LUDWIG BOLTZMANN

The Man Who Trusted Atoms

By Carlo Cercignani

Oxford University Press, 1998. 329 pages.

This journal has been inexcusably dilatory in not yet having reviewed this important book, written by a distinguished and long-time member of its editorial board. But better late than never, I suppose. Cercignani's book joins other "scientific biographies," a popular genre of late, whose reviews are planned in TTSP.1 One of the questions which must be addressed is whether or not scientific biographies are worthwhile.

The problem with this type of book is the identification of the readers for whom it is written. One would presumably like a wide readership among non-experts in the subject's discipline. This suggests the omission of much of the technical material which defines the term "scientific" in a scientific biography. On the other hand, one would also like the experts to study the book in an effort to learn the genesis of the field, particularly when the subject is Ludwig Boltzmann (1844-1906) who virtually invented statistical mechanics.2 Cercignani has found a happy medium by relegating the most difficult technical material to a sequence of Appendices.3

After a nice foreword by Sir Roger Penrose and a brief introduction by the author, we come to Chapter 1 entitled "A Short Biography of Ludwig Boltzmann." (44 pages.) To me this was one of the most interesting parts of the book; (another was Boltzmann's own description of his trip to California in 1905, pp. 231-250). Chapter 1 would have been even more interesting if additional details of Boltzmann's life had been presented, especially further speculation on what led him to commit suicide. About three pages (34-37) are devoted to this topic ("The tragic fate of a great scientist") but there is little in the way of hard facts other than that Boltzmann suffered from poor eyesight and nervousness, perhaps depression. Then on page 46 we learn that suicide was rife in Vienna in those days, claiming such victims as Crown Prince Rudolph (in a celebrated double suicide with his mistress at Mayerling)4 as well the three elder brothers of Ludwig Wittgenstein and others. Again, on page 49, the author tells us "We may see his suicide as the inevitable conclusion of his life, as inescapable as Fate in a Greek tragedy:" I confess that I left this chapter unsatisfied. I did learn, however, something about Boltzmann's education and his four teaching gigs5 and a little bit about his family life.

Chapters 2 and 3 are devoted to "Physics before Boltzmann" and "Kinetic Theory before Boltzmann" respectively. The most interesting portions of these two chapters describe the work of James Clerk Maxwell (18310-1879) and show how Maxwell set the stage, with his derivation of the Maxwell distribution (now properly called the Maxwell-Boltzmann distribution) for Boltzmann's later work. In fact, in Sec. 3.2 we find "With his transfer equations, Maxwell had come very close to an evolution equation for the distribution function, but this last step [8] must beyond any doubt be credited to Ludwig Boltzmann…" Ref. [8] here refers to the celebrated 1872 paper of Boltzmann in which the eponymous equation was introduced. The derivation of this equation is discussed in Chapter 4 with mathematical details relegated to Appendices 4.1-4.5. This nonlinear equation was the first ever mathematical description of the evolution of a probability.

In the Introduction, Cercignani states about Boltzmann, "His fame will be forever related to two basic contributions to science: the interpretation of the concept of entropy as a mathematically well-defined measure of what one can call the "disorder" of atoms, and the equation aptly known as the Boltzmann equation." The author's discussion of the entropy contribution is, in my opinion, the most important and compelling part of the book, perhaps because it introduced more concepts that were new to me.6 Beginning with Chapter 5 "Time irreversibility and the H-theorem"7 and continuing with Chapter 6 "Boltzmann's relation and the statistical theory of entropy" Cercignani provides a deep analysis of the philosophical, mathematical and physical questions involved in irreversibility, entropy and the "arrow of time." The question, of course, which scientists and philosophers are still arguing about is how completely reversible equations like Newton's and Schrödinger's can lead to an irreversible universe.8

The first important topic discussed in Chapter 5 is "Loschmidt's paradox." This merely asserts that as likely as water flowing over a waterfall is the time-reversed process in which "…the bursting bubble at the foot of a waterfall would reunite and ascend into the water." In general, all sorts of "backward phenomena" which we never see should occur. Boltzmann refuted this paradox using probabilistic arguments. In similar fashion he tried to explain away "Poincaré recurrence" and the so-called Zermelo paradox. In Secs. 5.4and 5.5 Cercignani gives his own ideas on the resolution of these paradoxes (including a discussion of the expanding universe) followed by a particularly intriguing section entitled "Is irreversibility objective or subjective?" This section tells us what the philosophers and psychologists have to say about this vexing question. The answer advanced by Boltzmann: Irreversibility is introduced by the boundary conditions, and these in turn come about probabilistically.

Chapter 6 discusses the statistical interpretation of entropy, and justifies the famous equation derived by Boltzmann S = klogW. (This equation is sculpted on his gravestone.) The only Appendix to this chapter shows Boltzmann's derivation of the microcanonical ensemble (he didn't call it that) from combinatorics.

As already mentioned in endnote 2, Chapter 7 is all about Boltzmann v. Gibbs and, in particular, ergodic theory.9 Chapter 8 is devoted to polyatomic molecules. There one of the most pressing questions was that of specific heats Since the equipartition theorem assigns an average energy ˝kT to every degree of freedom, it was necessary to explain, for example, why diatomic molecules had a specific heat of 5/2 k (instead of 3k) at room temperature, and why this specific heat decreased at low temperature. (The latter question could not be resolved until quantum mechanics arrived, of course.)

Chapter 9 is entitles "Boltzmann's contributions to other branches of physics." I have previously mentioned (endnote 5)his studies in radiation theory leading to the famous Stefan-Boltzmann law. He also worked in electromagnetic theory and the foundations of mechanics where his work already suggested some of the ideas used a few years later by Einstein in special relativity.

Chapter 10, " Boltzmann as a philosopher" in my opinion might well have been left out of the book. At least I found it boring. Perhaps I'm one of these "physicists of today" to whom Cercignani refers on page 192: "Boltzmann had an exceptional preparation in philosophy, compared with a physicist of today (especially if young and/or American)…Well, I'm certainly not young, but I am American!

One thing not mentioned so far is the fact that Boltzmann's work eventually led recalcitrant scientists to accept the existence of atoms (although the final nail was not hammered into the coffin until Einstein's 1905 paper on Brownian motion). Chapter 11 "Boltzmann and his contemporaries" goes into some of the differing points of views which Boltzmann and his contemporaries shared, not only on the question of atoms but, more fascinatingly, on the whole problem of whether or not classical mechanics had any validity. As hard as it is to believe, there was an influential school of scientists at that time, led by the famous Ernst Mach and Wilhelm Ostwald, which held that the only meaningful quantity in physics was the energy, and that all physical phenomena could be derived from a simple conservation-of-energy relation. So these scientists not only denied the existence of atoms, but of force, momentum and certainly entropy. Read about this dispute in Sec. 11.4

The extent and quality of Cercignani's scholarship is extraordinary. One wonders how, in a lifetime, he could have managed to read all of the books and papers he cites. The book is fluently written, if at times the English syntax is a little clumsy (as might perhaps be expected from a person who doesn't have the opportunity to speak the language every day). This clumsiness is especially noticeable in the various lengthy quotations attributed to Boltzmann, for example the one near the bottom of page 178 beginning "Just so people…" I assume that these quotes have been translated by the author as no further attribution is given, and their syntax may derive from the convoluted German syntax we all know and love so well10 Margaret Malt's translation of Boltzmann's description of his California trip mentioned above. provides a much more idiomatic text. Still, one has to applaud Cercignani for his excellent book, written in English with translations from German (not to mention Greek and Latin as well).

Endnotes:

 

1. They are scientific biographies of Julian Schwinger and of Richard Feynman, both scheduled for reviews in this journal.

2. And Cercignani makes a good case for the fact that much of the creative work in the field usually credited to J. Willard Gibbs had been anticipated by Boltzmann: "…it is he [Boltzmann] and not Josiah Willard Gibbs (1839-1903) who should be considered as the founder of equilibrium statistical mechanics and of the method of ensembles." This claim is backed up in Chapter 7, especially Sec. 7.3 (I suppose that no one will dispute Boltzmann's being called the founder of non-equilibrium statistical mechanics.)

3. For example the KAM theorem. Cf. Appendix 8.2. There is a certain amount of inconsistency in his treatment, however. For example in Appendix 5.1, page 269, he feels constrained to define the Kroneker delta symbol for his putative expert readership.

4. Our older readers may have seen the 1969 movie Mayerling starring Omar Sharif and Catherine Deneuve. (Also in the stellar cast were James Mason and Ava Gardner.) Anyone interested in purchasing the video will find it online at amazon.com. The double suicide at Mayerling (which may not have been a suicide at all, but rather a double murder) took place in 1888, some 18 years before Boltzmann's.

5. He did his doctorate under Josef Stefan at Vienna, r4eceiving his Ph.D. in (1866) and had two teaching appointments at Vienna and one each in Graz and Munich. He was offered a chair in Berlin, but eventually declined after having first accepted. Stefan was not his thesis adviser, incidentally, as theses were not required at Vienna in those days. The famous Stefan-Boltzmann law (cf. Appendix 9.1 for its derivation) did not hail from Boltzmann's student days; it was published in 1884 during his tenure at Graz. Cf. Sec. 9.4.

6. Like many readers of this journal I have spent most of my professional life trying to solve various versions of the Boltzmann equation, but I never worried too much about things like entropy.

7. The H-theorem is "derived" in Appendix 5.3. The quotation marks (called 'inverted commas' in this book) refer to the mathematical difficulties involved when the space-inhomogeneous case is considered.

8. The laws of quantum mechanics are not really reversible, since the wave packet "collapses" when a measurement is made. Roger Penrose is quoted (page 118) as having suggested this as a possible origin of irreversibility. I suggested the same thing, thirty years ago, in the discussion period after a colloquium by Eugene Wigner. Wigner completely pooh-poohed my idea.

9. We recall that Gibbs assumed the ergodic hypothesis to justify his use of ensemble averages in place of time averages. (This hypothesis asserted that a system, represented by a point in phase-space, passes, in the fullness of time, through every point in the space). In he early 20th century mathematicians realized that this hypothesis had to be false, on the basis of measure-theoretical arguments, but it was replaced with the "quasi-ergodic" hypothesis, that the point comes arbitrarily close to every point. This was proved by Garrett Birkhoff. (More precisely, as Cercignani explains on page 150, an ergodic system is one which does not possess any sets which are invariant under the dynamics except sets of measure zero. Ergodic theory has become a topic studied by pure mathematicians. To learn more visit www.math.psu.edu/kra)

10. Including such linguistic delights as the "double infinitive." For example "When he not to travel to be able has, so have I him lunch every day brought." (Wann er nicht fahren können hat, so habe ich ihm Mittagessen täglich gebracht.) This syntax, which contributes to the poetic beauty of the German language, sounds absurd in English.