# Boltzmann’s contributions to other branches of physics

# Boltzmann’s contributions to other branches of physics

# Abstract and Keywords

Ludwig Boltzmann's scientific activities were mainly devoted to statistical mechanics: in large measure to kinetic theory, but he also founded the formalism of equilibrium statistical mechanics for systems of a general kind, the paternity of which is usually, in a more or less explicit way, attributed to Josiah Willard Gibbs. About half of Boltzmann's publications deal with contributions to diverse areas, ranging over the fields of physics, chemistry, mathematics, and philosophy. This chapter examines this secondary but not negligible aspect of Boltzmann's scientific activity. Having fathomed the meaning of the theory and implications of James Clerk Maxwell's equations and in particular the unification between electromagnetism and optics, Boltzmann wrote his first paper on electrodynamics. He also studied the law that according to the Maxwell picture related the dielectric constant and the refractive index of a given material. This chapter looks at Boltzmann's contributions regarding Maxwell's theory of electromagnetism, hereditary mechanics, theoretical physics, mathematics, and foundations of mechanics.

*Keywords:*
optics, theoretical physics, electromagnetism, James Clerk Maxwell, optics, hereditary mechanics, mathematics, mechanics, dielectric constant

# 9.1 Boltzmann’s testing of Maxwell’s theory of electromagnetism

Boltzmann’s scientific activities were mainly devoted to statistical mechanics: in large measure to kinetic theory, but he also founded, as we saw in Chapter 7, the formalism of equilibrium statistical mechanics for systems of a general kind, the paternity of which is usually, in a more or less explicit way, attributed to Gibbs.

We also incidentally remarked in Chapter 1 that Boltzmann was at ease with all the live themes of the physics of his time. Actually, about half his publications deal with contributions to diverse areas, ranging over the fields of physics, chemistry, mathematics, and philosophy. In this chapter we want to examine this secondary but not negligible aspect of Boltzmann’s scientific activity. A common aspect of this secondary activity of his seems evident: most of what Boltzmann wrote in science represents some kind of response to an interaction with other scientists or with his students.

We recall from Chapter 1 that, having fathomed the meaning of the theory and implications of Maxwell’s equations and in particular the unification between electromagnetism and optics, Boltzmann wrote his first paper on electrodynamics [1] and, at the same time as he was developing his theoretical ideas about the basic integro-differential equation that bears his name, he was also busy with an experimental study on the law that according to the Maxwell picture related the dielectric constant and the refractive index of a given material. As a matter of fact, he rightly thought that this relation between the optical and electrical properties, if ascertained beyond any doubt, would reveal a new route to appreciating the nature of electricity, which appeared to him a rather obscure subject, with absolute certainty.

From a paper published in 1874 [2], one can conclude that the experiments made by Boltzmann on different materials show that the aforementioned law turns out to be correct within the margins of experimental error.

A study of Boltzmann’s scientific output reveals a series of papers on the same subject up to 1876. It is notable that these papers deal with both gases and solid bodies. The method used for gases is based on comparison of the capacities of two identical electrical condensers, filled with different gases; for solids, Boltzmann makes use of the attraction (p.161) exerted by a charged conducting sphere on a non-conducting one and thus shows that for a sulphur crystal the dielectric constant is anisotropic, like the refractive index, and its change with orientation is as predicted by Maxwell’s relation.

During the next few years, Boltzmann began an experimental investigation on diamagnetism.

# 9.2 Boltzmann lays the foundations of hereditary mechanics

Around 1874, Boltzmann’s scientific interests turned to the delayed effects (or memory phenomena) in the elasticity of glass, thus laying the foundations for hereditary mechanics [3]. From these papers by Boltzmann one obtains a vivid idea, not so immediate when reading his much more well-known papers on kinetic theory, of the relevance he gave to the experimental foundation of a theory. The phenomenon was not unknown in the literature; by a study of the stretching and torsion of wires, such outstanding scientists as W. Weber [4] and F. Kolrausch [5, 6] had been led to introduce the assumption that the strain at a given instant of time depends not only on the state of stress at the same instant, but on previous states as well. Boltzmann accepts the basic approach of these authors, but he also criticizes the lack of generality both in their papers and in the almost contemporaneous work of O.E. Mayer [7]. After recalling the classical approach of elasticity theory, which he takes as a paradigm of the generality that he had in mind, Boltzmann discusses how one should modify this approach in order to include hereditary phenomena. After considering the classical parallelepiped, with its edges parallel to the coordinate axes, well known from any treatment of the foundations of the mechanics of continua, he says [3]:

The forces acting on the faces of the parallelepiped at a given instant depend not only upon the strain of the body at that instant but also on the previous strains, under the assumption, however, that, for a given strain, the more remote is the instant at which the latter took place, the smaller is the effect produced; in other words the force required to produce a given strain is smaller if a strain of the same kind has already occurred. I want to call

decrease of force caused by the previous strainthe circumstance in which a strain that occurred previously reduces the force required to produce a strain of the same kind.

In this way, Boltzmann introduces the concept of “hereditary phenomenon” (the term would be first used by E. Picard in 1907 [8]) in a form which later became standard after the papers [9, 10] of V. Volterra appeared; and one finds here a clear statement of the concept of “fading memory”—i.e. the circumstance that the delayed effects tend to zero when time *t* tends to infinity—a property that Volterra would consider as a basic axiom of hereditary elasticity.

Concerning the “constitutive relations” for continua with memory, Boltzmann essentially assumes that they correspond to a sort of time-distributed elasticity. Thus, along with a term which describes the by then classical constitutive relation between stresses and strains of an elastic continuum, containing the Lamé coefficients λ and μ, Boltzmann introduces the contribution of each previous instant τ with infinitesimal coefficients, given by –Φ(*t* – τ) dτ and –Ψ(*t* – τ) dτ (where the minus sign accounts for the fact that the required force is decreased and not increased by the presence of the memory effects).
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Of course these contributions appear in an integral, where τ varies between –∞ and *t*; an obvious change of variable from τ to ω = *t* – τ leads to another integral where the integration runs from 0 to ∞.

He then expresses the strain components by the derivatives of the components (*u, v, w*) of the displacement with respect to three orthogonal Cartesian axes *x, y*, and *z*. In this way the problem of the mechanics of a continuum with memory is clearly laid down in a completely general manner. What is left to experiment, of course, is to determine the elasticity coefficients λ and μ, together with the functions Φ(ω) and Ψ(ω) which describe memory effects.

Boltzmann passes on to describe some typical situations, considering as the system to be studied a cylindrical solid body (a deformable wire) subject to a specified torque which varies in time by a defined law, in such a way that one can determine the Lamé coefficient μ and the corresponding memory function Ψ(ω); he then compares his theoretical analysis with the experimental results obtained by different authors. For his calculations he mainly uses Fourier series series expansions.

A detailed study on the development and historical setting of this paper by Boltzmann has been published by Ianniello and Israel [11].

We end this section by remarking that Boltzmann’s paper [3] contains “the first successful theory of rheology” according to an authoritative article by H. Markovitz [12].

# 9.3 Back to electromagnetism

Certainly Boltzmann had various further interests, but after the paper we have just discussed, his activities both theoretical and experimental appear to be almost exclusively oriented towards statistical mechanics and in particular the kinetic theory of gases. We can infer from a couple of theses of which he was mentor that he had an interest in both ionized gases and irreversible processes in electrolytic conduction, but from his papers we have definite evidence of his interest in the determination of the viscosity coefficient of gases and in diffusion processes in mixtures.

Another subject that interested Boltzmann throughout his scientific activity was Maxwell’s electromagnetic theory, the research theme with which he had started out. As we said in Chapter 1, beginning in 1886, deeply impressed by Hertz’s experimental verification of the equivalence between electromagnetic waves and light that was predicted by Maxwell’s theory, Boltzmann spent considerable effort in redoing Hertz’s experiments. These are documented in the last publication he wrote before leaving Graz [13].

Not only did he give a course on Maxwell’s electromagnetic theory in Graz in 1890 and in Munich in 1891, but he also published the text of these lectures [14]. This revival of interest in Maxwell’s theory was perhaps a result of Hertz’s experiments confirming the existence of electromagnetic waves. Also to be ascribed, at least partly, to this interest is his activity in the area of the mechanical models to be used as illustrations of various phenomena. Among these we can recall wave machines, condensers, and a mechanical model called the *Bicykel* which has already been mentioned in Chapter 1 and was built to illustrate the effect of one electric circuit on another; as mentioned in
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Chapter 1, the two specimens actually built, one in Graz, the other in Munich, were lost during the Second World War, but we can surmise that they were more or less analogous to the differential gear of a modern car.

In 1895 he also published an annotated German edition of Maxwell’s paper “On Faraday’s lines of force” in Ostwald’s series *Klassiker der exacten Wissenschaften*. It is widely recognized that Boltzmann played a very important role in the eventual acceptance of Maxwell’s theory in mainland Europe. It is true that, curiously enough, he did not advance the theory itself as much as Lorentz did, though he had all the necessary tools in his hands. As a consequence, he did not grapple with the difficulties of Maxwell’s theory that eventually led to Einstein’s theory of relativity.

Boltzmann was the first to prove the property of time reversal in electromagnetic theory: the Maxwell equations are invariant under the joint inversion of the direction of time and of the magnetic field, the electric field being left unaltered [15].

# 9.4 A true pearl of theoretical physics

Before concluding our consideration of Boltzmann’s contributions to physics and giving a short description of his position *vis-á-vis* mathematics, we must recall a basic result of his on the thermodynamics of heat radiation. This new area of investigation had been opened up by Kirchhoff, who in 1859 introduced the concepts of emissive and absorptive powers of a body and showed that their ratio depends only on the temperature of the body, and not on its nature. He also introduced the concept of a *black body*, defined as a body with absorptive power equal to unity. Hence a black body absorbs the whole of the heat radiation falling upon it, and the radiation it emits is a function of temperature alone. It is easy to produce black-body radiation experimentally: if there is a small aperture in a wall of a furnace at uniform temperature, the radiation escaping through it will be black-body radiation. This is due to the fact that the chance of emerging from the furnace is small and thus the radiation inside strikes the walls of the oven repeatedly and is eventually completely absorbed by them.

In 1879 Boltzmann’s former mentor J. Stefan [16] had established, or perhaps conjectured, from an analysis of rather rudimentary experimental data in the case of an enclosure which approximates a black body, the proportionality between the density *e* of the radiation energy and the fourth power of the absolute temperature *T*. In 1883, Boltzmann was preparing an abstract of a paper by H.T. Eddy on radiant heat as a possible exception to the Second Law of Thermodynamics, for Wiedemann’s *Beiblätter*, and learned of a work by the Italian physicist Adolfo Bartoli (1851–96) on radiation pressure. Bartoli’s arguments stimulated Boltzmann to work out a theoretical derivation [17] of the same relation on theoretical grounds. Boltzmann’s argument combined ideas from two disciplines rather modern at that time—thermodynamics and Maxwell’s equations—using in particular a notion now known even by undergraduates but a great novelty in those days, i.e. the fact that electromagnetic waves exert a pressure on the walls of an enclosure filled with radiation. In addition, the Maxwell equations imply the existence of electromagnetic energy distributed within the enclosure.

(p.164) To put this achievement in its proper historical perspective, it seems appropriate to point out that Boltzmann did this work three years before Hertz demonstrated the existence of electromagnetic waves. Lorentz, then, did not exaggerate when he referred to it as a “a true pearl of theoretical physics” [18], nor did Planck when he wrote that “Maxwell’s theory received powerful support through the short but now famous contribution by Boltzmann on the temperature variation of the heat radiation of a black body” [19].

Boltzmann applied what would now be called “the usual thermodynamic relations”, using the fact that in equilibrium conditions, and hence those of homogeneity and isotropy, the radiation pressure *p* equals *e*/3 and depends on the temperature alone. In particular, he was the first to introduce the entropy of radiation. Then it was easy for him, by using the First and Second Laws of Thermodynamics for equilibrium states, to obtain (see Appendix 9.1) that *e* = σ*T* ^{4}.

Although at the time the result obtained by Boltzmann for radiation seemed to be an isolated achievement with no further consequences, it did at least show a possible connection between thermodynamics and electromagnetism that was to lead to quantum theory and to an important aspect of modern astrophysics, i.e. the role of radiation pressure in the study of the equilibrium of stellar atmospheres. Boltzmann’s argument was pushed to its widest possible scope by Wien, who, applying thermodynamics to radiation of each single frequency and, taking due account of the Doppler effect, deduced the famous displacement law that carries his name [20] (Appendix 9.2). This is the most advanced result that can be obtained through thermodynamics and the general properties of classical radiation without introducing a detailed model or a more advanced statistical assumption. As is well known, the assumption of classical oscillators subject to an equipartition law leads to the Rayleigh-Jeans law (in complete disagreement with experiment). Hence Planck was led to proposing his law and the assumption of light quanta (see Chapter 12).

# 9.5 Mathematics and foundations of mechanics

As a final topic for this chapter, we want to deal briefly, as already hinted at, with Boltzmann’s position with regard to mathematics and the basic axioms of mechanics.

To start with, we must explode the myth of a Boltzmann not completely at ease with derivatives and integrals. Boltzmann managed very well; otherwise, how would he have dared to be the first to perform complicated calculations in thermodynamics and to introduce integro-differential equations? It is true however that he had a finitist conception of mathematics. Derivatives were for him convenient tools, but the *significant* things were the incremental ratios. There is no lack of passages on this point [21]:

[…] if I tell somebody to sum the series 1 + 1/2 + 1/4 + 1/8 + … really to the extent of infinitely many terms, he will be unable to do it; but if I tell him to sum so many terms that a further increase will no longer noticeably influence the result, I have given him a clear and executable prescription, and all proofs that the sum of infinitely many terms equals 2 merely signify that if (p.165) you add countless thousands of further terms you will never exceed 2, though you will approach it more and more.

And a little further on, in a footnote:

The concepts of differential and integral calculus divorced from any atomist notions are typically metaphysical, if following an apposite definition of Mach we mean by this the kind of notion of which we have forgotten how we obtained it.

And again:

When carrying out the by now customary manipulations with the symbols of integral calculus, one may temporarily forget that in forming these concepts we based ourselves on starting with a finite number of elements, but we cannot really circumvent this assumption. That, too, seems to be the reason why groups of mutually interacting atoms of an elastic body are intuitively much clearer than interacting volume elements. This naturally does not exclude that, once we have become used to the abstraction of volume elements and other symbols of integral calculus and have practised the methods of operating with them, it might be convenient and expedient no longer to remember the peculiar atomistic meaning of these abstractions when we derive certain formulae that Volkmann calls those for coarser phenomena. These abstractions constitute a general schema for all cases where we may imagine the number of elements in a cubic millimetre to be 10

^{10}or 10^{1010}or milliards of times more still; hence they are indispensable especially in geometry, which must of course be equally applicable to the most varied physical cases where the number of elements can be very different. In using any such schemata it is often expedient to leave aside the basic idea from which they have sprung or even forget it for a while; but I think it would nevertheless be erroneous to believe that one had thus got rid of it.

We can also recall that Boltzmann was concerned with the mathematical problems arising from the atomic models of matter. We have already mentioned (see Chapter 5) a paper in which he gave a mathematical proof that the collision invariants are exactly those that one would expect on physical grounds. Here we mention that an early paper of his, entitled “On the integrals of linear differential equations with periodic coefficients” [22], turns out to be an investigation of the validity of Cauchy’s theorem on differential equations with periodic coefficients, which is needed to justify the application of the equations for an elastic continuum to a crystalline solid in which the local properties vary periodically from one atom to the next.

In the last years of his life, Boltzmann became interested in non-Euclidean geometries and Cantor’s set theory, on which he even lectured [23], thus originating the verses in a student rag, that we have quoted in Chapter 1.

One of Boltzmann’s not negligible interests was of course the axiomatic foundations of mechanics, to which he was being pushed by the criticism of the concept of force and the new exposition of the principles of classical mechanics proposed by Hertz [24]. Although we shall have an opportunity to comment on the philosophical aspects of this interest in the next chapter, we cannot refrain here from remarking that Boltzmann felt the need to write a book, in two volumes, on classical mechanics [25] as a sort of interlude between the two volumes of his lectures on gas theory. For the benefit of the reader we mention that some important parts of this book have been translated into English [26].

In the first volume we find a conceptual contribution by Boltzmann concerning the definition of distinguishability of particles. What does it mean that two particles are identical but distinguishable? It seems that Boltzmann was the first to feel the need to specify what we mean when we say “the same mass point”. To make the issue clear, suppose that we show somebody two identical balls lying on a table and then ask this person to close his/her eyes and a little later to open them again. We then ask whether or not the two balls have been switched around in the meanwhile. He/she cannot tell, because the balls are perfectly identical. Yet we know the answer. If we have switched the balls, then we have been able to follow the continuous motion that took them from their initial to their final positions. This elementary example illustrates the first axiom of classical mechanics laid down by Boltzmann, which essentially states that identical material particles which cannot occupy the same point of space at the same time can be distinguished by their initial conditions and by the continuity of their motion. This assumption alone, Boltzmann underlines, “enables us to recognize the same material point at different times” [25, Vol.1, p.9; 26, p.230]. This property acquires a particularly pioneering aspect, because it shows what we understand by classical identical particles, as opposed to the indistinguishable particles of modern quantum mechanics.

Boltzmann is also willing to consider rather unusual ideas concerning atoms. Thus we read [25, vol.I, p.4; 26, p.227]:

Nor must one ever seek metaphysical reasons for the picture nor draw hasty inferences from it, for example that chemical atoms are material points. Nor should we lose from sight the possibility (p.167) that it might one day be displaced by quite different pictures, let us say, to avoid appearing small-hearted, ones taken from manifolds that lack even the properties of our three-dimensional space, so that for example simple geometrical constructions of atomism would have to be replaced by manipulations with numbers forming a complicated manifold.

More pioneering aspects are contained in the discussion of the possibility of changing the axioms of mechanics. Hans Motz [27] suggested that these passages were known to Einstein and paved the way to General Relativity. Here we shall restrict ourselves to bare facts.

When discussing inertial systems and the possibility of identifying them, and having said that “we are going to think that the world is finite”, Boltzmann continues [25, Vol.II, p.334; 26, p.264]:

Quite independently of this there is the question whether the mechanical equations here developed and therefore also the law of inertia might perhaps be only approximately correct and whether, by formulating them more correctly, the improbability or rather inhomogeneity of having to adopt into the picture a co-ordinate system as well as material points would disappear of itself.

After this remarkable sentence admitting that the principles of Newton’s mechanics, and in particular one of its pillars, might be only approximately correct, Boltzmann goes on to discuss what we call Mach’s principle [25, VO1.II, p.334; 26, p.264]:

Here perhaps “relative position” would be a better term than distance.Here Mach pointed to the possibility of a more correct picture, obtained by assuming that only the acceleration of the change of distance between any two material particles is determined mainly by the neighbouring masses, its velocity being determined by a formula in which very distant masses are decisive. This naturally avoids the adopting of any co-ordinate system into the picture, since now it is only a question of distances.

Boltzmann notes that, with this suggestion, Mach introduces other difficulties which he, at variance with some other physicists, does not consider “so particularly great”, though, he remarks, they seem “to exclude all empirical test forever”. This is another remarkable statement if read nowadays, when astrophysicists are looking for the “missing mass” needed to confirm Mach’s principle.

But this is not the end of it. After a few further comments, Boltzmann says [25, Vol.II:, p.334; 26, p.264]:

At all events I think that such an extension of our vision, by pointing out that what we regard as most certain and obvious may perhaps be only approximately correct, is most valuable. It is in line with the suggestion that the distances of fixed stars may perhaps be constructed only in a non-Euclidean space of very small curvature, which is of course connected with the law of gravity in that a moving body not acted on by forces would then after aeons have to return to its previous position if the curvature is positive.

This is the most extraordinary passage, which according to Motz [27] seems to contain the germ the General Relativity. Unfortunately the English translation, as in other places, is unfaithful, as stressed by Motz [27]. The most remarkable deviation from the German text concerns the statement that the space curvature is *of course* connected with the law of *gravity*. The German text says that the curvature is connected with the law of *inertia (in einem nichteuklidischen Raume von enorm geringer Krümmung […], was*
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*ja insofern auch mit den Trägheitsgesetze zusammenhängt*). Even if we ignore this incredible “discovery,” introduced only by the hindsight of the translator, Boltzmann makes a number of explicit statements: space may be non-Euclidean with (very small!) curvature which is connected with the law of inertia. The possibility that space might be non-Euclidean had been considered by both Gauss and Riemann, and the latter had suggested that this assumption might be useful to describe physical phenomena (see Chapter 2). What appears to be new with Boltzmann is the connection between Mach’s principle and curvature and the possibility that in a finite space the problem of inertial frames would be automatically solved, the laws being the same for all observers: two of the basic ideas of Einstein’s General Relativity. What Boltzmann of course lacks is the idea that space must be replaced by space–time, a concept which needed the ideas of Special Relativity to start with.

What about Special Relativity? Had Boltzmann any connection with its birth? This point has been investigated by Siegfried Wagner [28]. One point that he stresses is that Boltzmann was the first to discuss in a textbook a displacement of the reference system, although a few authors (Ludwig Lange, Streintz, Neumann) had used related terminology. Boltzmann is well aware of the basic problems concerning space and time; he explicitly assumes the need for rigid rods and a universal clock and discusses in detail the change of a reference frame. The next author to do this was Einstein when he laid down the foundations of Special Relativity [29]; the only change was of course that there was no absolute clock but the speed of light was absolute.

Boltzmann discusses in detail the use of a coordinate system. Here we just quote a few sentences [25, Vol.I, p.7–8; 26, pp.229–30]:

More straightforwardly, Einstein defines a system of coordinates as “three rigid material lines, perpendicular to one another, and issuing from a point” [29].To define the position at any time we imagine that at all times there is in space a definite rectangular system. […] The co-ordinate system is of course nothing real, but this offers no difficulty according to the views we base ourselves on here, since we are at present concerned only with construction and mental pictures. […] the cause why the above pictures are clear is obvious: they are prescriptions for thinking spatial circumstances that everybody can palpably represent for himself in approximation, by means of ruler and pencil or wooden sticks and knitting needles…

In the second volume of Boltzmann’s lectures, published one year before Einstein’s paper, at the end of the discussion on the law of inertia quoted above in connection with Mach’s principle, we read an interesting sentence [25, Vol.II, p.335; 26, p.265]:

However, the law of inertia does not hold for the particles of aether itself; Maxwell’s equations would have to be formulated in such a way as that they determine only the mutual actions of adjacent volume elements so that we need no absolute space to formulate them. A working out of this as yet quite undeveloped theory is no concern of ours here.

Again some remarks on the English translation are in order. First, let us remark that the word *Lichtäther* (luminiferous aether) is simply translated as “aether”. This is of course reasonable, but this comment is required for what we are going to say. Second, the last part of the passage is rather unfaithful to the original, which ends with *liegt uns hier ferne* (“lies far from us now”). As remarked by Wagner [28], the development foreseen by Boltzmann occurred just one year later. This indicates that Boltzmann was a very
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good physicist, but no prophet. We remark that at about that time Lorentz and Poincaré, who are rightly considered as forerunners of Special Relativity, were still using either an ether or an absolute space.

Now we wish to compare, as Wagner [28] suggests, the above passage by Boltzmann with a sentence from the second page of Einstein’s epoch-making paper [29]:

The introduction of a “luminiferous ether” will prove to be superfluous inasmuch as the view here to be developed will not require an “absolutely stationary space” provided with special properties, nor assign a velocity-vector to a point of the empty space in which electromagnetic processes take place.

The similarity is striking; in particular the word *Lichtäther* is used by both Boltzmann and Einstein, whereas as remarked by Wagner [28], usually *Äther* would suffice; thus we cannot blame the translator in this case.

Are these coincidences just accidental? According to Wagner [28] they are not. Although Einstein does not explicitly mention Boltzmann as an author that he had studied when he was a student, his authorized biography by Philipp Frank [30] confirms that he had in fact done so. In particular, Frank says that from the works of Hertz and Boltzmann Einstein “learned how one builds up the mathematical framework and then with its help constructs the edifice of physics”.

After what we have just said here (and in Chapter 6) we shall return to the connection between Einstein and Boltzmann in Chapter 12.