Ludwig Boltzmann

In Memory of Ludwig Boltzmann*

L. Flamm

Men of genius point the way for coming generations, but their creations often find universal recognition only after their own death. So it was with the world-famous Austrian physicist Ludwig Boltzmann, whose fiftieth anniversary of death we mark today. With his outstanding works, classical thermodynamics was brought to completion.

Boltzmann’s activity belongs to a time when it was believed that all of physics could be reduced to Newtonian mechanics. Newtonian mechanics knew only reversible motion. As for irreversible processes—such, for example, as the transfer of heat, which occurs only in one direction, from a higher to a lower temperature, or chemical reactions proceeding only in a definite sequence—they could not be explained by means of Newtonian mechanics.

At that time the regularities of irreversible processes were treated purely formally, with the aid of “entropy”—a quantity introduced by Clausius which, in a closed system, can only increase or remain unchanged, but can never decrease. Thermodynamic equilibrium is determined by a maximum of entropy. But why this is so remained unclear.

The explanation was given by Boltzmann’s famous principle, according to which entropy is proportional to the logarithm of the probability of a state; hence the increase of entropy meant the transition of a system from an improbable state to a more probable one. If, however, the most probable state is reached in a closed system, then the state of equilibrium sets in.

The probability of a state is calculated by means of Boltzmann statistics. It is determined by the number of complexions, i.e., by the number of atomic or, correspondingly, molecular statistically equivalent states of a substance; the calculation, in modern language, proceeds from quantum atomic or molecular states. The number of atoms or molecules in discrete states determines the number of complexions.

What was also new in Boltzmann’s principle was that the law of the behavior of entropy in a closed system is not a law of nature in the usual sense, but corresponds only to the most probable behavior of matter with a sufficiently large number of atoms or molecules, realized only with a certain degree of accuracy.

This remarkable discovery took shape as Boltzmann consistently deepened the development of the kinetic theory of gases. For monatomic ideal gases the Maxwellian law of distribution of velo—

* A report delivered on September 5, 1956, at a meeting of the Division of Physico-Mathematical Sciences of the Academy of Sciences of the USSR, devoted to the fiftieth anniversary of Ludwig Boltzmann’s death.

growth, which determined the distribution of gas molecules by velocities in the state of thermodynamic equilibrium. Boltzmann generalized Maxwell’s formula for polyatomic molecules, taking into account not only the energy of the translational motion of the molecules, but also the energy of their rotation, the vibrational energy of the atoms in the molecule, and also the external forces acting on the molecules. He gave a theoretical foundation to the very abstract treatment of statistical questions in Gibbs’s book Statistical Mechanics.

Boltzmann published two volumes of lectures on the theory of gases; he also published two volumes of lectures on mechanics, in which classical mechanics was set forth axiomatically and impeccably; he did this in order to create a rigorous foundation for his theory of gases. Even earlier there had appeared two volumes of Boltzmann’s lectures devoted to Maxwell’s theory. Boltzmann had already tried to develop Maxwell’s theory before that; he carried out experimental work on measuring the dielectric constant of sulfur crystals. In his lectures he tried to improve the mechanical models by means of which Maxwell had developed his theory. Boltzmann’s apparatus for demonstrating induction phenomena of electric currents, resembling the differential gear of an automobile that was created later, is well known.

Boltzmann’s selected works comprise three volumes. Physicists throughout the world valued Boltzmann highly: he was a member of 39 academies and learned societies, and three times was invited to America with lectures and to give occasional courses, which at that time was a rarity. Later, however, the predominance of an unfavorable philosophical current prevented his works from gaining general recognition. Boltzmann’s genius was recognized early. This is evidenced by his academic career.

Boltzmann was born on February 20, 1844, in Vienna. He was the son of an Austrian official, who was later transferred to Wels and then to Linz. From 1863 to 1866 he studied mathematics and physics at the University of Vienna; after completing his studies, he received the doctoral degree and passed the examinations for a gymnasium teacher. The following year he became an assistant to Prof. Stefan in the Physical Institute, where a close friendship arose between him and Privatdozent Loschmidt. A year later, in the autumn of 1869, Boltzmann received an invitation from the University of Graz to take the post of professor of mathematical physics, with the obligation to teach the course “Elements of Higher Mathematics.” In 1873 he became the successor to his teacher Mott in the post of professor of mathematics at the University of Vienna. But in 1876 he returned again to Graz, now already as professor of experimental physics and director of the newly opened Physical Institute. For that time, on the continent, it was the best-equipped physical institute; many foreign scholars were drawn to it. Nernst and Svante Arrhenius studied and worked here for a considerable time. Boltzmann, who married at the moment of taking up this post, did not leave Graz for 14 years.

Boltzmann’s growing worldwide fame as a theorist brought him an invitation to the post of professor of theoretical physics in Munich, which he accepted in 1890. In 1896 Boltzmann’s teacher Stefan died in Vienna, and Boltzmann returned in 1894 to Vienna as Stefan’s successor in the post of professor of theoretical physics. However, Vienna was no longer what it had been in the days of his youth. Loschmidt had retired and died in 1895. In the same year Ernst Mach, who had devoted himself to the history and theory of inductive science, became professor of philosophy at the University of Vienna. As a positivist, Mach recognized as real only sensory perceptions. It was difficult for students of the physics faculty to take Boltzmann’s lectures on the theory of gases seriously, while, in parallel with Boltzmann’s lectures, guided by theoretical-cognitive considerations, Ernst Mach in his lectures denied atomistics. Boltzmann had bitterly to regret that he had left Munich.

Therefore he readily accepted an offer to go to Germany and in 1900 became professor of theoretical physics at the University of Leipzig. But things were no better there. There, the positivist and physical chemist Wilhelm Ostwald also denied atomism. Ostwald passionately defended his “energetics,” a fantastic theory according to which all of physics fits within the framework of a single “law of energy.” In Vienna, however, Ernst Mach withdrew from teaching because of illness, and in 1902 Boltzmann again accepted an invitation to return to Vienna to his still unoccupied chair of theoretical physics. In addition, in 1903 he accepted the chair of natural philosophy, succeeding Mach. This secured for him the possibility of scientific work.

In 1905 Boltzmann published Popular Writings, which included the story “The Journey of a German Professor to Eldorado,” in which he recounts his trip to America, just undertaken. Instead of resting after the arduous journey, Boltzmann urgently prepared the Encyclopedia of Mathematical Sciences for publication. A severe nervous illness led him, on September 5, 1906, to a tragic death. Soon after his death, atomism received universal recognition.

Of outstanding importance is Boltzmann’s justification of the law of radiation, whose foundation had been laid by Stefan’s experimental work. Boltzmann used Kirchhoff’s law of radiation and Maxwell’s law of radiation pressure, as well as both principles of thermodynamics. At the turn of the two centuries Lord Rayleigh developed this theory further, applying Boltzmann’s statistics to the proper oscillations of the electromagnetic field in order to obtain the spectral distribution of blackbody radiation. However, the radiation formula obtained proved valid only in the region of long waves. Soon thereafter Max Planck was able to prove that complete agreement with experiment could be obtained if one assigned a finite value to the energy intervals in Boltzmann’s statistics; thus there arose Planck’s famous quantum of action. It may be said that quantum theory was born on the basis of Boltzmann’s ideas; its role is well known.

The development of quantum theory was completed many years after Boltzmann’s death. And does it not sound like an irony of fate that the new positivism declared precisely Boltzmann’s spiritual legacy to be the only reality, since it admits only statistical statements?

The positivists, of course, rejected Boltzmann’s visual pictures of atomism. For the elementary particles of matter that have been discovered since then (electrons, protons, neutrons, etc.), coordinates and momenta no longer possess strict definiteness. But ever louder are the voices of those who believe that such an extreme point of view is unfavorable for the development of the theory of microparticles, so that Boltzmann’s ideas may perhaps exert an ever-increasing influence on the development of physics.