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Boltzmann: a disordered genius

09 Apr 1999

Ludwig Boltzmann: The Man Who Trusted Atoms
Carlo Cercignani
1998 Oxford University Press 348pp £29.50/$49.95hb

The 1860s and 1870s form one of the most exciting periods in physics, probably on a par with the 1920s and 1930s when quantum mechanics was developed. James Clerk Maxwell was working on his theory of the electromagnetic field, Rudolf Clausius introduced the concept of entropy in thermodynamics, and kinetic theory was starting to become fashionable. It was also during this period that the Austrian physicist, mathematician and philosopher Ludwig Boltzmann began his scientific career and wrote some of his most famous papers.

Born in Vienna on 20 February 1844 during the night between Shrove Tuesday and Ash Wednesday, Boltzmann used to say that this was why his mood could swing so violently from one of great happiness to one of deep depression. It was a tragedy that Boltzmann did not live to experience the glory of his pioneering ideas, for he committed suicide during one such depression in September 1906.

Carlo Cercignani starts his book with a short biography of Boltzmann, and includes many original quotations from him and his contemporaries. Among these are a jocular poem by Boltzmann called “Beethoven in Heaven” and a quotation from Robert Musil’s novel The Man Without Qualities, which characterizes the Austro-Hungarian empire in the early 1900s. The chapter on “physics before Boltzmann” also contains numerous quotations and biographical notes about scientists such as Isaac Newton, Roger Josef Boscovich, Sady Carnot and Michael Faraday. Cercignani is clearly a man of wide reading, and this is obvious not just in his parenthetical remarks in the main part of the book, which covers Boltzmann’s scientific work, but throughout the other chapters as well.

Boltzmann studied at Vienna University from 1863 to 1866, where Josef Stefan acquainted his gifted student with the most recent developments in physics. Although Boltzmann’s first scientific paper was on electrodynamics, his second was devoted to the mechanical meaning of the second law of thermodynamics – a topic that Boltzmann would return to again and again throughout his life, and to which he eventually gave an exhaustive answer.

By the time he was 25 years old, Boltzmann was already a full professor of mathematical physics at the University of Graz. It was there that he met his wife Henriette von Aigentler, who was the first female student at the university. His scientific career blossomed at Graz, and his marriage was a happy one. But when his mother died in 1885, Boltzmann suffered his first deep depression. He did not write a single letter that year, and published just one paper on statistical mechanics, linked to a paper by the leading German physicist Hermann von Helmholtz.

From that time on Boltzmann led a restless life. Following his scientific liaison with Helmholtz in Berlin, Boltzmann agreed to succeed Gustaf Kirchoff as professor of physics there in 1888, but withdrew even though the German emperor Wilhelm I had already agreed his appointment. Two years later Boltzmann moved instead to the University of Munich, where he was able to devote his lectures entirely to theoretical physics. He enjoyed the academic life at Munich, but the university could not provide for his retirement, so in 1895 Boltzmann moved back to Vienna to succeed Stefan. However, he soon regretted that decision, and in 1900 left for the University of Leipzig. He did not last long there either, and within two years was back in Vienna once more.

One characteristic of Boltzmann’s work was that many of his papers were worked out in great detail. This verbosity was hardly surprising, for he believed that matters of elegance ought to be left to the tailor and the cobbler. (Albert Einstein said that he adhered scrupulously to the principle in 1920 when preparing a popular lecture on special and general relativity.) For example, Boltzmann’s 1872 paper, which contains the so-called “H-theorem” and the transport equation that is now known as the “Boltzmann equation”, is some 87 pages long.

The H-theorem gave the first explicit probabilistic expression for the entropy of an ideal gas, which is proportional to Boltzmann’s functional, H. It was also the first time that probability had been introduced in such a basic piece of physics as the second law of thermodynamics, and most of Boltzmann’s contemporaries were unable to accept this departure from strict determinism. Boltzmann wrote to his mother in Vienna in September 1872 saying that he had given a lecture on the theorem to the Physical Society in Berlin, but that hardly anyone was able to follow him – apart from Helmholtz, with whom an interesting discussion developed.

Even the ingenious Maxwell, who in 1859 had described the velocity distribution of gas molecules in thermal equilibrium, wrote the following in a letter to his colleague Peter Tait in 1873: “By the study of Boltzmann I have been unable to understand him. He could not understand me on account of my shortness, and his length was and is an equal stumbling block to me.” If Maxwell found Boltzmann’s papers difficult, it is hardly surprising that many other physicists found them difficult as well! Cercignani thinks that this is one reason why Boltzmann does not – even today – receive as much credit as he deserves, particularly since most physicists have never read his original papers.

Although many physicists at the end of the 19th century rejected the atomic hypothesis and the kinetic theory that was based on it, Boltzmann insisted that it did indeed provide a unified world picture. However, today, statistical mechanics is best known through the work of the great American physicist Josiah Willard Gibbs, who originally coined the term “statistical mechanics” and published an in-depth analysis of the field in 1902. Yet the microcanonical and the canonical ensemble of Gibbs can be found in papers that Boltzmann published nearly 18 years previously, albeit under the names “ergode” and “holode”. It is to the great credit of Cercignani – a world expert in statistical mechanics – that he has provided such a conscientious introduction to Boltzmann’s ideas. The book is also easy to read, even though the author is not afraid to include some basic formulae. More subtle derivations are provided in the appendices, which also link Boltzmann’s work to more recent progress in this field.

Boltzmann developed his statistical interpretation of the second law of thermodynamics in his famous paper of 1877 on the second law and probability calculus. He wrote the paper in response to objections from Josef Loschmidt, who said that the H-theorem singled out the direction in time in which H decreases, whereas the underlying mechanics was the same whether time flowed forwards or backwards. It was in this paper that Boltzmann’s famous relation, S = k log W, first appeared. This equation links the entropy, S, to the logarithm of the number of microstates, W, that a given macroscopic state of the system can have, with k now being known as Boltzmann’s constant. Since W is proportional to the probability of the macroscopic state, then the equilibrium state is by far the most probable. The transition from a non-equilibrium to the equilibrium state is simply a transition from a very unprobable to the most probable state. This relationship – on which Max Planck based the derivation of his formula for the intensity distribution of black-body radiation – was later called Boltzmann’s principle by Einstein. It is also engraved on Boltzmann’s tombstone in Vienna.

The book also addresses Boltzmann’s contributions to mathematics and the philosophy of science. Boltzmann’s work posed a number of problems for mathematicians. For example, in 1912 the great mathematician David Hilbert indicated how to obtain approximate solutions of the Boltzmann equation, a method that was later generalized by David Enskog. Meanwhile, “ergodic theory”, which is based on Boltzmann’s concepts in statistical mechanics, is now a special branch of mathematics. And in the philosophy of science, Boltzmann anticipated the views of Thomas Kuhn on scientific revolutions. He also applied Darwin’s theory to the evolution of the human mind, anticipating many aspects of the so-called “evolutionary epistemology” and theory of science that were proposed by Konrad Lorentz and Sir Karl Popper, respectively.

I can warmly recommend the book to everybody who is interested in the history of science.

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