q}, \qquad -\beta\Omega=\frac{\partial S}{\partial J}.$$

The canonical partition function is formally

$$Z(\beta)=\int dE\,e^{S(E)-\beta E},$$

and a saddle satisfies $S'(E)=\beta$. Its Gaussian stability requires

$$S''(E)=-\frac{1}{T^2C}<0,$$

so $C>0$. The microcanonical problem instead fixes $E$ and extremizes entropy over all remaining degrees of freedom. A branch with $C<0$ can therefore be present microcanonically even though its energy fluctuations make it unavailable as a stable canonical saddle.

With charge or angular momentum, replace the scalar curvature by the Hessian on the allowed fluctuation space. Which variables fluctuate is dictated by the boundary action: fixed potential, fixed flux, fixed angular velocity, and fixed angular momentum are different problems.

Small AdS black hole as the representative case

For spherical AdS–Schwarzschild,

$$T(r_h)=\frac{d r_h^2+(d-2)L^2}{4\pi L^2r_h}.$$

The small branch below the temperature minimum has $dT/dr_h<0$ and hence negative heat capacity. At fixed temperature it is a local canonical instability. At fixed energy, however, the same stationary geometry can contribute to a microcanonical density of states and may dominate over other configurations in a restricted energy range.

This does not guarantee full microcanonical stability: one must still compare all configurations at the same charges and test perturbations that preserve them. Radiation, localized black holes, internal-space modes, or multi-object configurations can have greater entropy.

Convex envelopes and missing branches

The canonical free energy is the Legendre–Fenchel transform of the entropy. If $S(E)$ contains a nonconcave segment, the canonical ensemble replaces it by its concave envelope; energies in that segment need not appear as stable canonical equilibria. The inverse transform then recovers the envelope, not the original nonconcave function. This is the mathematical origin of ensemble inequivalence in long-range systems such as gravity.

In a boundary QFT on noncompact space with additive local dynamics, ordinary thermodynamic limits often restore equivalence away from coexistence. Holographic gravitational saddles may nevertheless display metastable or constrained branches. The correct conclusion depends on the boundary theory’s volume limit, not on bulk intuition alone.

State ensembles versus theory ensembles

A canonical or microcanonical ensemble averages states in one Hamiltonian. An ensemble of boundary Hamiltonians or couplings is a different object. Connected multi-boundary gravitational amplitudes can raise the latter question, but nothing about a black hole’s negative heat capacity implies averaging over theories.

Keeping these meanings separate prevents a common category error: a Legendre transform between $E$ and $\beta$ changes thermodynamic control variables, not the microscopic theory.

Adversarial controls

Apply the wrong Hessian. Test a fixed-charge saddle using charge fluctuations. The apparent instability may be forbidden by the ensemble; conversely, suppressing allowed fluctuations can hide a real one.

Invert a canonical free energy naively. A simple Legendre transform across a first-order transition returns the concave envelope and erases microcanonical branches. The missing information cannot be reconstructed without additional data.

Confuse constrained existence with dominance. A fixed-energy black-hole solution may exist but lose to another topology or fragmentation channel with larger entropy.

Evidence ceiling

The constrained variational problem and Hessian determine local stability in a declared state ensemble. Comparing all known saddles at identical conserved quantities determines dominance within that set. Neither proves completeness of the saddle set, and neither licenses a claim about an ensemble of theories.

The original Hawking–Page saddle comparison is canonical-ensemble data Hawking and Page 1983; using it in a microcanonical argument requires the constrained transform and branch analysis developed on this page.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Chamblin, Andrew, Roberto Emparan, Clifford V. Johnson, and Robert C. Myers. “Charged AdS Black Holes and Catastrophic Holography.” Physical Review D 60, 064018 (1999). DOI; arXiv:hep-th/9902170.
• Hawking, Stephen W., and Don N. Page. “Thermodynamics of Black Holes in Anti-de Sitter Space.” Communications in Mathematical Physics 87, 577–588 (1983). DOI.
• York, James W. “Black-Hole Thermodynamics and the Euclidean Einstein Action.” Physical Review D 33, 2092–2099 (1986). DOI.