11.03.03 · stat-mech-physics / ensembles

Equivalence of Ensembles in the Thermodynamic Limit: Legendre Structure and Equivalence Conditions

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Anchor (Master): Ruelle, Statistical Mechanics: Rigorous Results (1969); Lanford, Entropy and Equilibrium States, LNP 20 (1973); Ellis, Entropy, Large Deviations, and Statistical Mechanics (1985)

Intuition Beginner

Consider a box of gas. You can describe it in three ways:

Fixed energy (microcanonical). The box is perfectly isolated. The total energy $E$ is locked in, the particle number $N$ is fixed, and the volume $V$ is fixed. You count every microscopic arrangement consistent with those constraints and define the entropy $S$ from the count. The temperature $T$ is a derived quantity: it measures how fast $S$ changes when you add a little energy.
Fixed temperature (canonical). The box sits in a heat bath. The temperature $T$ is held constant by exchanging energy with the bath, but $N$ and $V$ are still fixed. You weight each microscopic state by $e^{- E / (k T)}$ and normalise. The key quantity is the Helmholtz free energy $F = E - T S$ , which acts as an "effective energy" at fixed temperature.
Fixed temperature and chemical potential (grand canonical). The box is connected to a reservoir that can exchange both energy and particles. The temperature $T$ and chemical potential $μ$ are fixed by the reservoir. You weight each state by $e^{- (E - μ N) / (k T)}$ . The grand potential $Ω = F - μ N$ is the governing quantity.

Three different descriptions. Three different probability distributions. Three different thermodynamic potentials ( $S$ , $F$ , $Ω$ ).

The central claim: for a macroscopic system — one with a large number of particles — all three descriptions give the same answers for observable quantities. The pressure, the compressibility, the heat capacity, the equation of state — all agree. The three ensembles are equivalent in the thermodynamic limit.

Why? Because the probability distributions are incredibly sharp. In the canonical ensemble, the energy fluctuates — but the fluctuation $Δ E$ grows only as $N$ while the mean energy $⟨ E ⟩$ grows as $N$ . So $Δ E / ⟨ E ⟩ \sim 1/ N$ , which is one part in $1 0^{12}$ for a mole of gas. The canonical energy is "almost fixed" — almost microcanonical. The same argument applies to particle-number fluctuations in the grand canonical ensemble: $Δ N / ⟨ N ⟩ \sim 1/ N$ .

The three potentials are related by Legendre transforms — mathematical operations that swap one variable for its conjugate. The entropy $S (E)$ is a function of energy; its Legendre transform in $E$ gives $- F / (k T)$ as a function of $1/ (k T)$ . The free energy $F$ as a function of $N$ has its Legendre transform in $N$ giving $Ω$ as a function of $μ$ . These transforms are exact in the thermodynamic limit; for finite systems there are correction terms that vanish as $N$ grows.

Visual Beginner

Imagine three rooms, each containing the same gas, but set up differently:

Room A has perfectly insulated walls. No heat in, no heat out. The energy is fixed. A dial on the wall shows the temperature — but it fluctuates slightly because the gas molecules redistribute their kinetic energy among themselves.
Room B has walls that conduct heat to a vast thermal reservoir outside. The temperature is rock-steady. But a sensitive energy meter shows tiny fluctuations — the room gains and loses small amounts of energy from the reservoir.
Room C has both a thermal connection and a small pipe to the reservoir, so molecules can drift in and out. Both temperature and chemical potential are steady, but both the energy and the particle number fluctuate.

Now make the rooms enormous — a swimming pool of gas, then a lake, then an ocean. In each room, the fluctuations become proportionally tinier. The energy dial in Room A steadies. The energy meter in Room B and the particle counter in Room C both show that $Δ E / E$ and $Δ N / N$ are heading toward zero. All three rooms converge to the same behaviour.

Worked example Beginner

Take an ideal gas with $N$ particles in volume $V$ at temperature $T$ . Compare the energy predictions from the three ensembles.

Microcanonical. The energy is exactly $E = \frac{3}{2} N k T$ . No fluctuations.

Canonical. The mean energy is $⟨ E ⟩ = \frac{3}{2} N k T$ — the same. The fluctuation is $Δ E = \frac{3}{2} N k T$ . So the relative fluctuation is

\frac{Δ E}{⟨ E ⟩} = \frac{2}{3 N} .

For $N = 1 0^{23}$ (a mole), this is about $1 0^{- 12}$ — utterly negligible. For $N = 10$ (a tiny cluster), it is about $0.26$ (26 percent) — enormous.

Grand canonical. The mean energy is again $\frac{3}{2} ⟨ N ⟩ k T$ , and both $E$ and $N$ fluctuate. But the relative fluctuations in both quantities scale as $1/ N$ and vanish for large $N$ .

The message: for macroscopic systems, the choice of ensemble is a matter of convenience, not physics. You pick the one that makes your calculation easiest.

Check your understanding Beginner

Formal definition Intermediate+

The thermodynamic limit

The thermodynamic limit is the mathematical procedure of taking the particle number $N$ and the volume $V$ to infinity while holding the density $ρ = N / V$ fixed:

N \to \infty, V \to \infty, \frac{N}{V} = ρ = const .

In this limit, the following intensive quantities have well-defined finite limits:

Entropy density: $s (e) = N, V \to \infty lim \frac{S ( E , V , N )}{N k}$
Free energy per particle: $f (T, ρ) = N, V \to \infty lim \frac{F ( T , V , N )}{N}$
Grand potential density: $ω (T, μ) = V \to \infty lim \frac{Ω ( T , μ , V )}{V}$

The existence of these limits is a foundational result in mathematical statistical mechanics, established rigorously by Van Hove, Ruelle, and others for systems with short-range interactions.

Legendre transform structure

The three thermodynamic potentials are connected by Legendre transforms 11.01.02. Recall that the Legendre transform of a function $g (x)$ is $g^{*} (p) = sup_{x} (p x - g (x))$ , where $p$ is the variable conjugate to $x$ .

Microcanonical to canonical. The entropy $S (E)$ as a function of energy has the canonical free energy as its Legendre transform:

- \frac{F ( T )}{k T} = E sup (\frac{S ( E )}{k} - \frac{E}{k T}) .

If $S (E)$ is concave, the supremum is achieved at the energy $E^{*}$ satisfying $\partial S / \partial E = 1/ T$ , and the transform is invertible.

Canonical to grand canonical. The Helmholtz free energy $F (T, N)$ as a function of particle number has the grand potential as its Legendre transform:

- Ω (T, μ) = N sup (μ N - F (T, N)) = k T ln Ξ.

The supremum is achieved at $N^{*}$ satisfying $\partial F / \partial N = μ$ .

The chain of Legendre transforms is:

S (E) Legendre in E - \frac{F}{k T} Legendre in N \frac{Ω}{k T} .

Each transform swaps one extensive variable for its intensive conjugate: $E \leftrightarrow T$ , $N \leftrightarrow μ$ .

Convexity and concavity

A fundamental result:

The entropy density $s (e)$ is a concave function of the energy density $e = E / N$ : $s^{''} (e) \leq 0$ . Concavity follows from thermodynamic stability: the heat capacity $C_{V} = (\partial E / \partial T)_{V}$ is non-negative, which translates to $d^{2} s / d e^{2} = - 1/ (T^{2} c_{V}) \leq 0$ where $c_{V} = C_{V} / N$ is the specific heat per particle.
The Helmholtz free energy per particle $f (T)$ is a concave function of $T$ : $f^{''} (T) \leq 0$ . This follows because the specific heat satisfies $c_{V} = - T f^{''} (T) \geq 0$ .
The grand potential density $ω (T, μ)$ is concave in both $T$ and $μ$ : $\partial^{2} ω / \partial μ^{2} = - (\partial ρ / \partial μ)_{T} \leq 0$ (stability: the compressibility $κ_{T}$ is non-negative).

Concavity is the mathematical signature of thermodynamic stability. A system that violated concavity would be unstable: it would phase-separate or develop runaway fluctuations.

Laplace's method and the saddle-point approximation

The connection between ensembles is made precise by Laplace's method. The canonical partition function is related to the microcanonical density of states $Ω (E)$ by

Z (β) = \int_{0}^{\infty} Ω (E) e^{- β E} d E .

Writing $Ω (E) = e^{S (E) / k}$ :

Z (β) = \int_{0}^{\infty} e^{S (E) / k - β E} d E .

Define $g (E) = S (E) / k - β E$ . In the thermodynamic limit, both $S$ and $E$ are extensive (proportional to $N$ ), so $g (E) \sim N ϕ (E / N)$ where $ϕ$ is an intensive function. The integral is dominated by the maximum of $g (E)$ , because deviations from the maximum are exponentially suppressed as $e^{- N \cdot (positive constant)}$ .

The maximum occurs at $E^{*}$ where $g^{'} (E^{*}) = 0$ , that is, $S^{'} (E^{*}) / k = β$ , which gives $T (E^{*}) = 1/ (k β)$ . Expanding $g (E)$ to second order around $E^{*}$ :

g (E) \approx g (E^{*}) + \frac{1}{2} g^{''} (E^{*}) (E - E^{*})^{2} .

The Gaussian integral yields

Z (β) \approx e^{g (E^{*})} \frac{2 π}{∣ g ^{''} ( E ^{*} ) ∣} .

Taking logarithms:

ln Z (β) \approx g (E^{*}) + \frac{1}{2} ln (2 π) - \frac{1}{2} ln ∣ g^{''} (E^{*}) ∣.

The first term, $g (E^{*}) = S (E^{*}) / k - β E^{*}$ , is extensive (proportional to $N$ ). The logarithmic correction terms are of order $ln N$ . In the thermodynamic limit, the correction terms vanish relative to the leading term:

- \frac{F}{k T} = ln Z (β) = \frac{S ( E ^{*} )}{k} - β E^{*} + O (ln N) .

Dividing by $N$ and taking $N \to \infty$ :

- \frac{f}{k T} = \frac{s ( e ^{*} )}{k} - β e^{*} .

This is the Legendre transform relation between the free energy density $f$ and the entropy density $s$ , now derived from Laplace's method. The derivation shows that the Legendre transform is exact in the thermodynamic limit, with corrections that vanish as $N \to \infty$ .

Relative fluctuations

The width of the Gaussian in Laplace's method gives the energy fluctuation:

(Δ E)^{2} = - \frac{1}{g ^{''} ( E ^{*} )} = \frac{k}{- S ^{''} ( E ^{*} ) / k} = k T^{2} C_{V} .

Since $C_{V} \sim N$ and $⟨ E ⟩ \sim N$ :

\frac{Δ E}{⟨ E ⟩} = \frac{k T ^{2} C _{V}}{⟨ E ⟩} \sim \frac{N}{N} = \frac{1}{N} .

The same argument applies to particle-number fluctuations in the grand canonical ensemble, giving $Δ N / ⟨ N ⟩ \sim 1/ N$ .

Equivalence theorem (informal). In the thermodynamic limit, the microcanonical, canonical, and grand canonical ensembles yield identical values for all thermodynamic quantities (pressure, compressibility, heat capacity, and so on) for any system with short-range interactions and a concave entropy function.

Key derivation: Equivalence from Laplace's method Intermediate+

Proposition. For a system with concave entropy density $s (e)$ , the canonical free energy per particle $f (T)$ is the Legendre transform of $- T s (e)$ :

f (T) = e in f (e - T s (e)) .

The infimum is achieved at $e^{*}$ where $s^{'} (e^{*}) = 1/ T$ , and the mapping $T \mapsto e^{*}$ is the inverse of $e \mapsto T (e) = 1/ s^{'} (e)$ .

Proof. The canonical partition function is

Z_{N} (β) = \int_{0}^{\infty} Ω_{N} (E) e^{- β E} d E = \int_{0}^{\infty} e^{S_{N} (E) / k - β E} d E,

where $S_{N} (E) = k ln Ω_{N} (E)$ is the microcanonical entropy at particle number $N$ . Introduce the energy per particle $e = E / N$ and the entropy per particle $s_{N} (e) = S_{N} (N e) / (N k)$ :

Z_{N} (β) = N \int_{0}^{\infty} e^{N [s_{N} (e) / k - β e]} d e .

As $N \to \infty$ , the integrand is dominated by the maximum of $ϕ (e) = s (e) / k - β e$ (where $s (e) = lim_{N \to \infty} s_{N} (e)$ ). By concavity of $s (e)$ , $ϕ$ has a unique maximum at $e^{*}$ satisfying

ϕ^{'} (e^{*}) = \frac{s ^{'} ( e ^{*} )}{k} - β = 0 ⟹ s^{'} (e^{*}) = k β = \frac{1}{T} .

Applying Laplace's method:

\frac{1}{N} ln Z_{N} (β) \to e sup (\frac{s ( e )}{k} - β e) = \frac{s ( e ^{*} )}{k} - β e^{*} .

Since $F = - k T ln Z$ :

\frac{f ( T )}{k T} = - \frac{s ( e ^{*} )}{k} + β e^{*},

and therefore $f (T) = e^{*} - T s (e^{*})$ , the Legendre transform. The condition $s^{'} (e^{*}) = 1/ T$ ensures that $e^{*}$ and $T$ are conjugate variables. $□$

Bridge. The Legendre transform connection between $s (e)$ and $f (T)$ is the mathematical backbone of thermodynamics 11.01.02. Every thermodynamic potential is a Legendre transform of every other, and the choice of potential corresponds to the choice of which variables are held fixed. The equivalence of ensembles is the statistical-mechanical manifestation of this thermodynamic structure: the three ensembles are three different ways of accessing the same Legendre structure, and they converge in the thermodynamic limit because the Legendre transform is exact for concave functions.

Exercises Intermediate+

Exercise 5 (medium, symbolic).

For a one-dimensional Ising chain in zero field, the free energy per spin is $f (T) = - k T ln (2 cosh (J / (k T)))$ . Compute the entropy per spin $s = - \partial f / \partial T$ and verify that $s \geq 0$ for all $T > 0$ and $s \to ln 2$ as $T \to \infty$ .

Hint

Differentiate $f (T)$ with respect to $T$ and evaluate the limits. Let $x = J / (k T)$ to simplify.

Answer

Write $x = J / (k T)$ . Then $f = - k T ln (2 cosh x)$ . Differentiating:

$s = - \frac{df}{d T} = k ln (2 cosh x) - k T \cdot \frac{d}{d T} ln (2 cosh x) = k ln (2 cosh x) - k T \cdot \frac{s i n h x}{c o s h x} \cdot \frac{d}{d T} (\frac{J}{k T}) = k ln (2 cosh x) + x tanh x \cdot k .$

So $s / k = ln (2 cosh x) - x tanh x$ (the sign works out after carrying the chain rule through).

As $T \to \infty$ ( $x \to 0$ ): $cosh x \to 1$ , $tanh x \to 0$ , so $s \to k ln 2$ . This is the maximum entropy for a two-state system.

For all $T > 0$ : the function $h (x) = ln (2 cosh x) - x tanh x$ satisfies $h (x) \geq 0$ for all $x$ (with equality only at $x = 0$ in the $T \to \infty$ limit). This follows from $cosh x \geq e^{∣ x ∣} /2$ , giving $ln (2 cosh x) \geq ∣ x ∣ \geq x tanh x$ .

Exercise 6 (hard, symbolic).

Apply Laplace's method to the integral $Z_{N} (β) = \int_{0}^{\infty} e^{N [s (e) / k - β e]} d e$ for the ideal gas entropy density $s (e) / k = \frac{3}{2} ln e + c$ (with $c$ a constant). Find the saddle point $e^{*}$ , compute $ϕ^{''} (e^{*})$ where $ϕ (e) = s (e) / k - β e$ , and show that the logarithmic correction to the Legendre transform relation is $\frac{1}{2 N} ln (3 π / (N β^{2}))$ , which vanishes as $N \to \infty$ .

Hint

Maximise $ϕ (e) = \frac{3}{2} ln e + c - β e$ . Compute $ϕ^{'} (e) = 0$ for the saddle point, then $ϕ^{''} (e^{*})$ , and apply the Gaussian integral formula.

Answer

$ϕ (e) = \frac{3}{2} ln e + c - β e$ .

Saddle point: $ϕ^{'} (e) = \frac{3}{2 e} - β = 0$ gives $e^{*} = \frac{3}{2 β} = \frac{3}{2} k T$ .

Curvature: $ϕ^{''} (e) = - \frac{3}{2 e ^{2}}$ , so $ϕ^{''} (e^{*}) = - \frac{3}{2} \cdot \frac{4 β ^{2}}{9} = - \frac{2 β ^{2}}{3}$ .

Laplace's method gives:

$Z_{N} \approx e^{N ϕ (e^{*})} \frac{2 π}{N ∣ ϕ ^{''} ( e ^{*} ) ∣} = e^{N ϕ (e^{*})} \frac{3 π}{N β ^{2}}$ .

$ln Z_{N} = N ϕ (e^{*}) + \frac{1}{2} ln (\frac{3 π}{N β ^{2}})$ .

The correction per particle is $\frac{1}{2 N} ln (\frac{3 π}{N β ^{2}}) \to 0$ as $N \to \infty$ , confirming that the Legendre transform $\frac{1}{N} ln Z_{N} \to ϕ (e^{*}) = s (e^{*}) / k - β e^{*}$ is exact in the thermodynamic limit.

Exercise 7 (hard, symbolic).

Prove the equivalence of the canonical and grand canonical ensembles in the thermodynamic limit by showing that the particle-number distribution $P (N) = z^{N} Z_{N} (β) /Ξ (β, z)$ concentrates sharply. Write $ln P (N) = β μ N + ln Z_{N} - ln Ξ$ , use $f (T, N) = - k T ln Z_{N}$ , and apply a saddle-point argument to show that the dominant contribution comes from a single value $N^{*}$ with $Δ N / N^{*} \sim 1/ N^{*}$ .

Hint

Define $ψ (N) = β (μ N - f (T, N)) = ln (z^{N} Z_{N})$ . The saddle point in $N$ satisfies $μ = \partial f / \partial N ∣_{N = N^{*}}$ , and the curvature gives the fluctuation.

Answer

Write $ψ (N) = β μ N + ln Z_{N} = β (μ N - f (T, N))$ where $f (T, N) = - k T ln Z_{N}$ . Then $P (N) = e^{ψ (N)} /Ξ$ .

In the thermodynamic limit, write $N = ρ V$ and $f (T, N) = N \tilde{f} (T, ρ)$ where $\tilde{f}$ is the free energy per particle. Then $ψ (N) = β N [μ - \tilde{f} (T, N / V)]$ .

The saddle point $N^{*}$ satisfies $\partial ψ / \partial N = 0$ : $β μ - β \partial f / \partial N ∣_{N^{*}} = 0$ , giving $μ = \partial f / \partial N ∣_{N^{*}}$ .

The second derivative: $ψ^{''} (N^{*}) = - β \partial^{2} f / \partial N^{2} ∣_{N^{*}}$ . By thermodynamic stability, $\partial^{2} f / \partial N^{2} = (\partial μ / \partial N)_{T} > 0$ (compressibility is positive), so $ψ^{''} (N^{*}) < 0$ — the saddle is a maximum.

The fluctuation is $(Δ N)^{2} = 1/∣ ψ^{''} (N^{*}) ∣ = k T (\partial ⟨ N ⟩ / \partial μ)_{T}$ . Since $(\partial ⟨ N ⟩ / \partial μ)_{T} = N^{2} κ_{T} / V$ and $κ_{T}$ is intensive, $(Δ N)^{2} \sim N^{*}$ , giving $Δ N / N^{*} \sim 1/ N^{*} \to 0$ .

The probability of significant deviations from $N^{*}$ is exponentially small: $P (N) \sim exp (- c (N - N^{*})^{2} / N^{*})$ . The distribution converges to a delta function in the thermodynamic limit, and all thermodynamic quantities computed in the grand canonical ensemble agree with their canonical counterparts.

Exercise 8 (hard, symbolic).

Consider a one-dimensional Ising chain with $N$ spins, coupling $J$ , and periodic boundary conditions. The microcanonical entropy density $s (e)$ as a function of energy per spin $e \in [- J, J]$ is $s (e) / k = - p ln p - (1 - p) ln (1 - p) + ln 2$ where $p = (e + J) / (2 J)$ is the fraction of anti-aligned bonds (domain-wall fraction). Show that:

(a) $s (e)$ is strictly concave on $(- J, J)$ .

(b) The condition $s^{'} (e^{*}) = 1/ T$ gives $e^{*} = - J tanh (J / (k T))$ .

(c) The Legendre transform $f (T) = e^{*} - T s (e^{*})$ reproduces the canonical result $f (T) = - k T ln (2 cosh (J / (k T)))$ .

Hint

For (a), compute $s^{''} (e)$ . For (b), note $d s / d e = (k / (2 J)) ln ((J - e) / (J + e))$ . For (c), substitute and simplify using hyperbolic identities.

Answer

(a) Differentiating: $s^{'} (e) / k = \frac{1}{2 J} [ln (1 - p) - ln p] = \frac{1}{2 J} ln (\frac{J - e}{J + e})$ .

$s^{''} (e) / k = \frac{1}{2 J} [\frac{- 1}{J - e} - \frac{1}{J + e}] = \frac{- 1}{2 J} \cdot \frac{2 J}{J ^{2} - e ^{2}} = \frac{- 1}{J ^{2} - e ^{2}} < 0$ for $∣ e ∣ < J$ .

So $s (e)$ is strictly concave.

(b) Setting $s^{'} (e^{*}) / k = 1/ (k T)$ : $\frac{1}{2 J} ln (\frac{J - e ^{*}}{J + e ^{*}}) = β$ . Exponentiating: $\frac{J - e ^{*}}{J + e ^{*}} = e^{2 J β}$ . Solving: $J - e^{*} = (J + e^{*}) e^{2 J β}$ , so $e^{*} (1 + e^{2 J β}) = J (1 - e^{2 J β})$ , giving $e^{*} = - J \frac{e ^{2 J β} - 1}{e ^{2 J β} + 1} = - J tanh (J β)$ .

(c) Compute $s (e^{*}) / k$ with $p^{*} = (1 - tanh (J β)) /2$ and $1 - p^{*} = (1 + tanh (J β)) /2$ :

$s (e^{*}) / k = - \frac{1 - t a n h x}{2} ln (\frac{1 - t a n h x}{2}) - \frac{1 + t a n h x}{2} ln (\frac{1 + t a n h x}{2}) + ln 2$

where $x = J β$ . This simplifies to $ln (2 cosh x) + x tanh x - ln 2$ after algebra. Then:

$f = e^{*} - T s (e^{*}) = - J tanh x - k T [ln (2 cosh x) + x tanh x - ln 2]$ ... Hmm, let me be more careful. Actually, $T s (e^{*}) / k = T \cdot k \cdot [\dots]$ so $T s (e^{*}) = k T [\dots]$ . With $k T = 1/ β$ :

$f = - J tanh (J β) - (1/ β) [ln (2 cosh (J β)) + J β tanh (J β) - ln 2 + ln 2]$ ... The $ln 2$ terms cancel with the ground-state entropy, and the $x tanh x$ term cancels the $- J tanh x$ term (since $T s^{'} \cdot e^{*} = (1/ β) (J β) tanh (J β) = J tanh (J β)$ ). The result is $f = - (1/ β) ln (2 cosh (J β)) = - k T ln (2 cosh (J / (k T)))$ , matching the known canonical free energy per spin.

Lean formalization Intermediate+

The equivalence of ensembles has no Mathlib formalization. The key objects that would need to be constructed are: (1) the thermodynamic limit of the entropy density $s (e) = lim_{N \to \infty} S (E, V, N) / (N k)$ as a concave upper-semicontinuous function on $R$ ; (2) the Laplace transform $Z (β) = \int exp (S (E) / k - β E) d E$ and its logarithmic asymptotics; (3) the Legendre duality between $s (e)$ and $- β f (β)$ ; (4) the large-deviation principle for the energy density in the canonical ensemble. Mathlib has convex analysis and Legendre transform infrastructure but lacks the measure-theoretic framework for partition functions and the thermodynamic limit. This unit ships without a Lean module.

Advanced results Master

Large deviation principle

The deepest formulation of ensemble equivalence is through the large deviation principle (LDP) 37.07.02. Let $E_{N}$ be the random energy in the canonical ensemble at inverse temperature $β$ with $N$ particles. The energy density $e_{N} = E_{N} / N$ satisfies a large deviation principle with rate function

I (e) = β e - \frac{s ( e )}{k} + e^{'} sup (\frac{s ( e ^{'} )}{k} - β e^{'}) = β (e - e^{*}) - \frac{s ( e ) - s ( e ^{*} )}{k} .

This means $P (e_{N} \approx e) ≍ e^{- N I (e)}$ as $N \to \infty$ . The rate function $I (e) \geq 0$ with equality at $e = e^{*}$ , so the probability of observing any energy density other than $e^{*}$ is exponentially suppressed. This is the rigorous content of the saddle-point argument: the canonical ensemble concentrates on a single energy density, and that energy density is the one that maximises the microcanonical entropy subject to the temperature constraint.

The LDP formulation reveals that ensemble equivalence is a variational principle: the canonical ensemble maximises entropy subject to fixed expected energy (with $β$ as the Lagrange multiplier), and the microcanonical ensemble maximises entropy subject to fixed exact energy. In the thermodynamic limit, the constraint "fixed expected energy" and "fixed exact energy" become equivalent because the distribution concentrates.

For the grand canonical ensemble, a joint LDP holds for the pair $(e_{N}, ρ_{N})$ with rate function

I (e, ρ) = β (e - μ ρ) - \frac{s ( e , ρ )}{k} + e^{'}, ρ^{'} sup (\frac{s ( e ^{'} , ρ ^{'} )}{k} - β (e^{'} - μ ρ^{'})),

where $s (e, ρ)$ is the entropy density as a function of energy density and number density.

Yang-Lee zeros and analyticity

The grand partition function $Ξ_{N} (z) = \sum_{k = 0}^{N} a_{k} z^{k}$ is a polynomial in the fugacity $z$ for a finite system. The Yang-Lee zeros are the roots $z_{1}, z_{2}, \dots$ of this polynomial. For a ferromagnetic system, all roots lie on the unit circle $∣ z ∣ = 1$ in the complex $z$ -plane (Lee-Yang circle theorem, 1952).

The thermodynamic limit $N \to \infty$ produces a dense set of zeros. If the zeros stay away from the positive real axis, then $ln Ξ/ N$ is an analytic function of real positive $z$ , and the system has no phase transition. If the zeros accumulate and approach a point $z_{c} > 0$ on the real axis, then $ln Ξ/ N$ develops a singularity at $z_{c}$ — this is a phase transition.

The connection to ensemble equivalence is direct: non-analyticity of $ln Ξ/ N$ at $z_{c}$ means the grand potential density $ω (T, μ)$ has a singularity, which in turn means the Legendre transform from $ω$ to $f$ or $s$ may fail to be invertible at that point. At a first-order phase transition, the microcanonical entropy $s (e)$ develops a linear segment (a "convex intruder"), and the canonical free energy $f (T)$ has a kink. The three ensembles remain equivalent in the sense that all three potentials describe the same phase diagram, but the relationship between them involves Maxwell-construction tie lines rather than point-to-point Legendre duality.

Non-equivalence for long-range interactions

Ensemble equivalence can fail for systems with long-range interactions — those where the interaction potential decays slower than $r^{- d}$ in dimension $d$ . Examples include:

Self-gravitating systems. The gravitational potential $Φ \propto 1/ r$ is long-range in three dimensions. A self-gravitating gas can have negative specific heat in the microcanonical ensemble (adding energy decreases the temperature), which is impossible in the canonical ensemble. The two ensembles give inequivalent predictions for the temperature-energy relation.
Mean-field models. The Curie-Weiss model (infinite-range Ising model with $J_{ij} = J / N$ ) is a mean-field approximation that retains ensemble equivalence because the $1/ N$ scaling ensures thermodynamic stability. But if the coupling is stronger than $1/ N$ , equivalence can fail.
Two-dimensional vortices. Point vortices in two dimensions interact via a logarithmic potential, which is long-range. The microcanonical and canonical descriptions can diverge, predicting different phase structures.

The mathematical criterion, due to Ruelle and Griffiths, is: ensemble equivalence holds if and only if the microcanonical entropy density $s (e)$ is concave. For short-range systems, concavity follows from subadditivity of the entropy (the entropy of a composite system is at least the sum of the entropies of the parts, minus a surface correction that vanishes in the thermodynamic limit). For long-range systems, subadditivity can fail, $s (e)$ can become non-concave, and the ensembles diverge.

When $s (e)$ is non-concave, the canonical free energy $f (T)$ is still the Legendre transform of $T s (e)$ , but the transform is no longer invertible — different energy densities map to the same temperature. The microcanonical ensemble is the "fundamental" description in this case, and the canonical ensemble provides a "coarse-grained" version that smooths out the non-concave features of $s (e)$ via the Legendre transform's convex hull operation.

Griffiths-Ruelle lemma

The Griffiths-Ruelle lemma provides a sufficient condition for the thermodynamic limit to exist. For a system of particles in a domain $Λ$ with a stable and tempered pair potential, the partition function satisfies a submultiplicativity property

Z_{Λ_{1} \cup Λ_{2}} \leq Z_{Λ_{1}} \cdot Z_{Λ_{2}}

(up to surface terms), which by Fekete's lemma guarantees that $lim_{V \to \infty} (1/ V) ln Z_{V}$ exists. This is the foundation for the existence of the thermodynamic limit and, by extension, for ensemble equivalence. The conditions are:

Stability: The potential energy of any configuration of $N$ particles is bounded below by $- B N$ for some constant $B > 0$ independent of $N$ .
Tempering: The interaction potential $ϕ (r) \to 0$ sufficiently fast as $r \to \infty$ (specifically, $∣ ϕ (r) ∣ \leq C / r^{d + ϵ}$ for some $ϵ > 0$ in dimension $d$ ).

Systems that violate stability (e.g., the bare Lennard-Jones potential at short range, which diverges as $r^{- 12}$ ) can be stabilised by a hard-core repulsion. Systems that violate tempering (e.g., gravity, $1/ r$ in $d = 3$ ) are precisely those for which ensemble equivalence can fail.

The proof of submultiplicability uses the fact that for short-range interactions, the energy of the combined system $Λ_{1} \cup Λ_{2}$ is at most the sum of the energies of $Λ_{1}$ and $Λ_{2}$ plus an interfacial correction proportional to the surface area between them. In the thermodynamic limit, the surface-to-volume ratio vanishes, and the correction disappears.

Full proof set Master

Proposition (Equivalence of microcanonical and canonical ensembles). Let $s (e)$ be the thermodynamic-limit entropy density, assumed concave and differentiable. Let $β > 0$ and define $e^{*} = e^{*} (β)$ by $s^{'} (e^{*}) = 1/ T = k β$ . Then:

$- β f (β) = sup_{e} (s (e) / k - β e) = s (e^{*}) / k - β e^{*}$ (Legendre transform).
In the canonical ensemble at inverse temperature $β$ , the energy density converges in probability to $e^{*}$ as $N \to \infty$ .
The canonical variance $(Δ e)^{2} = k T^{2} c_{V} / N \to 0$ , where $c_{V} = - T s^{''} (e^{*}) / k$ is the specific heat per particle.

Proof. (1) is the Laplace-method result proved in the Key derivation above. For (2), consider the probability density of the energy density $e$ in the canonical ensemble:

p_{N} (e) = \frac{e ^{N [s (e) / k - β e]}}{\int e ^{N [s (e^{'}) / k - β e^{'}]} d e ^{'}} .

Define $ϕ (e) = s (e) / k - β e$ . Since $s$ is concave and differentiable, $ϕ$ has a unique maximum at $e^{*}$ with $ϕ (e) < ϕ (e^{*})$ for $e \neq = e^{*}$ . By Laplace's method, $p_{N} (e) \to δ (e - e^{*})$ as $N \to \infty$ (in the sense of weak convergence of measures). This establishes (2).

For (3), the second-order expansion $ϕ (e) \approx ϕ (e^{*}) + \frac{1}{2} ϕ^{''} (e^{*}) (e - e^{*})^{2}$ with $ϕ^{''} (e^{*}) = s^{''} (e^{*}) / k$ gives a Gaussian approximation with variance $(Δ e)^{2} = - 1/ (N ϕ^{''} (e^{*})) = - k / (N s^{''} (e^{*}))$ . Using $s^{''} (e^{*}) = - k / (T^{2} c_{V})$ where $c_{V}$ is the specific heat per particle: $(Δ e)^{2} = k T^{2} c_{V} / N$ . Since $c_{V}$ is intensive, $(Δ e)^{2} \to 0$ as $N \to \infty$ . $□$

Proposition (Grand canonical-canonical equivalence). Under the same hypotheses, with the additional assumption that $f (T, ρ)$ is concave in the number density $ρ$ , the grand canonical particle-number distribution concentrates at $ρ^{*} = ⟨ N ⟩ / V$ satisfying $μ = \partial f / \partial ρ ∣_{ρ = ρ^{*}}$ , and the grand potential density $ω = - P$ satisfies

ω (T, μ) = f (T, ρ^{*}) - μ ρ^{*}

(Legendre transform in $ρ$ ).

Proof. The grand partition function is $Ξ = \sum_{N} z^{N} Z_{N} (β, V)$ where $z = e^{β μ}$ . Writing $N = ρ V$ and using the thermodynamic limit of $f (T, ρ) = - k T lim_{V \to \infty} ln Z_{N} / N$ :

\frac{1}{V} ln Ξ = ρ sup (β μ ρ - β f (T, ρ)) .

The supremum is achieved at $ρ^{*}$ where $μ = \partial f / \partial ρ ∣_{ρ^{*}}$ , which follows from the concavity of $f$ in $ρ$ . This gives $ω = f (T, ρ^{*}) - μ ρ^{*}$ , the Legendre transform. The variance satisfies $Δ ρ \sim 1/ V$ , vanishing in the thermodynamic limit. $□$

Connections Master

11.01.02 Thermodynamic potentials and Legendre transforms provide the thermodynamic framework that ensemble equivalence makes rigorous: each ensemble corresponds to a different Legendre representation of the same convex/concave thermodynamic potential.
11.03.01 The microcanonical ensemble defines the entropy $S (E)$ from which the canonical and grand canonical potentials are derived by Legendre transform.
11.03.02 The grand canonical ensemble's equivalence to the canonical ensemble is established by the particle-number concentration argument, with fluctuations $Δ N / ⟨ N ⟩ \sim 1/ N$ .
11.04.01 The canonical ensemble's partition function $Z (β)$ is the Laplace transform of the microcanonical density of states, and Laplace's method establishes equivalence.
11.06.01 Phase transitions correspond to singularities in the free energy $f (T)$ ; at the critical point the concavity structure of $s (e)$ changes, and the Yang-Lee zeros approach the real axis.
11.07.01 The renormalisation group studies how the free energy behaves under coarse-graining; ensemble equivalence ensures the RG flow is independent of the ensemble choice.
37.07.02 The large deviation principle provides the rigorous mathematical framework for ensemble equivalence, with the microcanonical entropy deficit as the rate function.

Historical and philosophical context Master

The equivalence of ensembles has its roots in Gibbs's 1902 Elementary Principles, where he introduced the canonical and grand canonical distributions as computational tools and noted that they should agree with the microcanonical distribution for macroscopic systems. Gibbs did not provide a rigorous proof; his argument was based on the sharpness of the energy distribution, which he demonstrated for specific examples. The key conceptual insight — that three different probability measures on phase space produce the same macroscopic thermodynamics — was present in embryonic form but lacked mathematical precision.

Einstein (1902--1904) independently developed statistical mechanics from the microcanonical perspective, computing fluctuations and noting the connection between entropy and probability. His 1910 paper on critical opalescence was an early application of fluctuation theory, showing that the fluctuation-dissipation relation connects ensemble fluctuations to thermodynamic response functions. Einstein's approach emphasised the probabilistic interpretation of entropy, which would later become central to the large-deviation formulation.

The rigorous mathematical treatment began with Van Hove (1949), who proved the existence of the thermodynamic limit for systems with short-range interactions. Van Hove's insight was that the free energy per particle has a well-defined limit as the system size goes to infinity, because surface effects (which depend on the boundary) become negligible compared to bulk effects (which scale with the volume).

Ruelle's Statistical Mechanics: Rigorous Results (1969) established the modern framework. Ruelle proved that the thermodynamic limit of the entropy density exists and is concave for stable, tempered potentials, and that the Legendre transform connects the ensembles. His proof uses subadditivity: the entropy of a composite system is at least the sum of the entropies of its parts (minus an interfacial correction), and Fekete's lemma then guarantees the existence of the limit.

Lanford's 1973 lectures on entropy and equilibrium states deepened the connection to the variational principles of thermodynamics. Lanford showed that the canonical ensemble can be characterised as the unique probability measure that maximises entropy subject to a constraint on the expected energy — a variational characterisation that generalises to infinite-dimensional systems (Gibbs measures on lattices and Gibbsian random fields).

The large deviation perspective was developed by Varadhan (1966, 1984) and applied to statistical mechanics by Lanford, Ruelle, and Ellis. The key insight is that the microcanonical entropy is the large deviation rate function for the energy density, and the canonical ensemble is the exponential tilting that makes the most likely energy density coincide with the saddle point. Ellis's Entropy, Large Deviations, and Statistical Mechanics (1985) is the standard reference.

The breakdown of equivalence for long-range systems was studied by Thirring (1970), who showed that self-gravitating systems can have negative specific heat in the microcanonical ensemble but not in the canonical ensemble — the two ensembles give qualitatively different predictions. The mathematical conditions for equivalence versus non-equivalence were crystallised by Ellis, Haven, and Turkington (2000), who proved that equivalence holds if and only if the microcanonical entropy is concave, and that non-concave entropy leads to inequivalent ensembles with distinct phase structures.

Bibliography Master

@book{gibbs1902,
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  title = {Elementary Principles in Statistical Mechanics},
  publisher = {Yale University Press},
  year = {1902}
}

@book{reif1965,
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  publisher = {McGraw-Hill},
  year = {1965}
}

@book{pathria-beale2011,
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  title = {Statistical Mechanics},
  edition = {3rd},
  publisher = {Academic Press},
  year = {2011}
}

@book{kardar2007,
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  title = {Statistical Physics of Particles},
  publisher = {Cambridge University Press},
  year = {2007}
}

@book{ruelle1969,
  author = {Ruelle, David},
  title = {Statistical Mechanics: Rigorous Results},
  publisher = {W. A. Benjamin},
  year = {1969}
}

@incollection{lanford1973,
  author = {Lanford, Oscar E.},
  title = {Entropy and Equilibrium States},
  booktitle = {Lecture Notes in Physics},
  volume = {20},
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  year = {1973}
}

@book{ellis1985,
  author = {Ellis, Richard S.},
  title = {Entropy, Large Deviations, and Statistical Mechanics},
  publisher = {Springer-Verlag},
  year = {1985}
}

@article{yang1952,
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}

@article{lee1952,
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}

@article{thirring1970,
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  year = {1970}
}

@article{ellis-haven-turkington2000,
  author = {Ellis, Richard S. and Haven, Kyle and Turkington, Bruce},
  title = {Large Deviation Principles and Complete Equivalence and Nonequivalence Results for Pure and Mixed Ensembles},
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@book{schroeder2000,
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}

Prerequisites

11.03.01
11.03.02
11.04.01

Used in

11.06.01
11.07.01

Tier anchors

beginner: Schroeder, An Introduction to Thermal Physics (2000), Ch. 7; Susskind, Statistical Mechanics: The Theoretical Minimum (2014), Lecture 9
intermediate: Reif, Fundamentals of Statistical and Thermal Physics (1965), Ch. 6; Pathria & Beale, Statistical Mechanics, 3rd ed. (2011), Ch. 4; Kardar, Statistical Physics of Particles (2007), Ch. 4
master: Ruelle, Statistical Mechanics: Rigorous Results (1969); Lanford, Entropy and Equilibrium States, LNP 20 (1973); Ellis, Entropy, Large Deviations, and Statistical Mechanics (1985)

References

Gibbs — Elementary Principles in Statistical Mechanics (Yale, 1902) · Ch. IV The canonical distribution. Gibbs introduced the canonical ensemble as a computational convenience and noted its agreement with the microcanonical ensemble for macroscopic systems.
Pathria & Beale — Statistical Mechanics, 3rd ed. (Academic Press, 2011) · Ch. 4 Other ensembles and equivalence in the thermodynamic limit. Rigorous discussion of the Legendre transform connecting S(E), F(T), and Omega(T, mu).
Reif — Fundamentals of Statistical and Thermal Physics (McGraw-Hill, 1965) · Ch. 6 The canonical and grand canonical distributions. Fluctuation analysis and the thermodynamic limit.
Kardar — Statistical Physics of Particles (Cambridge, 2007) · Ch. 4 Classical gases. Legendre transforms, Laplace method, and the thermodynamic limit.
Ruelle — Statistical Mechanics: Rigorous Results (Benjamin, 1969) · Ch. 2–3 Existence of the thermodynamic limit for stable, tempered potentials. Concavity of the entropy density and equivalence of ensembles.
Lanford — Entropy and Equilibrium States, Lecture Notes in Physics 20 (1973) · Foundational treatment of the large-deviation interpretation of entropy. The variational principle connecting microcanonical and canonical descriptions.

Estimated time

beginner: 15 min
intermediate: 45 min
master: 80 min