Interpretation of $W$ in Boltzmann's entropy formula

Question

Background

In the famous Boltzmann's entropy formula, carved in the physicist's tombstone, a mysterious quantity $W$ appears (a):

$$S=k_B \log W\label{1}\tag{1}$$

We often hear that $W$ represents "the number of microstates corresponding to the macrostate of a system". However, this definition is problematic in classical physics, as noted for example by Fermi (bold is mine):

We have seen that a thermodynamical state of a system is not a sharply defined state of the system, because it cor- responds to a large number of dynamical states. This consideration led to the Boltzmann relation: $$S = k \log \pi$$ where $\pi$ is called the probability of the state. Strictly speaking, $\pi$ is not the probability of the state, but is actually the number of dynamical states that correspond to the given thermodynamical state. This seems at first sight to give rise to a serious difficulty, since a given thermodynamical state corresponds to an infinite number of dynamical states.

-E. Fermi, Thermodynamics (1937), Chapter VIII

Fermi then proceeds to propose a solution for this inconsistency in the framework of classical physics:

The phase space is divided into a number of very small cells all of which have the same hyper-volume $\tau$; the state is then characterized by specifying the cell to which the point representing the state belongs. Thus, states whose representative points all lie in the same cell are not considered as being different. This representation of the state of a system would evidently become exact if the cells were made infinitesimal. The cell representation of the dynamical states of a system introduces a discontinuity in the concept of the state of a system which enables us to calculate $\pi$ by the methods of combinatory analysis, and, hence, with the aid of the Boltzmann relation, to give a statistical definition of the entropy.

-E. Fermi, Thermodynamics (1937), Chapter VIII

This solution also has a problem, however, since the arbitrariness in the choice of $\tau$ leads to an arbitrary additive constant in $S$. Fermi claims that this inconsistency can ultimately be solved only by quantum mechanics:

[...] quantum theory introduces a discontinuity quite naturally into the definition of the dynamical state of a system (the discrete quantum states) without having to make use of the arbitrary division of the phase space into cells. It can be shown that this discontinuity is equivalent, for statistical purposes, to the division of the phase space into cells having a hyper-volume equal to $h^f$, where $h$ is Planck's constant ($h = 6.55 \cdot 10^{-27}$ cm$^2$ gm sec$^{-1}$) and $f$ is the number of degrees of freedom of the system.

-E. Fermi, Thermodynamics (1937), Chapter VIII

However, in modern statistical mechanics, it seems that the problem of the infinity of the microstates is not present (or ignored?)(b): in the microcanonical ensemble, entropy is simply defined as (c)

$$S= k_B \log \left(\int d^{3N}\mathbf p \ d^{3N} \mathbf q \ \rho({\{\mathbf p, \mathbf q\})} \right)\label{2}\tag{2}$$

where $\rho(\{\mathbf p, \mathbf q\})$ is the microcanonical probability density:

$$ \rho(\{\mathbf p, \mathbf q\})= \begin{cases} \text{const.} & E<\mathcal H (\{\mathbf p, \mathbf q\}) <E+\Delta E\\ 0 & \text{otherwise} \end{cases} \tag{3}\label{3} $$

In Eq.\ref{3}, $\mathcal H$ is the system's hamiltonian, $E$ is its energy and $\Delta E$ is chosen such that $\Delta E / E \ll 1$.

Eq.\ref{2} does not present the problem discussed by Fermi, since the integral appearing in it is just an hypervolume in phase space, which is finite if $E$ is finite. It appears therefore that Fermi's problem only appears if we (loosely?) define $W$ of Eq.\ref{1} as "the number of microstates of the system" without having a precise notion of what a microsatate is in statistical mechanics. However, this inconsistency is avoided if a definition like \ref{2} is chosen, where there is no mention of "microstates".

My question

What was the original interpretation Boltzmann gave of the $W$ appearing in Eq.\ref{1}?

Did he see it as a "number of microstates"? If so, was he conscious of the ambiguity of this concept, as pointed out by Fermi? Or did he see it, in more modern terms, as an hypervolume in phase space?

---

(a) Even if it appears that the equation was actually reformulated in this way later by Planck.

(b) But thr problem of the indeterminate additive constant remains.

(c) See for example K. Huang, Statistical Mechanics.

Answer 1

One should remember the context of the time when Boltzmann wrote his 1877 paper. Weierstrass just created a rigorous version of mathematical analysis and probability theory was not even close to Kolmogorov's formalization. Kinetic theory of gases was highly controversial, with Mach and Ostwald rejecting it altogether. Boltzmann's earlier publication (1872) of the "$H$-theorem" about entropy was contravened by the Loshmidt's paradox. He could not be expected to have as many mathematical scruples as Fermi, and his focus was on demonstrating the viability and fruitfulness of the kinetic program.

Boltzmann did not write the formula as $S=k\ln W$ in his 1877 paper. In place of $W$ he had the "probability of a state" $P$, and $k$ was generally ignored in 19th century calculations starting with Maxwell, it was seen as merely a choice of units. Planck introduced it explicitly when it became clear that it was important for precision estimates of the Avogadro constant. Since probability is merely proportional to the number of microstates it is clear that Boltzmann did not much worry about the additive constant either. He did not put much weight even on the logarithm, in his verbal formulations it is not even mentioned:

"The initial state of a system will be, in most cases, a not so probable state and the system will tend always towards more probable states, until it will reach the most probable state, i.e. the state of thermodynamic equilibrium. If we apply this to the second law of thermodynamics, we can identify the quantity that is usually called entropy with the probability of the corresponding state. [...] the total entropy of the system cannot but increase. In our present interpretation, this has no other meaning than the fact that the probability of the global state of the bodies of the system must continuously increase: the system cannot but pass from a state to a more probable state."

Boltzmann starts from a gas of $n$ particles with discrete energies $(0, e, 2e, ..., pe)$, "an unrealizable fiction" by his own admission, and fixed total energy, and calls the collection of integers $n_k$, the number of particles with energy $ke$, "state distribution". Repartitions in which each particle has an assigned energy are called "complexions" and the usual multifactorial formula for the probability of a complexion is then derived. Here is from Cercignani's Boltzmann's biography, which discusses this issue in detail:

"Boltzmann took the number of complexions... as a measure of the likelihood of a given distribution, underlining that in this way he considered the complexions as a priori equiprobable, in the same way as the probability that the sequence 1, 2, 3, 4, 5 will come out of the hat in a game of lotto is no different from that of any preassigned sequence of five numbers. In this way, the problem is reduced to the search for a constrained maximum of P... The convenience of working with the logarithm of P is then clear."

To get the continuous case he discretizes it and passes to the limit. This is done quite heuristically, sums are replaced by integrals. It turns out that the analog of $-\log P$ is the $H$ from his 1872 paper, given by the integral of $f\log f$ and identified with entropy there, and this is how the Boltzmann relation is arrived at. It is interesting that already Einstein in 1905 criticized Boltzmann's handling of probabilities in his calculations with complexions:

"The word probability is used in a sense that does not conform to its definition as given in the theory of probability. In particular, "cases of equal probability" are often hypothetically defined in instances where the theoretical pictures used are sufficiently definite to give a deduction rather than a hypothetical assertion." [quoted from Cercignani]

The original paper is: L. Boltzmann (1877). Uber die Beziehung eines allgemeine mechanischen Satzes zum zweiten Hauptsatze der Warmetheorie. Sitzungsberichte der Akademie der Wissenschaften, Wien, II, 75, 67-73 [English translation in: S.G. Brush, Kinetic theory, Vol. 2, Irreversible processes, pp. 188-202, Pergamon Press, Oxford (1966)].