Thermodynamic Equivalence of Ensembles

Welcome to the first lesson of our fifth module. In the previous module, we laid the groundwork for the canonical ensemble, culminating in the equipartition theorem—a powerful classical result for calculating average energies. We also noted its limitations, which underscored the necessity of quantum mechanics. A recurring theme has been that for macroscopic systems, fluctuations around average values are small. This observation is the physical intuition behind one of the most fundamental concepts in statistical mechanics: the thermodynamic equivalence of ensembles.

In this lesson, we will formalize this idea. Our goal is to understand why, for a vast class of physical systems in the thermodynamic limit ( $N \to \infty$ ), it doesn't matter whether we describe the system as isolated (microcanonical ensemble), in contact with a heat bath (canonical ensemble), or in contact with a heat and particle bath (grand canonical ensemble). The ability to choose the most mathematically convenient ensemble for a given problem is a cornerstone of a theorist's toolkit. We will explore this equivalence from both a physical perspective, by analyzing fluctuations, and a mathematical one, by seeing how thermodynamic potentials are related through the Legendre transform—a connection made explicit by the saddle-point approximation you were interested in.

1. What is Ensemble Equivalence?

Before diving into proofs, it's crucial to define precisely what we mean by "equivalence." It's not a single, monolithic concept but can be understood on several levels. A system might be equivalent in one sense but not another.

Advanced Statistical Physics: 1. Introduction - LPTHE

This excerpt from lecture notes by Leticia Cugliandolo provides a concise and rigorous breakdown of the different tiers of ensemble equivalence. This framework will guide our discussion.

Please read the brief introduction to the three Gibbs ensembles in this paragraph. Then, jump to the section "Equivalence of ensembles" and read the three bullet points that define the different types of equivalence: thermodynamic, macrostate, and measure equivalence.

As you've just read, we are primarily concerned with thermodynamic equivalence (do the different thermodynamic potentials yield the same macroscopic predictions?) and macrostate equivalence (do the different ensembles predict the same equilibrium values for observables?). The key insight is that for most systems we encounter, these forms of equivalence hold in the thermodynamic limit.

2. The Physical Basis: Vanishing Fluctuations

The physical reason for ensemble equivalence is that as a system becomes very large, the probability of observing it with a macroscopic property far from its average value becomes vanishingly small. In the canonical ensemble, the energy $E$ is not fixed, but can fluctuate as the system exchanges energy with a heat bath. However, we will now demonstrate that the relative size of these fluctuations shrinks to zero as the system size $N$ grows.

In a previous lesson, we derived the relation between energy fluctuations and the heat capacity:

⟨(Δ E)^{2} ⟩ = ⟨ E^{2} ⟩ - ⟨ E ⟩^{2} = k_{B} T^{2} C_{V}

For a typical macroscopic system, both the average energy $⟨ E ⟩$ and the heat capacity $C_{V}$ are extensive quantities, meaning they are proportional to the number of particles $N$ . So, we have $⟨ E ⟩ \propto N$ and $C_{V} \propto N$ .

This implies that the standard deviation of the energy, $σ_{E} = ⟨(Δ E)^{2} ⟩$ , scales as $N$ . The relative fluctuation is then the ratio of the standard deviation to the mean:

\frac{σ _{E}}{⟨ E ⟩} \propto \frac{N}{N} = \frac{1}{N}

In the thermodynamic limit, as $N \to \infty$ , this ratio goes to zero.

N \to \infty lim \frac{σ _{E}}{⟨ E ⟩} = 0

This means the energy probability distribution $P (E)$ in the canonical ensemble becomes incredibly sharply peaked around its mean value $⟨ E ⟩$ . For all practical purposes, a large canonical system has a definite energy, just like a microcanonical system.

The following reading provides a more detailed derivation, showing that $P (E)$ approaches a Gaussian distribution whose relative width vanishes as $1/ N$ .

Equilibrium Statistical Mechanics

These notes from MIT provide a very clear, step-by-step argument for the equivalence of the microcanonical and canonical ensembles, grounded in the analysis of energy fluctuations.

Please start at the subsection "Probability distribution in the large size limit". First, follow the derivation showing how the canonical probability distribution for energy, P(E), can be written in terms of the microcanonical entropy S(E) as this expression. The key step is recognizing that the partition function Z is essentially a Laplace transform of the density of states \Omega(E) = e^{S(E)/k_B}. The text then argues that for large N, the integral for Z is dominated by the energy E^* that maximizes the exponent. Read the subsequent paragraphs that use this saddle-point argument to show that P(E) converges to a Dirac delta function, which is the microcanonical distribution. This convergence from a sharp Gaussian to a delta function is the formal statement of "measure equivalence".

This convergence demonstrates that in the thermodynamic limit, making a measurement on a canonical system is overwhelmingly likely to yield the same result as making a measurement on a microcanonical system prepared with energy $E = ⟨ E ⟩$ .

3. The Mathematical Structure: Legendre Transforms

The equivalence of ensembles is not just a numerical coincidence in a specific limit; it reflects a deep mathematical structure. The thermodynamic potentials associated with different ensembles (entropy $S$ for microcanonical, Helmholtz free energy $F$ for canonical, etc.) are related to each other by Legendre transforms.

You'll recall from classical mechanics that the Lagrangian $L (q, q ˙)$ and Hamiltonian $H (q, p)$ are Legendre transforms of each other. The transform involves switching from a function of a variable ( $q ˙$ ) to a function of its derivative ( $p = \partial L / \partial q ˙$ ). The same structure applies here.

The connection is made via the partition function integral, evaluated in the thermodynamic limit using the saddle-point approximation—a method you've expressed interest in. The canonical partition function is:

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

Using the Boltzmann relation $S (E) = k_{B} ln Ω (E)$ , we can write this as:

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

Since both $S$ and $E$ are extensive, we can write $S (E, N) \approx N s (ϵ)$ and $E = N ϵ$ , where $s$ is the entropy per particle and $ϵ$ is the energy per particle. The integral becomes:

Z (β) \approx \int e^{N (s (ϵ) / k_{B} - β ϵ)} d ϵ

This integral is of the form $\int e^{N ϕ (x)} d x$ , which for large $N$ is perfectly suited for the saddle-point approximation. The integral is dominated by the value $ϵ^{*}$ that maximizes the exponent $ϕ (ϵ) = s (ϵ) / k_{B} - β ϵ$ . The condition for this maximum is:

\frac{d ϕ}{d ϵ}_{ϵ^{*}} = 0 ⟹ \frac{1}{k _{B}} \frac{d s}{d ϵ}_{ϵ^{*}} - β = 0 ⟹ \frac{\partial S}{\partial E}_{E^{*}} = \frac{1}{T}

This is precisely the microcanonical definition of temperature! The saddle-point condition selects the energy $E^{*}$ that corresponds to the temperature $T = 1/ (k_{B} β)$ of the heat bath.

The value of the integral is then approximately $e^{N ϕ (ϵ^{*})}$ (times some prefactors that are sub-leading in $N$ ). Taking the logarithm, we find the Helmholtz free energy $F = - k_{B} T ln Z$ :

F = - N k_{B} T (s (ϵ^{*}) / k_{B} - β ϵ^{*}) = N ϵ^{*} - T (N s (ϵ^{*})) = E^{*} - T S (E^{*})

This is the familiar thermodynamic relation connecting $F$ and $S$ . The saddle-point evaluation has shown that $- F / T$ is the Legendre transform of $S$ .

Advanced Statistical Physics: 1. Introduction - LPTHE

This reading formalizes the argument we just outlined. It shows how the canonical free energy density arises as the Legendre transform of the microcanonical entropy density.

Please read the final part of the section "Equivalence of ensembles", starting from the proof. This derivation explicitly uses the saddle-point method to connect Z and S, demonstrating the Legendre transform relationship between the free energy f and the entropy s.

4. Conditions and Breakdown of Equivalence

So far, we have assumed equivalence holds. But when does it? The derivations above contain a crucial, often implicit, assumption: additivity. When we consider a small system in contact with a large reservoir, we assume the total energy is just the sum of the energies of the two parts, $E_{t o t a l} = E_{sy s} + E_{r es}$ , neglecting the interaction energy at the interface. This is a very good approximation for systems with short-range interactions, where interactions are local. For such systems, the energy of a subsystem is an extensive property, scaling with its volume, while the interaction energy scales with the surface area. In the thermodynamic limit, the surface-to-volume ratio goes to zero, and the interaction energy becomes negligible.

However, for systems with long-range interactions (like gravity or unscreened Coulomb forces), this assumption breaks down. Every particle interacts with every other particle, and the energy is no longer additive.

Advanced Statistical Physics: 1. Introduction - LPTHE

This reading defines the key properties of additivity and extensivity and shows how their failure in systems with long-range interactions invalidates the standard derivation of the canonical ensemble.

Please read the section "Extensivity and additivity". Pay close attention to the distinction between short-range and long-range forces and how this impacts the additivity of energy. The examples of the fully-connected Ising model and power-law potentials are particularly illuminating. Then, read the next section, "The canonical ensemble". This shows the standard textbook derivation of the canonical distribution from the microcanonical one, explicitly highlighting the assumptions of additivity and independence. Note the concluding sentence: when these assumptions fail, the derivation collapses.

The failure of additivity has profound consequences. It leads to a situation where the microcanonical entropy $S (E)$ is not a globally concave function. A concave function has a non-increasing first derivative and a non-positive second derivative ( $\partial^{2} S / \partial E^{2} \leq 0$ ). The concavity of entropy ensures thermodynamic stability. For example, the microcanonical specific heat is given by:

C_{V}^{micro} = (\frac{\partial E}{\partial T})_{V} = - \frac{( \partial S / \partial E ) ^{2}}{\partial ^{2} S / \partial E ^{2}}

If $S (E)$ is concave, then $\partial^{2} S / \partial E^{2} \leq 0$ , and $C_{V}^{micro} \geq 0$ . But if there is a region where $S (E)$ is convex ( $\partial^{2} S / \partial E^{2} > 0$ ), the microcanonical specific heat becomes negative! This means adding energy to the system makes it colder. While counterintuitive, this can happen in systems like self-gravitating star clusters.

A negative specific heat is strictly forbidden in the canonical ensemble, where $C_{V}^{cano} = ⟨(Δ E)^{2} ⟩ / (k_{B} T^{2}) > 0$ . This discrepancy is a clear signature of ensemble inequivalence. The canonical ensemble simply cannot access the states in the region where the entropy function is convex. This region corresponds to a first-order phase transition in the canonical description.

This graph illustrates the geometric meaning of the Legendre transform. The original function is $F(x)$ (blue curve), and its transform is $G(s)$, where $s$ is the slope of the tangent line. The value of $G(s)$ is the negative of the y-intercept of the tangent line. If $F(x)$ has a convex "bump" (a region of non-concavity), there will be a range of y-intercepts that are inaccessible via single tangent lines, corresponding to the phase transition region in the transformed ensemble.

Conclusion

In this lesson, we have established the conditions under which different statistical ensembles provide an equivalent description of a macroscopic system.

Key Takeaways:

Ensemble equivalence means that in the thermodynamic limit ( $N \to \infty$ ), macroscopic observables calculated in the microcanonical, canonical, or grand canonical ensembles yield the same results.
The physical origin of equivalence is the vanishing of relative fluctuations of extensive quantities, which scale as $1/ N$ . This makes the energy distribution in the canonical ensemble sharply peaked, mimicking the fixed energy of the microcanonical ensemble.
The mathematical basis is that the thermodynamic potentials (S, F, etc.) are Legendre transforms of one another. The saddle-point approximation of the partition function makes this connection explicit.
Equivalence holds for systems with short-range interactions, where energy is additive and the entropy function $S (E)$ is concave.
Equivalence can fail for systems with long-range interactions. This is associated with a non-concave entropy function, leading to phenomena like negative microcanonical specific heat, which are inaccessible in the canonical ensemble.

We have seen that the probability of observing a state with energy $E$ in the canonical ensemble is $P (E) \propto e^{S (E) / k_{B} - β E}$ , and that this distribution becomes sharply peaked as $N \to \infty$ . In our next lesson, we will formalize this concentration of measure using the tools of large deviation theory. We will show how the canonical free energy emerges as the "rate function" that governs the exponentially small probability of observing macroscopic energy fluctuations away from the equilibrium value. This will provide an even deeper, more rigorous perspective on the relationship between ensembles.

Recommended resources

1. What is Ensemble Equivalence?

2. The Physical Basis: Vanishing Fluctuations

3. The Mathematical Structure: Legendre Transforms

4. Conditions and Breakdown of Equivalence

Conclusion