1 Introduction

The postulates of statistical mechanics have been examined and debated ever since the beginnings of the field in the nineteenth century. A central postulate in equilibrium thermodynamics, put in place by Boltzmann, is that there is equal a priori probability that an isolated system will be found in any one of its microstates which are compatible with the overall constraints placed on the system. In the words of Planck [1], “all microscopic states are equally probable in dynamics”.

The equal-probability assumption has been rationalized in several ways. One can simply appeal to the Laplacian stance of insufficient reason. The observer’s knowledge of the system does not yield a distinction among the microstates, so no distinction can legitimately be introduced via their probabilities of occurrence [2]. Jaynes [3] replaced this assumption with a maximum-entropy principle (a principle of “maximum noncommitment with respect to missing information”) in order to derive the canonical (Boltzmann) distribution in the microcanonical ensemble. Goldstein et al. [4] proved that, for quantum systems, the canonical distribution arises for almost all wave functions of the universe (system plus heat bath). Popescu et al. [5] showed that, even without energy constraints, a “general canonical principle” can be established for quantum systems, under which a system will almost always behave as if the universe is in the equal-probability state.

In this note, we take a different route (for classical systems). We replace the equal-probability postulate with two physically interpretable axioms, which we show characterize the canonical (Boltzmann) distribution.

2 Axioms

In the usual (textbook) derivation, one fixes a heat bath \(\mathbb {B}\) at a temperature T and a system \(\mathbb {S}\) with possible states \(s_{i}\), for \(i=1,2,\ldots ,n\). The system \(\mathbb {S}\) specifies an energy level \(E_{i}\) for each state \(s_{i}\). (See Fig. 1.) The probability assigned to state \(s_{i}\) depends on the system \(\mathbb {S}\) and the heat bath \(\mathbb {B}\) and can therefore be written as \(p_{\mathbb {S}}(s_{i},\mathbb {B})\). One then appeals to the equal-probability postulate to write the ratios of probabilities of states as

$$\begin{aligned} \frac{p_{\mathbb {S}}(s_{i};\mathbb {B})}{p_{\mathbb {S}}(s_{j};\mathbb {B})}=\frac{\varOmega _{\mathbb {B}}(E_{\mathrm {total}}-E_{i})}{\varOmega _{\mathbb {B}}(E_{\mathrm {total}}-E_{j})}, \end{aligned}$$

where \(E_{\mathrm {total}}\) is the total energy of the composite \(\mathbb {S}+\mathbb {B},\) so that \(\varOmega _{\mathbb {B}}(E_{\mathrm {total}}-E_{i})\) is then the number of microstates of \(\mathbb {B}\). A Taylor expansion of the entropy \(S_{\mathbb {B}}(E_{\mathrm {total}}-E_{i})=k\ln \varOmega _{\mathbb {B}}\) of \(\mathbb {B}\) (where k is the Boltzmann constant), and use of the formula \(\partial S_{\mathbb {B}}/\partial E_{\mathrm {total}}=1/T\), yields the Boltmann distribution

$$\begin{aligned} p_{\mathbb {S}}(s_{i};\mathbb {B})=\frac{1}{Z}e^{-\frac{E_{i}}{kT}}, \end{aligned}$$

where \(Z={\textstyle {\textstyle \varSigma _{j}}}e^{-E_{j}/kT}\) is the partition function (e.g., Mandl [2], pp. 52–56).

Fig. 1
figure 1

System plus heat bath

Our derivation will also begin with ratios of probabilities, as in Eq. (1), but will not assume the equal-probability postulate. Our axioms are stated over a family \(\{\mathbb {S},\mathbb {S}^{\prime },\mathbb {S}^{\prime \prime },\ldots \}\) of systems and a family \(\{\mathbb {B},\mathbb {B}^{\prime },\mathbb {B}^{\prime \prime },\ldots \}\) of heat baths. All systems are defined on the same fixed underlying finite set of states \(\{s_{1},s_{2},\ldots ,s_{n}\}\).

Axiom 1

(Thermal Equilibrium) Associated with each heat bath \(\mathbb {B}\) there is a strictly increasing function \(G_{\mathbb {B}}:(0,\infty )\rightarrow (0,\infty )\) such that for any system \(\mathbb {S}\) and pair of states \(s_{i}\) and \(s_{j}\), the ratio equation

$$\begin{aligned} G_{\mathbb {B}}\left( \frac{p_{\mathbb {S}}(s_{i};\mathbb {B})}{p_{\mathbb {S}}(s_{j};\mathbb {B})}\right) =G_{\mathbb {\mathbb {B}^{\prime }}}\left( \frac{p_{\mathbb {S}}(s_{i};\mathbb {\mathbb {\mathbb {B}^{\prime }}})}{p_{\mathbb {S}}(s_{j};\mathbb {\mathbb {B}^{\prime }})}\right) \end{aligned}$$

is satisfied.

Our first axiom ensures that the probabilistic ranking of states of the system does not differ with changes in the heat bath. This is physically correct, since we are considering systems in equilibrium and, therefore, population inversions are not possible. If state \(s_{i}\) is more likely than another state \(s_{j}\), this is because \(s_{i}\) is lower energy than \(s_{j}\). In thermal equilibrium, the same probabilistic ranking of states will hold whether the heat bath is \(\mathbb {B}\) or \(\mathbb {B}{}^{\prime }\). Lemma 1 below states this formally. The axiom does allow the actual probability of a state of the system to depend on the particular heat bath \(\mathbb {B}\) to which the system is attached. This is the role of the \(G_{\mathbb {B}}\)-functions. Again, this is physically correct.

Lemma 1

If \(p_{\mathbb {S}}(s_{i};\mathbb {B})\ge p_{\mathbb {S}}(s_{j};\mathbb {B})\), then \(p_{\mathbb {S}}(s_{i};\mathbb {\mathbb {B}{}^{\prime }})\ge p_{\mathbb {S}}(s_{j};\mathbb {\mathbb {B}{}^{\prime }})\).


We can write

$$\begin{aligned} \frac{p_{\mathbb {S}}(s_{i};\mathbb {B})}{p_{\mathbb {S}}(s_{j};\mathbb {B})}\ge \frac{p_{\mathbb {S}}(s_{j};\mathbb {\mathbb {B}})}{p_{\mathbb {S}}(s_{i};\mathbb {\mathbb {B}})}, \end{aligned}$$

so that, since \(G_{\mathbb {B}}\) is increasing,

$$\begin{aligned} G_{\mathbb {B}}\left( \frac{p_{\mathbb {S}}(s_{i};\mathbb {B})}{p_{\mathbb {S}}(s_{j};\mathbb {B})}\right) \ge G_{\mathbb {\mathbb {\mathbb {B}}}}\left( \frac{p_{\mathbb {S}}(s_{j};\mathbb {\mathbb {B}})}{p_{\mathbb {S}}(s_{i};\mathbb {\mathbb {B}})}\right) . \end{aligned}$$

But, using Eq. (3),

$$\begin{aligned} G_{\mathbb {B}}\left( \frac{p_{\mathbb {S}}(s_{i};\mathbb {\mathbb {B}})}{p_{\mathbb {S}}(s_{j};\mathbb {\mathbb {B}})}\right) =G_{\mathbb {\mathbb {\mathbb {B}{}^{\prime }}}}\left( \frac{p_{\mathbb {S}}(s_{i};\mathbb {\mathbb {B}{}^{\prime }})}{p_{\mathbb {S}}(s_{j};\mathbb {\mathbb {B}{}^{\prime }})}\right) \text{ and } G_{\mathbb {\mathbb {\mathbb {B}}}}\left( \frac{p_{\mathbb {S}}(s_{j};\mathbb {\mathbb {B}})}{p_{\mathbb {S}}(s_{i};\mathbb {\mathbb {B}})}\right) =G_{\mathbb {\mathbb {\mathbb {B}{}^{\prime }}}}\left( \frac{p_{\mathbb {S}}(s_{j};\mathbb {\mathbb {B}{}^{\prime }})}{p_{\mathbb {S}}(s_{i};\mathbb {\mathbb {B}{}^{\prime }})}\right) , \end{aligned}$$

and, therefore,

$$\begin{aligned} G_{\mathbb {\mathbb {\mathbb {B}{}^{\prime }}}}\left( \frac{p_{\mathbb {S}}(s_{i};\mathbb {\mathbb {B}{}^{\prime }})}{p_{\mathbb {S}}(s_{j};\mathbb {\mathbb {B}{}^{\prime }})}\right) \ge G_{\mathbb {\mathbb {\mathbb {B}{}^{\prime }}}}\left( \frac{p_{\mathbb {S}}(s_{j};\mathbb {\mathbb {B}{}^{\prime }})}{p_{\mathbb {S}}(s_{i};\mathbb {\mathbb {B}{}^{\prime }})}\right) , \end{aligned}$$

from which, since \(G_{\mathbb {\mathbb {\mathbb {B}{}^{\prime }}}}\) is increasing,

$$\begin{aligned} \frac{p_{\mathbb {S}}(s_{i};\mathbb {\mathbb {B}{}^{\prime }})}{p_{\mathbb {S}}(s_{j};\mathbb {\mathbb {B}{}^{\prime }})}\ge \frac{p_{\mathbb {S}}(s_{j};\mathbb {\mathbb {B}{}^{\prime }})}{p_{\mathbb {S}}(s_{i};\mathbb {\mathbb {B}{}^{\prime }})}, \end{aligned}$$

or \(p_{\mathbb {S}}(s_{i};\mathbb {\mathbb {B}{}^{\prime }})\ge p_{\mathbb {S}}(s_{j};\mathbb {\mathbb {B}{}^{\prime }})\), as required. \(\square \)

Our second axiom is designed to capture the fact that a heat bath \(\mathbb {B}\) is very large compared with a system \(\mathbb {S}\), so that any energy flows are possible between the two at the given temperature of the bath. We say this formally by fixing a heat bath \(\mathbb {B}\) and a probability distribution on the states \(\{s_{1},s_{2},\ldots ,s_{n}\}\). We then say that we can attach a system \(\mathbb {S}\) to \(\mathbb {B}\) so that the desired probabilities are obtained. Physically, we know we can do this. Indeed, Eq. (2) for the Boltzmann distribution tells us there are energy levels \(E_{i}\), for \(i=1,2,\ldots ,n\), that yield the probabilities in question. (If \(\lambda _{i}\) is the probability of state i, then we set \(E_{i}=-kT\ln \lambda _{i}\).) So, we attach a system \(\mathbb {S}\) with these energy levels to the heat bath \(\mathbb {B}\). Since \(\mathbb {B}\) is very large compared with \(\mathbb {S}\), we can always do this at the prevailing temperature T. Here is the formal statement. (We assume that \(\lambda \) has full support, i.e, that \(\lambda _{i}>0\) for all i. This guarantees that all ratios of probabilities are well-defined.)

Axiom 2

(Energy Exchange) For any heat bath \(\mathbb {B}\) and any full-support probability distribution \(\lambda =(\lambda _{1},\lambda _{2},\ldots ,\lambda _{n})\) on \(\{s_{1},s_{2},\ldots ,s_{n}\}\), there is a system \(\mathbb {S}\) such that \(p_{\mathbb {S}}(\cdot ;\mathbb {B})=\lambda \).

3 Result

We can now state our result, which is an axiomatic derivation of the Boltzmann distribution.

Theorem 1

Suppose Axioms 1 and 2 are satisfied. Then there are functions \(T:\{\mathbb {B},\mathbb {B}^{\prime },\mathbb {B}^{\prime \prime },\ldots \}\rightarrow (0,\infty )\) and \(E:\{s_{1},s_{2},\ldots ,s_{n}\}\times \{\mathbb {S},\mathbb {S}^{\prime },\mathbb {S}^{\prime \prime },\ldots \}\rightarrow (0,\infty )\) such that for each heat bath \(\mathbb {B}\) and system \(\mathbb {S}\), and for each \(i=1,2,\ldots ,n\),

$$\begin{aligned} p_{\mathbb {S}}(s_{i};\mathbb {B})=\frac{1}{Z(\mathbb {B},\mathbb {S})}e^{-\frac{E(s_{i},\mathbb {S})}{T(\mathbb {B})}}, \end{aligned}$$

where \(Z(\mathbb {B},\mathbb {S})={\textstyle {\textstyle \varSigma _{j}}}e^{-E(s_{j},\mathbb {S})/T(\mathbb {B})}\).

Equation (4) is the Boltzmann distribution, with temperature \(T(\cdot )\) (as a function of the heat bath) and energy levels \(E(s_{1},\cdot ),E(s_{2},\mathbb {\cdot }),\ldots ,E(s_{n},\mathbb {\cdot })\) (as a function of the system). (We get \(k=1\) since temperature and energy are not measured in physical units here). Notice that only positive temperatures are possible under our treatment. This makes sense, since we have assumed thermal equilibrium, and negative temperatures can arise only in systems which are (temporarily) out of equilibrium (e.g., Braun et al. [6]). Also, as expected in an abstract treatment, the fundamental quantity that emerges is \(E(\cdot ,\cdot )/T(\cdot )\), namely, entropy. We can be more precise about this last point by establishing the uniqueness properties of the functions T and E that represent a given heat bath and system.

Theorem 2

Assume that, for each heat bath \(\mathbb {B}\), it is not the case that all states have equal probability. Suppose a system \(\mathbb {S}\) satisfies Eq. (4) with functions E and T. Then \(\mathbb {S}\) satisfies Eq. (4) with functions \(\widetilde{E}\) and \(\widetilde{T}\) if and only if there are real numbers \(\alpha >0\) and \(\beta \) such that

$$\begin{aligned} E(s_{i},\mathbb {S})= & {} \alpha \widetilde{E}(s_{i},\mathbb {S})\,+\,\beta \hbox { for all states }s_{i},\\ T(\mathbb {B})= & {} \alpha \widetilde{T}(\mathbb {B})\hbox { for all heat baths }\mathbb {B}. \end{aligned}$$

(Physically speaking, the equal-probability case ruled out is that of infinite temperature.) Notice that the scaling factor for T is the same as the multiplicative factor in the affine transformation of E. It follows that, while the ratios \(E\left( \cdot ,\cdot \right) /T(\cdot )\) are not unique, the differences between these ratios, i.e., the entropy differences

$$\begin{aligned} \frac{E(s_{i},\mathbb {S})-E(s_{j},\mathbb {S})}{T\left( \mathbb {B}\right) } \end{aligned}$$

between states, are unique. Again, we expect this on physical grounds.

4 Summary

We have shown that two physically interpretable axioms can replace the traditional equal-probability postulate of equilibrium thermodynamics. The first axiom is an abstraction of the notion that the probabilistic ranking of states is the same across systems in equilibrium. The second axiom is an abstraction of the notion that all energy flows are possible between the system in question and a heat bath to which it is attached, at the given temperature of the bath. Together, these two axioms characterize the Boltzmann distribution. That is, we establish that the axioms identify the Boltzmann distribution—and the converse that the Boltzmann distribution satisfies the axioms.

Two extensions of this work would be interesting. The first extension would be to quantum systems, to see if our characterization goes through and to compare the resulting analysis with those of Goldstein et al. [4] and Popescu et al. [5]. A second extension would be to continuous probability distributions, where new mathematical issues may arise.