Canonical, Grand-Canonical, and Microcanonical Ensembles

A black-hole “ensemble” is a choice of boundary data, not merely a choice of thermodynamic vocabulary. Fixing energy defines a microcanonical question; fixing proper boundary temperature defines a canonical one; fixing angular velocity or electric potential additionally opens angular-momentum or charge exchange. The corresponding entropy, free energy, and Gibbs potential are Legendre transforms of one underlying equation of state, but their stability criteria and even their admissible gravitational saddles can differ.

Required background. Gravitational Actions and Boundary Terms supplies the boundary variations that distinguish fixed extensive from fixed intensive data.

Helpful background. Quantum Fields in Curved Spacetime supplies thermal density operators; Einstein Equations supplies the stationary saddles; and Black-Hole Geometry and Mechanics supplies mass, spin, charge, and horizon mechanics.

Boundary data define the ensemble

Let the equilibrium entropy be $S(E,J,Q)$. Its differential is

$$dS=\beta\,dE-\beta\Omega\,dJ-\beta\Phi\,dQ, \qquad \beta=\frac{1}{T}.$$

The variables held fixed determine both allowed fluctuations and the potential whose stationary point is tested:

Ensemble	Fixed boundary data	Thermodynamic function	Allowed exchanges
Microcanonical	$E,J,Q$	$S(E,J,Q)$	none of these charges
Canonical	$T,J,Q$	$F=E-TS$	energy
Grand canonical	$T,\Omega,\Phi$	$\mathcal G=E-TS-\Omega J-\Phi Q$	energy, angular momentum, and charge

Mixed ensembles are also possible: for example, one may fix $T$, $J$, and $\Phi$ while allowing charge but not angular momentum to fluctuate. Calling a calculation simply “canonical” is incomplete unless the treatment of $J$ and $Q$ is also stated.

In gravity, these choices appear as boundary conditions on the metric and gauge field. A Euclidean Dirichlet problem fixes the induced boundary geometry, including the proper thermal-circle length. Fixing a boundary electrostatic potential gives the grand-canonical electromagnetic variational problem; fixing charge requires the corresponding Legendre boundary term. The same logic applies to a rotating boundary frame.

Legendre transforms and stationary points

At fixed $T,\Omega,\Phi$, the saddle obeys

$$\delta\mathcal G =\delta E-T\delta S-\Omega\delta J-\Phi\delta Q=0.$$

The first variation recovers the first law. The second variation decides local stability only against fluctuations admitted by that ensemble. In the microcanonical description a fluctuation with $\delta E\ne0$ is forbidden; in the canonical description it is precisely the energy-exchange direction that produces the heat-capacity test. Hence an entropy maximum at fixed $E$ need not be a free-energy minimum at fixed $T$.

At leading saddle order the Euclidean partition function has the formal form

$$Z(\beta,\Omega,\Phi)\simeq \sum_s e^{-I_E[g_s,A_s]}, \qquad \mathcal G_s=\frac{I_E[g_s,A_s]}{\beta}.$$

The sum is meaningful only after the boundary geometry, reference subtraction or counterterms, gauge data, and integration contour have been fixed. Gibbons and Hawking identify the action/partition-function relation with its boundary contribution Gibbons and Hawking 1977, pp. 2752–2756. This is a semiclassical saddle expansion, not a nonperturbative definition of the gravitational path integral.

Why gravity makes the distinction sharp

An isolated asymptotically flat Schwarzschild black hole has $T=1/(8\pi GE)$ and therefore negative heat capacity. It can still be discussed microcanonically at fixed total energy, but it is not a stable canonical system coupled to an infinite heat bath. A finite reflecting cavity changes the question: the wall fixes a proper temperature and limits the available volume. York finds a small unstable and a large locally stable Schwarzschild branch for appropriate wall data York 1986, §§II–V and VIII.

Anti-de Sitter boundary conditions provide another canonical setting. There, thermal AdS and a sufficiently large black hole share the same conformal boundary data, so their renormalized on-shell actions can be compared. The dominant saddle changes at the Hawking–Page temperature Hawking and Page 1983, pp. 577–588. That global phase comparison is distinct from the sign of one branch’s local heat capacity.

For intuition, in a well-defined ordinary canonical ensemble, $C=\beta^2(\langle E^2\rangle-\langle E\rangle^2)\geq0$. A formal gravitational saddle with $C\lt0$ therefore cannot represent a stable canonical probability distribution by itself. It may still occur as an unstable saddle in the action analysis. This connects the boundary-value language to the familiar statement that a variance cannot be negative; the heat-capacity calculation shows how changing the boundary makes a stable branch possible.

These examples cannot be mixed. A cavity free energy depends on wall radius and wall temperature; an AdS free energy depends on the asymptotic AdS boundary; and an asymptotically flat microcanonical density fixes total energy. Subtracting different reference backgrounds or comparing saddles with different boundary metrics does not produce a thermodynamic phase transition.

Exercise

A system has entropy $S(E)$ with $S''(E)\gt0$ at an equilibrium energy. Explain why this point can be allowed microcanonically but unstable canonically, and relate the sign to the heat capacity.

Solution

Microcanonically, $E$ is fixed, so the energy direction is not an allowed fluctuation. Canonically, the exponent is $S(E)-\beta E$ and its second derivative is $S''(E)$. A stable Gaussian saddle requires this curvature to be negative. Since $S''=-1/(T^2C)$, positive $S''$ means $C\lt0$ and the canonical energy integral runs away from the stationary point even though the microcanonical state at that fixed energy remains definable.

References

• Gibbons, G. W., and S. W. Hawking (1977), “Action Integrals and Partition Functions in Quantum Gravity,” Physical Review D 15, 2752–2756. DOI.
• Hawking, S. W., and D. N. Page (1983), “Thermodynamics of Black Holes in Anti-de Sitter Space,” Communications in Mathematical Physics 87, 577–588. DOI.
• York, J. W. Jr. (1986), “Black-Hole Thermodynamics and the Euclidean Einstein Action,” Physical Review D 33, 2092–2099. DOI.