1. Introduction

There are two main theoretical frameworks in statistical mechanics (SM), one associated with Boltzmann and the other with Gibbs. One of the crucial differences is the characterization of equilibrium. While in the Boltzmannian framework equilibrium corresponds to the largest macroregion, the Gibbsian framework associates equilibrium with the stationary probability distribution of maximum entropy. The Boltzmannian picture is usually accepted as correct when it comes to answering foundational questions, in particular in connection with the approach to equilibrium. At the same time the Gibbsian framework is dismissed as “thoroughly misguided” (Goldstein Reference Goldstein, Bricmont, Dürr, Galavotti, Ghirardi, Petruccione and Zangh2001, 39) and stands accused of introducing theoretical instruments that we “neither have nor … need” (Lebowitz Reference Lebowitz1993, 38). Yet, the Gibbsian framework is the undisputed workhorse of the practitioner.

This would be not be particularly worrisome if the two formalisms were equivalent or intertranslatable as, for instance, the Schrödinger and the Heisenberg picture in quantum mechanics. But they are not. Not only do they disagree on how equilibrium is conceptualized; they do not even study the same entities. While the Boltzmannian framework investigates individual systems, the object of study in the Gibbs approach are ensembles, and there is no obvious way to translate results from one framework into the other. This creates an awkward situation: how can we think that the Boltzmannian framework gives the right foundational account of SM and at the same time rely on the Gibbsian framework for calculations if the two accounts are in fact at odds with each other?

A common response is to play down the severity of the problem with the argument that the two frameworks lead to the same results. One can, so the argument goes, use the Gibbs formalism as an effective tool for the calculation of equilibrium values of a wide spectrum of physical quantities because these coincide with the values that would come out of the Boltzmann machinery if one was able to make the calculations. However, the reasoning behind this assertion remains unclear. Either it is simply asserted as an obvious truism (Davey [Reference Davey2009] and Wallace quoted in Werndl and Frigg [Reference Werndl and Friggforthcominga]), or arguments are given only for special cases (Malament and Zabell Reference Malament and Zabell1980; Lavis Reference Lavis2005). So the claim that Gibbsian and Boltzmannian equilibrium values generally coincide has the status of an article of faith. Faith need not be wrong, but it can be. The project for this article is to investigate whether the purported equivalence of results really holds.

Our answer is negative. If understood as a general proposition, the claim is provably wrong. After briefly introducing the Boltzmannian and the Gibbsian notions of equilibrium (sec. 2), we present two examples in which Boltzmannian and Gibbsian equilibrium values come apart: the baker’s gas (sec. 3) and the Ising model (sec. 4). The differences between the two frameworks have gone unnoticed partly because Boltzmannian and Gibbsian calculations are seldom carried out on the same systems. We make a first step toward rectifying this situation by discussing our examples from both theoretical vantage points, which makes visible how the two frameworks differ. Things get worse still. Not only do equilibrium values fail to coincide; equilibriums do not necessarily exist under the same conditions. There are systems that have a Boltzmannian equilibrium but fail to have a Gibbsian equilibrium and vice versa (sec. 5). This raises the question whether there is even a grain of truth in common wisdom, and the good news is that there is. We prove a new theorem establishing that under certain special circumstances the results of the two formalisms indeed coincide (sec. 6). We describe these circumstances in some detail and give examples. We end by offering some conclusions (sec. 7).

2. Equilibrium in Gibbs and Boltzmann

In this section we briefly introduce the Boltzmannian and the Gibbsian notions of equilibrium (for details, see Uffink Reference Uffink, Butterfield and Earman2007; Frigg Reference Frigg and Rickles2008). Throughout the article we consider systems with a phase space X. Let μ_X denote the probability measure on X. Further, let T_t(x) denote the state of the system after t time units that starts in x. The dynamics T_t is usually deterministic. Sometimes, for instance, in the Ising model, dynamics are considered that are stochastic processes {Z_t}. For ease of presentation we state general definitions and results for the deterministic case, but all definitions have stochastic equivalents and the results equally also hold in the stochastic case (see Werndl and Frigg [Reference Werndl, Frigg, Massimi, Romeijn and Schurz2017] for a discussion of stochastic systems).

In Gibbsian SM the object of study is an ensemble, an infinite collection of independent systems that are all governed by the same equations but are in different states. The ensemble is described by a probability density ρ(x, t), $x\in X$, reflecting the probability of finding the state of a system chosen at random from the ensemble in a certain part of X at time t. Physical observables are associated with real-valued functions $f:X\times t\to ℝ$. The phase average of such a function is

$\begin{array}{}\text{(1)}& \overline{f}\left(t\right)={\int }_{X}f\left(x,t\right)\rho \left(x,t\right)\mathrm{dx}.\end{array}$

According to the standard understanding of the formalism, phase averages are observed in experiments. The Gibbs entropy of a distribution is

$\begin{array}{}\text{(2)}& S_G\left[\rho \right]=-k{\int }_X\rho \left(x,t\right)\log \left[\rho \left(x,t\right)\right]\mathrm{dx},\end{array}$

where k is Boltzmann’s constant. A distribution ρ(x, t) is stationary if it does not depend on time: $\rho \left(x,t\right)=\rho \left(x\right)$ for all t. In Gibbsian SM equilibrium is the property of an ensemble. The ensemble is in equilibrium if the distribution is stationary and has maximum entropy given the constraints imposed on the system. The most common equilibrium distributions are the microcanonical, canonical, and grand-canonical distributions (Uffink Reference Uffink, Butterfield and Earman2007).

It is customary to begin a presentation of Boltzmannian SM with the combinatorial argument. However, it is now recognized that combinatorial considerations do not provide a good general definition of Boltzmannian equilibrium (Uffink Reference Uffink, Butterfield and Earman2007), and we therefore work with our own alternative definition (Werndl and Frigg Reference Werndl and Frigg2015a, Reference Werndl and Frigg2015b).Footnote ¹ Consider the same dynamical system as above and assume that the measure μ_X is stationary. At the macrolevel the system is characterized by a set of macrovariables {v ₁, …, v_l} $\left(l\in \mathbb{N}\right)$. They are measurable functions $v_i:X\to ℝ$, associating a value with each $x\in X$. We use capital letters V_i to denote the values of v_i. A particular set of values {v ₁, …, v_l} defines a macrostate $M_{V_1},\dots {,}_{V_l}$.Footnote ² The region of phase space corresponding to the macrostate $M_{V_1},\dots {,}_{V_l}$, its macroregion, is denoted by $X_{M_{V_1},\dots {,}_{V_l}}$.

The equilibrium macrostate is defined as the state in which a system spends most of its time. Let LF_R be the fraction of time a system spends in region $R\in X$ in the long run:

$\begin{array}{}\text{(3)}& {\mathrm{LF}}_R\left(x\right)=\underset{t\to \infty }{\lim }\frac{1}{t}{\int }_0^t{1}_A\left(T_{\tau }\left(x\right)\right)d\tau ,\end{array}$

where 1_A(x) is the characteristic function of R: $1_A\left(x\right)=1$ for $x\in R$ and 0 otherwise. The notion of ‘most’ can be read in two different ways, leading to two different notions of equilibrium. The first introduces a lower bound of 1/2 for the fraction of time spent in equilibrium, leading to the notion of an α-ε-equilibrium:

The second reading takes ‘most’ to refer to the fact that the system spends more time in equilibrium than in any other state (this can be less that 50% of its time). This provides the γ-ε-equilibrium:

It is obvious that on both notions of equilibrium the value associated with the equilibrium macrostate is the observed value in equilibrium. It can be proven that equilibrium states thus defined are the largest states in the system in the following sense: their measure is larger than $\alpha \left(1-\epsilon \right)>1/2$ for an α-ε-equilibrium, and their measure is $\gamma -\epsilon$ larger than the measure of any other macroregion for a γ-ε-equilibrium.Footnote ⁴

3. Example 1: The Baker’s Gas

The baker’s gas consists of N identical particles that evolve independently according to the baker’s transformation (Lavis Reference Lavis2005). Its microstates are of the form $x=\left(b_1,c_1,\dots ,b_N,c_N\right)$, where $b_i\in \left[0,1\right]$ is the momentum and $c_i\in \left[0,1\right]$ is the position coordinate of the ith particle. The system’s phase space therefore is $X={\left[0,1\right]}^{2N}$, which is endowed with the uniform probability measure μ_X (the 2N-dimensional Lebesgue measure). Time is discrete, and the evolution to the next time step is given by applying the baker’s transformation to each coordinate: $x=\left(\dots b_i,c_i\dots \right)$ evolves into $\Lambda \left(x\right)=\left(\dots \theta \left(b_i,c_i\right)\dots \right)$, where

$\begin{array}{}\text{(4)}& \theta \left(b_i,c_i\right)=2b_i,\frac{c_i}{2}\text{ if }0\le b_i\le \frac{1}{2}\text{and}2b_i-1,\frac{c_i+1}{2}\text{otherwise}.\end{array}$

Let us begin with the Boltzmannian treatment of the baker’s gas. Here μ_X is the stationary measure. We now use combinatorial considerations to construct the system’s macrostate. Consider a partition of the unit square (the phase space for one particle) into cells of equal size δω whose dividing lines run parallel to the position and momentum axes. This results in a finite partition ${\Omega }_{\mathrm{bg}}≔\left\{{\omega }_1^{\mathrm{bg}},\dots ,{\omega }_k^{\mathrm{bg}}\right\}$, $k\in \mathbb{N}$. The coarse-grained microstate of a particle is the cell in which a particle’s state lies. An arrangement is given by a specification of the coarse-grained microstates of all the particles. A distribution is a specification of how many particles’ states lie in a given cell. Consider the distribution $D_{\mathrm{bg}}=\left(N_1,N_2,\dots ,N_k\right)$, where N_i is the number of particles in cell ω_i. The number G(D_bg) of arrangements that lead to the same distribution D_bg is $G\left(D_{\mathrm{bg}}\right)=N!/N_1!N_2!\dots ,N_k!$.

One can now define a partition on X by grouping together in one cell all points that have the same distribution. It is easy to see that the cell X_u corresponding to the uniform distribution, that is, where $N_i=N/k$,Footnote ⁵ is larger than any other cell. We now introduce a macrovariable V as follows: $V\left(x\right)=0$ for $x\in X_u$, and for all other cells of the partition V(x) takes values between 10⁶ and 10⁸ so that no two cells have the same value. The baker’s gas is ergodic (Lavis Reference Lavis2005) and, hence, spends more time in X_u than in any other cell. Therefore, the long-run fraction of time for which the value of V is 0 is larger than the long-run fraction for any other value. Thus, the macrostate defined by $V=0$ is a γ-0-equilibrium, and $V=0$ is the Boltzmannian equilibrium value.Footnote ⁶

Let us now turn to the Gibbsian treatment of the baker’s gas. The stationary maximum entropy distribution is the uniform distribution $\rho \left(x\right)=1$. The phase space average $\overline{V}$ for the macrovariable V will be greater than $\left(1-{\mu }_X\left(X_u\right)\right)\times {10}^6$. Lavis (Reference Lavis2005) has shown that ${\mu }_X\left(X_u\right)\approx 0.47$ (for large N), and hence $0.53\times {10}^6=530,000$ is a lower bound for $\overline{V}$.

So we find Boltzmannian and Gibbsian equilibrium values that are very different. The macrovariable of this system is admittedly contrived, so one might argue that the problem does not arise in practice. The grain of truth in this remark is that much depends on the choice of the macrovariable, and for sufficiently restrictive classes of macrovariables the problem can indeed be avoided (see sec. 6 for a characterization of such classes). However, there are real physical systems that do not fall into these classes. Thus, the relevant contrast is not between ‘mathematical contrivance’ and ‘sensible physics’. The example we discuss in the next section is a case in point.

4. Example 2: The Ising Model

The two-dimensional Ising model is an important system in SM. We here consider a version of the model with nearest-neighbor interactions in the absence of an external field. Despite being only two-dimensional, this model provides a realistic description of crystals such as K₂NF₄ and RB₂MnF₄, which have strong horizontal and weak vertical interactions (Baxter Reference Baxter1982).

Consider a regular two-dimensional lattice with N grid points. The lattice is assumed to lie on a two-dimensional torus, so that every grid point has exactly four nearest neighbors (allowing us to neglect border effects). At every grid point there is a spin pointing either up $\left(\sigma =1\right)$ or down $\left(\sigma =\mathrm{-1}\right)$. The system’s microstate is given by

$\begin{array}{}\text{(5)}& \sigma =\left\{{\sigma }_1,\dots ,{\sigma }_N\right\},\end{array}$

and its Hamiltonian is

$\begin{array}{}\text{(6)}& H\left(\sigma \right)=-J\sum _{\mathrm{nn}}{\sigma }_i{\sigma }_j.\end{array}$

The sum is over all nearest-neighbor pairs, and the constant $J\ge 0$ is the energy associated with the nearest-neighbor interaction (Baxter Reference Baxter1982).

We treat the model stochastically, and hence we first introduce probabilities. The probability distribution has the form of a Gibbsian distribution, but it is important to emphasise that at this point this is nothing more than a formal definition that is neither Gibbsian nor Boltzmannian. We begin with the partition function:

$\begin{array}{}\text{(7)}& Z=\sum _{\sigma }{e}^{-\beta H\left(\sigma \right)}.\end{array}$

The sum is taken over all possible configurations σ of the model, and $\beta =1/\mathrm{kT}$ is a constant, where T is the temperature and k is Boltzmann’s constant. The probability of finding the system in a certain configuration σ is given by the canonical distribution:

$\begin{array}{}\text{(8)}& P\left(\sigma \right)=\frac{e^{-\beta H\left(\sigma \right)}}{Z}.\end{array}$

For large values of β (low temperature), the probabilities of the lower energy states are dominant. For small values of β (high temperature), the probability distribution is flattened out and all configurations are more or less equally likely (Baxter Reference Baxter1982; Cipra Reference Cipra1987).

The Boltzmannian treatment is as follows. The probability P(σ) is the stationary measure that defines the dynamics of the stochastic dynamical system of section 2. It can be shown that this dynamics is in fact an irreducible Markov chain, and irreducibility is the stochastic equivalent to ergodicity (Berger Reference Berger2001). As the relevant macrovariable we choose the internal energy:

$\begin{array}{}\text{(9)}& E\left(\sigma \right)=\sum _{nn}-J{\sigma }_i{\sigma }_j,\end{array}$

where the sum is over all nearest-neighbor interactions. Let $\overline{\sigma }$ be the microstate for which ${\sigma }_i=1$ for all i, and let $\hat{\sigma }$ be the microstate for which ${\sigma }_j=\mathrm{-1}$ for all j. Then, $A≔\left\{\overline{\sigma },\hat{\sigma }\right\}$ is the macroregion for which the internal energy has value $E=\mathrm{-2}\mathrm{JN}$ (because there are 2N nearest-neighbor pairs for a quadratic lattice with N sites for $N\ge 9$). As noted above, the larger the β, the larger the probabilities of the lower energy states. Since the dynamics is irreducible, for large enough β the system spends most of its time in A. Hence, A is a Boltzmannian γ-0-equilibrium, and the value of the internal energy in equilibrium is −2JN.

Let us now turn to the Gibbsian treatment. Here P(σ) is the stationary measure of maximum entropy. As above the value of E is lowest for $\overline{\sigma }$ and $\hat{\sigma }$, namely, −2JN. Since all other microstates have higher energy, the Gibbsian phase average of E over all microstates is higher than −2JN (Baxter Reference Baxter1982; Cipra Reference Cipra1987; Secular Reference Secular2015).

It follows that the equilibrium value of the internal energy in a Boltzmann equilibrium (−2JN) is lower than in Gibbsian equilibrium. This difference arises because the macrovariable takes the lowest value in the largest macroregion and a higher value in all other macroregions.

The difference between the equilibrium values is not negligible. By way of illustration consider the case of a two-dimensional lattice with four grid points: $\sigma =\left({\sigma }_1,{\sigma }_2,{\sigma }_3,{\sigma }_4\right)$, where (σ₁, σ₂) constitute the first row and (σ₃, σ₄) constitute the second row. Let $\beta =1/2$ and $J=1$. There are four nearest-neighbor pairs. The Boltzmannian equilibrium macrostate is {(1, 1, 1, 1), (−1, −1, −1, −1)}, and the equilibrium value is −4. The value from phase space averaging can be obtained as follows. There are two states that have energy −4 and whose probability is $e^{4/2}/Z\approx 7.389/Z$: (1, 1, 1, 1), (−1, −1, −1, −1). There are 12 states that have energy 0 and whose probability is $e^{0/2}/Z=1/Z$: (−1, 1, 1, 1), (1, −1, 1, 1), (1, 1, −1, 1), (1, 1, 1, −1), (−1, −1, −1, 1), (−1, −1, 1, −1), (−1, 1, −1, −1), (1, −1, −1, −1), (1, 1, −1, −1), (−1, −1, 1, 1), (1, −1, 1, −1), (−1, 1, −1, 1). Finally, there are two states that have energy 4 and whose probability is $e^{-4/2}/Z\approx 0.1353/Z$: (−1, 1, −1, 1), (1, −1, 1, −1). Hence, the phase average is

$\begin{array}{}\text{(10)}& \frac{\mathrm{-4}\times 2\times 7.389+0\times 12\times 1+4\times 2\times 0.1353}{2\times 7.389+12\times 1+2\times 0.1353}=\mathrm{-2.1454.}\end{array}$

So the Gibbsian equilibrium value (−2.1454) is almost half of the Boltzmannian equilibrium value (−4).

Nothing depends on the choice $N=4$. It remains the case for large N that Gibbsian phase averages will be different from the Boltzmannian equilibrium values. Indeed this situation remains unchanged even in the limit $N\to \infty$, where one also finds that the Boltzmannian and Gibbsian values differ.Footnote ⁷

5. Existence under Different Conditions

We have shown that Gibbsian and Boltzmannian equilibrium values can fail to coincide. And things get worse: there are systems that have a Boltzmannian equilibrium but fail to have a Gibbsian equilibrium and vice versa.

6. When Boltzmann and Gibbs Agree

This section focuses on special cases in which the Boltzmannian and Gibbsian calculations agree. We first consider the main case discussed in the literature. Then we present a new theorem specifying a set of conditions under which Boltzmannian and Gibbsian values coincide. We show that many standard examples satisfy these conditions and that they fail in the cases discussed in previous sections.

Suppose that the relevant observable is such that it takes the value of the phase average nearly everywhere on phase space. Furthermore, assume that a Boltzmannian equilibrium exists. In such a situation it will be the case that the value of the observable in the Boltzmannian equilibrium macrostate is equal to the phase average. The question is under what circumstances something like this is the case. One such situation is described by Khinchin (Reference Khinchin1949). He argued that phase functions have to satisfy strong symmetry requirements and therefore ought to have small dispersion for systems with a large number of constituents (we refer to this as the ‘Khinchin condition’). This, or something like it, is often taken to be an explanation of why calculations in both frameworks coincide (Malament and Zabell Reference Malament and Zabell1980; Davey Reference Davey2009; Wallace quoted in Werndl and Frigg [Reference Werndl and Friggforthcominga]).

This is correct, but the question is how far it goes. The point to note is that the conditions rule out not only artificial examples but also realistic physical models. Examples of systems with macrovariables that do not satisfy the Khinchin condition include the Ising model with the internal energy or the magnetization as macrovariables, the six-vertex model with the internal energy or the polarization as macrovariables (cf. Baxter Reference Baxter1982), and the Kac ring with the standard macrostate structure of the number of black and white balls.

So we need other conditions next to the Khinchin condition. Let us look at a situation in which a system has both a Boltzmann and a Gibbs equilibrium. In this case the following theorem provides sufficient conditions for the equilibrium values of both equilibriums to coincide:

Equilibrium Equivalence Theorem (EET). Suppose that the system (X, T _t, μ_X) is composed of $N\ge 1$ constituents. That is, the state $x\in X$ is given by the N coordinates $x=\left(x_1,\dots ,x_N\right)$; $X=X_1\times X_2\dots \times X_N$, where $X_i=X_{\mathrm{oc}}$ for all $1\le i\le N$ (X_oc is the one-constituent space). Let μ_X be the product measure ${\mu }_{X_1}\times {\mu }_{X_2}\dots \times {\mu }_{X_N}$, where ${\mu }_{X_i}={\mu }_{X_{\mathrm{oc}}}$ is the measure on X_oc. Suppose that an observable κ is defined on the one-particle space X_oc and takes the values κ₁, … , κ_k with equal probability 1/k, $k\le N$.Footnote ⁹ Suppose that the macrovariable K is the sum of the one-component observable; that is, $K\left(x\right)={\sum }_{i=1}^N\kappa \left(x_i\right)$. Then the value corresponding to the largest macroregion as well as the value obtained by phase space averaging is

$\frac{N}{k}\left({\kappa }_1+{\kappa }_2+\dots {\kappa }_k\right).$

The proof is stated in full in the appendix; an intuitive sketch is as follows. Since the observable on the one-constituent space takes the values {κ₁,…, κ_k} with equal probability, combinatorial considerations show that the Boltzmannian equilibrium macroregion (i.e., the macroregion of largest size) is the one where there are N/k constituents taking the value κ₁, N/k constituents taking the value κ₂, …, N/k constituents taking the value κ_k. That is, the Boltzmannian equilibrium value is $\left({\kappa }_1+\dots +{\kappa }_k\right)N/k$. The Gibbsian equilibrium value obtained by phase averaging is also $\left({\kappa }_1+\dots +{\kappa }_k\right)N/k$ because one simply takes the average over all sequences of the {κ₁, …, κ_k} and the κ_i are equally probable.

The proof does not make any assumptions about the dynamics of the system; in particular, it does not assume that the system is ergodic. The crucial assumptions of the theorem are (i) that the macrovariable is a sum of the observable on the constituent space and that the system’s measure is the product measure of its constituents and (ii) that the macrovariable on the constituent space corresponds to a partition with cells of equal probability. This theorem is important because it applies to many examples in SM. Consider, for instance, the baker’s gas discussed in section 3. The gas involves a partition of the unit square (the phase space for one particle) into cells $\left\{{\omega }_1^{\mathrm{bg}},\dots ,{\omega }_k^{\mathrm{bg}}\right\}$, $k\in \mathbb{N}$, of equal size δω. Suppose that a particle in ${\omega }_i^{\mathrm{bg}}$ takes value κ_i for all $i=1,\dots ,k$. Now the macrovariables often considered for the baker’s gas are of the form $K\left(x\right)={\sum }_{i=1}^N\kappa \left(x_i\right)$ (e.g., Lavis Reference Lavis2005).Footnote ¹⁰ Then the assumptions of EET are satisfied. Thus, the Boltzmannian equilibrium value is the same as the value obtained by Gibbsian phase space averaging, namely, $\left({\kappa }_1+\dots +{\kappa }_k\right)N/k$.

The Kac ring with the standard macrostate structure given by the magnetization and the ideal gas with the standard macrostates afford further examples of systems satisfying the conditions of the theorem (these examples are discussed in Werndl and Frigg [Reference Werndl and Frigg2015b]). In the Boltzmannian framework, one often considers macrovariables of the type assumed in the theorem (Frigg Reference Frigg and Rickles2008, 110). In these cases the Gibbsian and Boltzmannian equilibrium calculations lead to the same results.

The examples discussed earlier in the article, showing that equilibrium calculations in Boltzmannian and Gibbsian SM do not lead to the same results, violate the relevant assumptions. In the baker’s system with the macrovariables as discussed in section 3 and in the Ising model with the internal energy as an observable significant chunks of the phase space are taken up by nonequilibrium states, which results in the Khinchin condition not being satisfied. These two examples do not satisfy the assumptions of EET either because the macrovariables of both the baker’s gas and the Ising model are not the sum of a one-component macrovariable whose outcomes have equal probability. What does the work in these two examples is that significant parts of the phase space are taken up by nonequilibrium states and that the Boltzmannian equilibrium state has the lowest value of the macrovariable. This results in the phase average being different from the lowest value of the macrovariable (the Boltzmannian equilibrium value).

These examples show that Gibbsian and Boltzmannian calculations can but need not provide the same results.Footnote ¹¹ An important task of the foundations of SM is to classify under which conditions the two frameworks lead to the same results and under which conditions they do not. The Khinchin condition and EET provide partial answers to this question. The answers are partial because both offer only sufficient but not necessary conditions for Gibbsian and Boltzmannian results being the same.

7. Conclusion

There is widespread belief that Gibbsian and Boltzmannian SM provide the same equilibrium values. We argued that this is false. There are important cases in which the Boltzmannian and Gibbsian equilibrium calculations will lead to different results. Furthermore, the conditions under which an equilibrium exists are different for Gibbsian and Boltzmannian SM. It is, however, true that the equilibrium values coincide under special circumstances. We proved a new theorem giving conditions under which this is the case.

This raises the question of what happens in cases in which the two do not coincide. Is the Gibbsian or the Boltzmannian equilibrium the correct one? For the Ising model the correct empirical conclusions follow from the Boltzmannian and not the Gibbsian calculations, and we suspect that this will be the same for other cases. Which framework provides the correct values and under which circumstances is a question for future research.