Breakdown of Ensemble Equivalence

For most systems the microcanonical, canonical, and grand canonical ensembles are three interchangeable languages for the same thermodynamics — they agree exactly in the thermodynamic limit. But "most" is not "all". There are physically important situations where the ensembles genuinely disagree, predicting different equations of state, different heat capacities, even different signs of the heat capacity. Understanding when and why equivalence fails is as instructive as understanding why it usually holds. The mathematical heart of the matter is a single condition: the entropy density $s(\varepsilon)$ must be a strictly concave function of the energy density $\varepsilon$ . When it is, the Legendre transform that carries the microcanonical $s(\varepsilon)$ to the canonical free energy is invertible, and the two ensembles carry identical information. When $s(\varepsilon)$ has a non-concave (convex) region — a "dip" — the Legendre transform silently replaces that dip by its concave envelope (the common-tangent construction). The canonical ensemble literally cannot see the non-concave region; the information is erased. In the erased window the microcanonical ensemble describes states the canonical ensemble skips over entirely. Three classes of system break concavity. First-order phase transitions produce a straight (marginally concave) segment where energy jumps discontinuously in the canonical ensemble but interpolates smoothly in the microcanonical one. Long-range interactions — self-gravitating stars, small nuclei, unscreened plasmas — make energy non-additive, so $s(\varepsilon)$ develops a genuine convex intruder and the microcanonical heat capacity can go negative, something the canonical ensemble forbids ( $C_V=\langle\Delta E^2\rangle/k_BT^2\ge0$ ). Small systems, where the surface is not negligible against the bulk, show residual ensemble differences of order $1/N$ . In each case the lesson is the same: ensemble equivalence is a theorem with hypotheses, and the hypotheses can fail.

Key takeaways

The three ensembles agree only when the entropy density $s(\varepsilon)$ is strictly concave; concavity is the hypothesis behind ensemble equivalence.
The canonical free energy is the Legendre transform of $s(\varepsilon)$ ; on a non-concave $s$ the transform returns the concave hull, erasing the dip — canonical cannot see it.
At a first-order transition $s(\varepsilon)$ has a straight segment: canonical energy jumps (latent heat), microcanonical energy interpolates through the forbidden gap.
Long-range/non-additive systems (self-gravity) develop convex regions where the microcanonical heat capacity is negative — impossible in the canonical ensemble, where $C_V=\langle\Delta E^2\rangle/k_BT^2\ge0$ .
Ensembles disagree exactly where fluctuations are macroscopic and $1/\sqrt{N}$ suppression fails.
Key idea: ensemble equivalence needs a concave entropy; a non-concave $s(\varepsilon)$ — from first-order transitions, long-range forces, or small size — makes the canonical Legendre transform lose information and the ensembles genuinely disagree.

The core insight

Ensemble equivalence is not automatic. It holds when the microcanonical entropy density $s(\varepsilon)$ is strictly concave. When $s$ is non-concave — a first-order transition (straight segment), a long-range non-additive interaction (convex intruder), or a small system — the canonical Legendre transform collapses $s$ to its concave hull, discards the non-concave window, and the microcanonical ensemble describes states (including negative-heat-capacity ones) the canonical ensemble cannot.

Where this lives today

Ensemble inequivalence is a live topic wherever interactions are long-range or systems are small. Astrophysics leans on the microcanonical ensemble because self-gravitating systems (star clusters, galaxies) have negative heat capacity and no canonical description — the gravothermal catastrophe is inequivalence in action. Nuclear multifragmentation experiments report negative microcanonical heat capacities in hot nuclei. Cold-atom and trapped-ion platforms now engineer tunable long-range interactions, letting experimentalists dial between equivalent and inequivalent regimes and watch the phase diagram change with the ensemble. In simulations, choosing the wrong ensemble near a first-order transition produces spurious van der Waals loops; the microcanonical ensemble is often preferred there precisely because it resolves the coexistence region the canonical ensemble collapses. The unifying modern statement, made rigorous by Touchette and others, is a large-deviation one: equivalence holds if and only if the microcanonical entropy is concave.

The physical picture

Start with the object that decides everything: the entropy density $s(\varepsilon)=\lim_{N\to\infty}\frac1N S(N\varepsilon)/k_B$ , plotted against the energy per particle $\varepsilon$ . For a normal short-range system this curve is a smooth cap — strictly concave, curving downward everywhere. The canonical ensemble asks a Legendre question of it: "at inverse temperature $\beta$ , which $\varepsilon$ maximizes $s(\varepsilon)-\beta\varepsilon$ ?" On a concave cap there is always exactly one answer, the tangent point moves smoothly as $\beta$ changes, and the two ensembles track each other perfectly. Now dent the cap — give $s(\varepsilon)$ a region that curves the wrong way (convex). The Legendre question now has the same answer for a whole range of $\beta$ : the maximizing $\varepsilon$ jumps across the dent from one shoulder to the other, never landing inside it. Geometrically, the canonical ensemble replaces the dented curve by the straight chord that bridges the dent — the concave hull. Every energy under that chord is invisible to the canonical ensemble. The microcanonical ensemble, which fixes $\varepsilon$ by hand, happily sits inside the dent.

Figure

A non-concave entropy density s(ε): the canonical Legendre transform replaces the convex dip by the straight common-tangent chord (its concave hull). Microcanonical states in the shaded gap are invisible to the canonical ensemble.

The punchline figure is the negative-heat-capacity signature. In the convex region the microcanonical temperature $T(\varepsilon)$ — defined by $1/T=\partial s/\partial\varepsilon$ — actually decreases as you add energy, so $C=d\varepsilon/dT<0$ . A cluster of stars is the classic example: add energy and it puffs out, cools, and its heat capacity is negative. The canonical ensemble, wired to a heat bath, can never reproduce this: there $C_V=\langle\Delta E^2\rangle/k_BT^2$ is a variance, manifestly non-negative.

Figure

Punchline: microcanonical temperature T(ε)=1/(∂s/∂ε) versus energy. In the convex region of s, T decreases as energy is added, so the heat capacity C=dε/dT is NEGATIVE — forbidden in the canonical ensemble where C_V=⟨ΔE²⟩/kT²≥0.

The core equations

Everything follows from the entropy density and its Legendre transform to the canonical free energy. The microcanonical ensemble fixes $\varepsilon$ and reads off $s$ ; the canonical ensemble fixes $\beta$ and reads off the free energy $\phi(\beta)$ . Equivalence is the statement that these two carry the same information.

s(\varepsilon)=\lim_{N\to\infty}\frac{1}{N}\frac{S(N,E=N\varepsilon)}{k_B},\qquad \frac{1}{k_BT}=\frac{\partial s}{\partial\varepsilon}\equiv\beta \qquad (1)

where: $s(\varepsilon)$ — entropy per particle at energy density $\varepsilon$ ; the microcanonical temperature is defined by the slope of $s$ . When $s$ is concave this slope decreases monotonically, so $T$ rises with $\varepsilon$ and $C>0$ . The canonical free energy is the Legendre transform of the entropy density. This is where equivalence is decided: the transform of a non-concave function equals the transform of its concave hull.

-\beta\phi(\beta)=\sup_{\varepsilon}\big[s(\varepsilon)-\beta\varepsilon\big] \qquad (2)

where: $\phi(\beta)$ — canonical free energy per particle; the supremum picks the energy that dominates the canonical partition sum. If two energies $\varepsilon_1<\varepsilon_2$ tie for the supremum at some $\beta^\ast$ , the canonical energy jumps between them — a first-order transition. Taking the transform twice recovers not $s$ but its concave hull $s^{\ast\ast}$ . Any non-concave dip is flattened; the microcanonical information there is lost to the canonical description.

s^{\ast\ast}(\varepsilon)=\inf_{\beta}\big[\beta\varepsilon-\beta\phi(\beta)\big]=\text{concave hull of }s(\varepsilon)\;\ge\;s(\varepsilon) \qquad (3)

where: $s^{\ast\ast}$ — double Legendre transform; it equals $s$ exactly where $s$ is concave and lies strictly above $s$ across any non-concave gap. Equivalence $\Leftrightarrow s=s^{\ast\ast}$ . The distinguishing thermodynamic signature is the sign of the heat capacity, which the two ensembles compute differently.

C^{\text{micro}}=\left(\frac{\partial\varepsilon}{\partial T}\right)^{-1}\!\!\cdot\!\left(\ldots\right)\;\text{can be}<0,\qquad C_V^{\text{can}}=\frac{\langle\Delta E^2\rangle}{k_BT^2}\ge0 \qquad (4)

where: the microcanonical $C$ inherits the sign of $-\partial^2 s/\partial\varepsilon^2$ and turns negative in a convex region; the canonical $C_V$ is a variance and cannot. A negative microcanonical heat capacity is the unmistakable fingerprint of ensemble inequivalence.

Derivation AWhy the canonical transform erases a non-concave region

The goal is to show precisely how the canonical ensemble loses the information in a non-concave portion of the entropy density, so that first-order transitions and long-range systems come out inequivalent. The strategy is to follow the supremum in the Legendre transform (2) as $\beta$ varies and watch the maximizing energy skip over any convex dip, then confirm by taking the transform a second time that the recovered curve is the concave hull, not the original.

Step 1. Fix $\beta$ and look for the energy $\varepsilon$ maximizing $s(\varepsilon)-\beta\varepsilon$ . The stationarity condition is $s'(\varepsilon)=\beta$ : the tangent line of slope $\beta$ touches the curve. On a concave curve there is exactly one such tangent point for each $\beta$ .

Step 2. Suppose $s$ has a convex dip between $\varepsilon_1$ and $\varepsilon_2$ . There is a special slope $\beta^\ast$ — the common tangent — whose line touches the curve at both $\varepsilon_1$ and $\varepsilon_2$ while lying above it in between. For $\beta$ near $\beta^\ast$ the supremum sits at $\varepsilon_1$ (for $\beta>\beta^\ast$ ) or jumps to $\varepsilon_2$ (for $\beta<\beta^\ast$ ).

Step 3. So the maximizing energy as a function of $\beta$ is discontinuous at $\beta^\ast$ : it never takes any value strictly inside $(\varepsilon_1,\varepsilon_2)$ . Those energies simply do not appear in the canonical ensemble at any temperature — this is the latent-heat gap of a first-order transition. Why the jump. The canonical weight $e^{Ns(\varepsilon)-N\beta\varepsilon}$ is exponentially dominated by the largest exponent; when two energies tie, probability splits between them, and the mean energy jumps as the balance tips. Intermediate energies are exponentially suppressed.

Step 4. Now take the Legendre transform a second time, (3). Because $\phi(\beta)$ was built only from tangent points, the double transform $s^{\ast\ast}$ reconstructs the upper concave envelope of $s$ — the chord across the dip — not $s$ itself. Across the gap, $s^{\ast\ast}>s$ : the canonical route has strictly overestimated the entropy of the intermediate energies (it replaced them by a mixture of the endpoints).

Step 5. Test numerically on a toy non-concave entropy $s(\varepsilon)=-\varepsilon^4+3\varepsilon^2$ , which is convex for $|\varepsilon|<1/\sqrt2$ . Sweeping $\beta$ and computing (2) then (3) reproduces the concave hull, strictly above $s$ inside the convex window, and the maximizing $\varepsilon(\beta)$ shows a finite jump. Verified: the double Legendre transform of $s=-\varepsilon^4+3\varepsilon^2$ equals its concave hull (≥ s everywhere, strictly greater in the |ε|<1/√2 gap), and argmax ε(β) jumps by more than 0.5 across the transition — all in verify_aq-sm-breakdown-ensemble-equivalence.py.

Go deeper · Derivation B · 1 minMicrocanonical negative heat capacity and its canonical impossibility

Derivation BMicrocanonical negative heat capacity and its canonical impossibility

The goal is to derive the sign of the microcanonical heat capacity from the curvature of $s(\varepsilon)$ , show it can be negative in a convex region, and prove that the canonical ensemble can never reproduce a negative heat capacity — the sharpest statement of inequivalence. The strategy is to differentiate the microcanonical temperature relation $1/T=\partial s/\partial\varepsilon$ and separately write the canonical heat capacity as an energy variance.

Step 1. From (1), $1/T=\partial s/\partial\varepsilon$ . Differentiate with respect to $\varepsilon$ : $\partial(1/T)/\partial\varepsilon=\partial^2 s/\partial\varepsilon^2=s''(\varepsilon)$ .

Step 2. In a concave region $s''<0$ , so $1/T$ decreases as $\varepsilon$ rises, i.e. $T$ increases with energy — the normal case, $C=d\varepsilon/dT>0$ . In a convex region $s''>0$ , so $1/T$ increases with $\varepsilon$ , meaning $T$ decreases as energy is added: $C=d\varepsilon/dT<0$ .

Step 3. A negative heat capacity is therefore the direct microcanonical reading of a convex entropy. Self-gravitating systems realize it: adding energy to a star cluster makes it expand, virialize, and cool — $C<0$ is measured in globular clusters and in nuclear multifragmentation. Why gravity does this. Long-range attraction makes the energy non-additive, $U(\lambda N)\propto\lambda^2 U(N)$ (degree 2, not degree 1), so entropy loses concavity — extensivity, the usual guarantee of concavity, fails.

Step 4. In the canonical ensemble the heat capacity is an equilibrium fluctuation, $C_V^{\text{can}}=\langle\Delta E^2\rangle/k_BT^2$ . A variance is non-negative, so $C_V^{\text{can}}\ge0$ always — the canonical ensemble is structurally incapable of a negative heat capacity.

Step 5. Hence in any convex-entropy window the two ensembles give opposite-sign heat capacities. This is not a subtle $1/N$ correction; it is a qualitative disagreement that survives the thermodynamic limit whenever additivity is broken. Verified: for the toy $s(\varepsilon)$ , the region where $s''=\partial^2 s/\partial\varepsilon^2>0$ coincides with $d T/d\varepsilon<0$ (negative $C$ ), while the canonical $C_V=\langle\Delta E^2\rangle/k_BT^2\ge0$ identically — verify script.

Limiting cases

Equivalence is the generic case; inequivalence appears at the boundaries where concavity or additivity fails.

Case	Ensembles	Reason
Short-range, single phase, $N\to\infty$	Equivalent	$s(\varepsilon)$ strictly concave; Legendre transform invertible.
First-order transition	Inequivalent in the gap	$s$ has a straight segment; canonical energy jumps, microcanonical interpolates.
Long-range / non-additive (self-gravity, small nuclei)	Inequivalent	$s$ convex region; microcanonical $C<0$ , impossible in canonical.
Small $N$ (surface not negligible)	Weakly inequivalent, $O(1/N)$	Finite-size corrections; equivalence restored as $N\to\infty$ .
Continuous transition (critical point)	Equivalent (but singular)	$s$ stays concave; fluctuations diverge but ensembles still agree.

Common trapCommon pitfall

Ensemble equivalence is often quoted as a universal fact — "just pick whichever ensemble is convenient." That is true only under its hypotheses: short-range interactions and a strictly concave entropy. It fails, genuinely and observably, at first-order transitions (where microcanonical and canonical disagree inside the coexistence gap) and for long-range or small systems (where the microcanonical heat capacity can be negative). A second trap is thinking a negative heat capacity violates thermodynamics: it does not — it is perfectly consistent microcanonically for a non-additive system, and its impossibility is a property of the canonical ensemble, not of nature. Choose the ensemble that matches the physical constraints: isolated system → microcanonical; heat bath → canonical.

Worked examples

Example 1 — Reading the sign of the heat capacity from curvature (enumerate, apply, boundary number). Take the toy entropy density $s(\varepsilon)=-\varepsilon^4+3\varepsilon^2$ (arbitrary units) and locate the negative-heat-capacity window.

Step 1. Enumerate the curvature: $s'(\varepsilon)=-4\varepsilon^3+6\varepsilon$ , $s''(\varepsilon)=-12\varepsilon^2+6$ .

Step 2. Apply the concavity test: $s''>0$ (convex) when $-12\varepsilon^2+6>0$ , i.e. $|\varepsilon|<\sqrt{6/12}=1/\sqrt2\approx\mathbf{0.707}$ .

Step 3. In that window $T(\varepsilon)$ decreases with energy, so the microcanonical heat capacity is negative for $|\varepsilon|<0.707$ and positive outside.

Step 4. Boundary number: the common-tangent (coexistence) points, where the canonical chord touches, satisfy $s'(\varepsilon)=0$ (by the symmetry of this $s$ ), giving $-4\varepsilon^3+6\varepsilon=0\Rightarrow\varepsilon=\pm\sqrt{3/2}\approx\pm\mathbf{1.225}$ . Between these the canonical ensemble uses the straight chord; the microcanonical curve dips below it, and the states in the shaded gap never appear canonically.

Example 2 — Non-additivity of a self-gravitating cluster. Estimate how the potential energy of $N$ mutually gravitating masses scales, to see why additivity (hence concavity) fails.

Step 1. The gravitational potential energy sums over all pairs: $U\sim-\dfrac{GM^2}{R}$ with total mass $M=Nm$ , so $U\propto N^2$ .

Step 2. Apply the additivity test: doubling the system, $N\to2N$ at fixed density, gives $U\to4U$ , i.e. $U(\lambda N)=\lambda^2 U(N)$ — degree 2, not degree 1.

Step 3. A short-range system would have $U\propto N$ (degree 1). The extra factor of $N$ means the energy is non-extensive: you cannot split the cluster into independent subsystems whose entropies add. Concavity of $s(\varepsilon)$ is not guaranteed, a convex region appears, and the microcanonical heat capacity can be negative — exactly what is observed for star clusters, which heat up as they lose energy.

Origins · 1 mingiovanni gallavotti and how this was discovered

giovanni gallavotti

Italian mathematical physicist, b. 1941 · rigorous statistical mechanics and ensembles

Giovanni Gallavotti (born 1941) is an Italian mathematical physicist known for placing statistical mechanics on rigorous foundations — the existence of the thermodynamic limit, the structure of Gibbs states, and the precise conditions under which ensembles are equivalent. His textbook Statistical Mechanics: A Short Treatise is a standard rigorous reference, and with E. G. D. Cohen he formulated the Gallavotti–Cohen fluctuation theorem (1995) for entropy production in chaotic non-equilibrium systems, extending the "chaotic hypothesis." The ensemble-equivalence question — when the microcanonical and canonical descriptions genuinely coincide, and the sharp counterexamples of first-order transitions and long-range forces where they do not — sits squarely in the tradition of rigorous results he helped build.

History

1902

— Gibbs's Elementary Principles proves ensemble equivalence for the systems he considers, and notes its subtleties.

1949

— van Hove establishes the existence of the thermodynamic limit for a class of short-range interactions.

1963

— Fisher and Ruelle prove rigorous existence-and-equivalence theorems, pinning down the short-range/stability hypotheses.

1968

— Lynden-Bell analyses the negative heat capacity of self-gravitating systems, a canonical example of inequivalence.

1995

— Gallavotti and Cohen formulate the fluctuation theorem, deepening the ensemble framework for non-equilibrium systems.

Going further

References. Pathria & Beale, Statistical Mechanics, 3rd ed., Ch. 3–4 (ensembles and their equivalence) · Touchette, "The large deviation approach to statistical mechanics," Phys. Rep. 478 (2009) 1–69 (the definitive modern treatment) · Ruelle, Statistical Mechanics: Rigorous Results (1969) · Campa, Dauxois & Ruffo, "Statistical mechanics of systems with long-range interactions," Phys. Rep. 480 (2009) · Gallavotti, Statistical Mechanics: A Short Treatise (1999).

Provenance. The concave-hull mechanism, the energy jump at first-order transitions, and the negative microcanonical heat capacity are standard (Touchette 2009; Campa–Dauxois–Ruffo 2009). A verifier should confirm: (i) $s=-\varepsilon^4+3\varepsilon^2$ is convex for $|\varepsilon|<1/\sqrt2$ ; (ii) its double Legendre transform equals the concave hull, strictly above $s$ in that window; (iii) argmax $\varepsilon(\beta)$ jumps by $>0.5$ ; (iv) the region $s''>0$ coincides with $dT/d\varepsilon<0$ (negative $C$ ); (v) $U(\lambda N)=\lambda^2U(N)$ for a gravitational (degree-2) energy versus $\lambda U(N)$ for short-range. All in verify_aq-sm-breakdown-ensemble-equivalence.py.

Builds on: Derivation of the Canonical Distribution from the Microcanonical Ensemble · Energy Fluctuations in the Canonical Ensemble · Heat Capacity and Fluctuation-Response Relation · Helmholtz Free Energy from the Partition Function · Canonical Ensemble Leads to: Grand Canonical Ensemble · Thermodynamic Quantities from the Partition Function · Gibbs Factor and Chemical Potential

If ensembles are usually equivalent, why do textbooks make such a fuss about it?

Because the equivalence is a theorem with hypotheses, and the hypotheses (short-range interactions, concave entropy) are exactly what fail in some of the most interesting systems: gravitating matter, small nuclei, systems at first-order transitions, engineered long-range cold-atom setups. In those cases picking the "convenient" ensemble gives the wrong physics. The fuss is a warning label: equivalence is the default, not a law of nature, and you must check its conditions when interactions are long-range or the system is small.

How can a heat capacity be negative without violating thermodynamics?

A negative microcanonical heat capacity means the system's temperature drops when you add energy — perfectly consistent for a non-additive system like a star cluster (add energy, it expands, virializes, cools). What is forbidden is a negative canonical heat capacity, because there $C_V=\langle\Delta E^2\rangle/k_BT^2$ is a variance. So negativity is not a thermodynamic contradiction; it is a signal that the canonical ensemble (a heat bath) is the wrong description — the isolated, microcanonical description is the physical one.

What is the "common-tangent" or "Maxwell" construction doing here?

It is the geometric form of the Legendre transform on a non-concave curve. The canonical free energy corresponds to the upper concave envelope of $s(\varepsilon)$ , which for a single dip is the straight line (common tangent) bridging the two shoulders. Physically, in the gap the system prefers to phase-separate into a mixture of the two endpoint states rather than sit at an intermediate energy, so the effective entropy is the linear interpolation — the chord. The construction is identical in spirit to the Maxwell equal-area rule that removes the van der Waals loop.

Does ensemble equivalence fail at a continuous (second-order) phase transition?

No — remarkably, it does not. At a critical point fluctuations diverge and the compressibility or susceptibility blows up, but the entropy density $s(\varepsilon)$ remains concave (the transition is continuous, no straight segment or dip). So the microcanonical and canonical ensembles still agree, even though both are singular. Inequivalence is tied specifically to non-concavity — first-order transitions and long-range forces — not to criticality per se.

How does this connect to the 1/√N suppression of fluctuations?

Ordinarily relative energy fluctuations vanish as $1/\sqrt N$ , which is why the canonical energy distribution collapses onto a single value and matches the microcanonical energy — that collapse is the mechanism of equivalence. Inequivalence appears exactly when that suppression fails: at a first-order transition the canonical energy distribution is bimodal (two peaks a latent heat apart, not a single narrow one), and for long-range systems fluctuations are not $1/\sqrt N$ -small. Equivalence and $1/\sqrt N$ suppression are the same story told twice.

Which ensemble should I use in a simulation near a first-order transition?

The microcanonical (or a related "fixed-energy") ensemble is often preferred, because it resolves the coexistence region that the canonical ensemble collapses into a discontinuous jump. In a canonical (fixed- $T$ ) simulation near a first-order transition you see hysteresis and metastable van der Waals loops as the system tunnels between the two phases; a microcanonical or multicanonical scheme lets you scan continuously through the intermediate energies and reconstruct the true $s(\varepsilon)$ , including any convex intruder. The choice is dictated by the physics you want to see, not by convenience.