LUDWIG BOLTZMANN*

N. N. Bogolyubov and Yu. V. Sanochkin

Fifty years separate us from the day of the tragic death of one of the greatest physicists of the nineteenth century, the author of fundamental studies in the kinetic theory of gases, thermodynamics, and the theory of radiation, a convinced and passionate fighter for atomistic views in science—Ludwig Boltzmann.

Boltzmann’s name is inseparably connected with the foundation and development of statistical physics: he discovered the fundamental law of the stationary “Boltzmann distribution,” which runs like a red thread through all branches of statistical physics; to him belong the statistical interpretation of the second law of thermodynamics and the famous \(H\)-theorem, which formed the basis of the theory of irreversible processes; he is the author of the kinetic equation on the basis of which modern physical kinetics has grown. He is the author of one of the laws of black-body radiation that led to the “ultraviolet catastrophe” of classical physics, the elimination of which in turn led to quantum theory.

The fate of Ludwig Boltzmann, as one of the founders of modern physics, can only be compared with the fate of the great creator of set theory—Georg Cantor. The ideas of both men were not understood or properly appreciated during their authors’ lifetimes, which tragically affected the destinies of these men of genius. But their works were understood and appreciated by subsequent generations, rose up as a rich harvest on the scientific field of physics and mathematics, and just as today one cannot conceive of modern mathematics without the methods of set theory, without the methods of the transfinite, whose creator was Georg Cantor, so one cannot conceive of modern physics without statistical methods, whose founder and zealous propagandist was Ludwig Boltzmann.

Boltzmann was born on February 20, 1844, in Vienna. His father, a financial commissioner, died early. His brother Albert, who according to contemporaries was also a highly gifted youth, died of pulmonary tuberculosis while still a gymnasium pupil. Boltzmann also had a sister, Hedwig, whose life likewise ended early and tragically—in mental derangement. Boltzmann completed the gymnasium course in Linz and in 1863 entered the University of Vienna. There, at that celebrated university, Boltzmann attended Stefan’s lectures.

Already at that time Boltzmann formed close and friendly relations with Loschmidt, who had a great influence on the formation of Boltzmann’s scientific interests. In his student years his first work appeared, “On the motion of electricity on curved surfaces” (1865), and a year later his second, “On the mechanical meaning of the second law of the theory of heat.” From the titles of these first two works one can see the two areas of physics to which

* Report read on September 5, 1956, at a meeting of the Division of Physical and Mathematical Sciences of the Academy of Sciences of the USSR, dedicated to the fiftieth anniversary of the death of Ludwig Boltzmann.

Boltzmann, thanks to the influence of his teachers and friends—Stefan and Loschmidt—directed his attention to the kinetic theory of gases and Maxwell’s theory of the electromagnetic field.

Boltzmann himself related that, in response to his question as to what he should study in order to “enter into” the doctrine of electricity, Stefan gave him an English grammar, which was to open for Boltzmann access to Maxwell’s works. Maxwell’s theory made an enormous impression on Boltzmann.

Boltzmann was interested in art. Many of his works are furnished with poetic epigraphs. Boltzmann especially noted Schiller’s influence on him: “Without Schiller there might, of course, have been a man with my nose and beard, but that man would not have been me.”

Having received the doctoral degree in 1866, at the age of 22, Boltzmann a year later took the position of Privatdozent in physics at the University of Vienna; before that he had been Stefan’s assistant.

In 1869 he moved to Graz to take the post of professor of theoretical physics. This period was the flowering of his scientific activity, when he carried out his most fundamental investigations.

Boltzmann remained in Graz until 1873, and in 1871–1872 he spent one semester each on leave, which he passed in Heidelberg and Berlin. During these trips he became close to the mathematician Königsberger, the physicists Kirchhoff and Helmholtz, the chemist Bunsen, and made the acquaintance of Sofia Kovalevskaya. By this time Boltzmann was already the author of a number of major works, and his name was beginning to become known. In 1873 Boltzmann took over the chair of his mathematics teacher Mott at the University of Vienna; however, in 1876 he again returned to Graz as professor of experimental physics. By this time he was already the author of the famous \(H\)-theorem and of the fundamental gas-kinetic equation. In the same year Boltzmann married Henriette von Aigentler, after which he lived without leaving Graz for a full 14 years. The only notable event during this time was his invitation, on Kirchhoff’s advice, to the University of Berlin, which he declined.

In 1890 Boltzmann left Graz and moved to Munich, where he held the chair of theoretical physics until 1894. In that year Stefan died, and Boltzmann inherited the chair of his teacher at the University of Vienna.

In 1900 he moved to Leipzig, where he held the chair of theoretical physics, but after two years he again returned to Vienna. The last years of his life were spent in Vienna, where he lectured on physics and natural philosophy.

On September 5, 1906, a tragic death cut short his life.

The scientific legacy of Ludwig Boltzmann is extraordinarily rich. His original works were published in three volumes (1909). In addition, he left a number of courses, of which the most fundamental is the course Lectures on Gas Theory. In these lectures his fundamental results on the kinetic theory of matter and statistical physics are systematically set forth. Among his courses one should also note the Lectures on Maxwell’s Theory of Electricity and Light, delivered in Munich and published in 1891–1893, in which he actively popularized the then still new teaching of Faraday and Maxwell on electricity and magnetism. He also wrote the lectures Principles of Mechanics, distinguished by depth of approach, and, jointly with his pupil Nabl, the article “Kinetic Theory of Matter” in the Encyclopedia of Mathematical Sciences.

Boltzmann’s methodological views found expression in his numerous public lectures and addresses.

Only now, half a century later, are we in a position to assess the significance of his contribution to science, for the achievements of a scholar are evaluated not only by his own results, but also by the results of followers who worked and developed

...his ideas. The higher and broader the building erected on the scientist’s fundamental ideas, the more valuable his contribution.

In his theoretical works Boltzmann was above all a passionate and convinced adherent of molecular theory. The notion that macroscopic material bodies are not a continuous medium but consist of an enormous number of the tiniest particles goes back, as is well known, to Democritus. But only in the nineteenth century did this notion lead to the creation of a completed physical theory—the kinetic theory of matter and statistical physics. It is Boltzmann, precisely, to whom belong the fundamental ideas that complete this line of development.

Initially, the main idea of the kinetic theory of matter was the molecular-kinetic interpretation of the first law of thermodynamics—the law of conservation of energy. Then, in the second half of the nineteenth century, in 1859, Maxwell discovered the equilibrium distribution of gas molecules. In 1868–1875 Boltzmann generalized Maxwell’s results to gases in an external force field. In this way he arrived at the formula of the “equilibrium Boltzmann distribution,” which subsequently became the basis of all classical statistical physics.

To understand the difficult situation in which the great physicist had to create, and the intense ideological struggle that he waged, one must take into account the situation that had developed in physics by the end of the last century. The flourishing of his scientific creativity falls in the period of dominance by the supporters of the theory of “pure description.” Classical physics, developing the ideas of Newton and Descartes, sought to be a continuation of mechanics and tried to reduce all phenomena to mechanical motion. For example, the universal law of conservation and transformation of energy was perceived and understood by the majority of physicists purely mechanically, as a consequence of a well-known theorem of mechanics, if one assumes that thermal and other phenomena have a mechanical nature. The situation was otherwise with the second law of thermodynamics, which stated the irreversibility of all real processes. The second law was formulated as a generalization of experience; moreover, its physical meaning remained unclear, as did the limits of its applicability. As a conclusion based on the straightforward application of the second law to the entire universe, one obtained the law of the dissipation of energy and, as its consequence, the “heat death” of the universe. This latter circumstance gave rise to the difficulties of classical mechanical physics. If all physical phenomena could be reduced to mechanical ones, then the law of the dissipation of energy would remain completely inexplicable. Hence arose the much-discussed opposition between the “energetic” (in general, phenomenological) and the mechanical worldviews.

The energetists (in general, the advocates of “pure description”) asserted that the task of science and theory should be, as far as possible, a simple description of phenomena. Theory should be the mathematical expression of relations given directly in experience. At the same time, theory should renounce hypotheses, model representations, and attempts to explain phenomena by reducing them to the simplest elements. By the very spirit of its construction, phenomenology attributed universality and absolute validity to the second law. Boltzmann’s statistical interpretation of the second principle of thermodynamics opened up the broadest possibilities for the fusion of thermodynamics with atomistics. It should be recalled that the atomistic hypothesis was not popular in the scientific world, for at that time there was still no direct experimental evidence in favor of atomistics, and throughout his life Boltzmann had to defend atomistics against countless attacks. “I am the last who thinks of denying the possibility of constructing another, non-atomistic picture of nature,” he writes with bitterness in his Principles of Mechanics. One of the heralds of phenomenology, Ostwald, only in 1908, two years after

Boltzmann, openly acknowledged the existence of atoms and molecules, under the influence of direct experiments.

Let us dwell somewhat more fully on Boltzmann’s methodological views.

The originality of the philosophical and methodological principles defended by Ludwig Boltzmann determined, to a considerable extent, the unusual fate of his work. He was a member of the leading Academies of the Old and New Worlds. A number of universities elected him an honorary doctor. Yet toward the end of his life his views were declared antiscientific, and he himself a scholar of the “old school.” To gain an idea of the situation that had arisen, it is enough to cite the statement of a German journal (1898) occasioned by the publication of Boltzmann’s now classic book Lectures on Gas Theory. This journal wrote (the quotation is given in the works of M. Smoluchowski, vol. III, p. 61): “The kinetic theory, as is well known, is just as erroneous as the various mechanical theories of gravitation; in particular, it mistakenly understands the principle of the conservation of energy; if, however, anyone wishes to become acquainted with it, let him take up Boltzmann’s book.”

Concerning this statement, Boltzmann’s great pupil, the Polish physicist Marian Smoluchowski, says: “It is highly instructive to follow the changeable fates of scientific theories. They are more interesting than the changeable fates of people, for each of them contains within itself something immortal, at least a particle of eternal truth.”

In the speech “On the Significance of Theories” Boltzmann writes: “I hold the view that the task of theory consists in constructing the reflection of the external world that exists within us, which must serve as a guiding star in all our thoughts and experiments. The originality of the human spirit lies precisely in the fact that it strives to create for itself such a reflection and to adapt it more and more to the external world... In our sense Columbus, R. Mayer, and Faraday are theoreticians of the purest kind, since not the search for practical utility, but the representation of nature in their mind served them as the guiding aim. The elaboration and constant improvement of this representation is the chief task of theory.”

In another article we find: “We draw conclusions regarding the existence of all things only from the impressions they produce upon our sense organs.”

In the article “Mathematics on Energetics” Boltzmann writes: “Distrust of representations derived from immediate sense perceptions has generally led to an extreme opposite to the former naïve belief. They say: only sense perceptions are given to us; therefore one cannot go a single step further.” And further: “If we do not wish to come to the conclusion that, in general, only the representations I have at the given moment exist, and nothing more—which is already refuted by the usefulness of knowledge for practical activity—then it is necessary for us to recognize, with all caution, also the capacity, on the basis of perceptions, to make conclusions about that which we do not perceive, although these conclusions we must constantly correct, for they come into contradiction with new perceptions.” If one is consistent, then from such arguments, adduced here, of the adherents of Mach and others, there follows the denial of the existence not only of all other beings besides one’s own self, but also of all representations that existed earlier. Thus Boltzmann refutes the idealistic philosophy of Machism, reducing it to solipsism. Strange as it may seem, in his time Boltzmann, in connection with his struggle against the absolutization of purely phenomenological theories, had to defend the usefulness and necessity of hypotheses for the development of science.

In the article “On Statistical Mechanics” Boltzmann writes: “Our theories are in no way constructed from logically irrefutable truths; on the contrary,

they consist of more or less arbitrary pictures—so-called hypotheses.

In his work on mechanics, Kirchhoff sets himself the task of describing natural phenomena as simply and vividly as possible, renouncing any explanation of them. His work initiated the “descriptive physics” that was fashionable at the time. Its adherents objected to the explanation of natural phenomena, since we are not in a position to explain them completely. The simplest regularities of nature cannot, at the present stage of the development of science, be the object of explanation. “Natural science merely decomposes complexes into simpler and homogeneous constituent parts, reduces more complex laws to more fundamental ones,” says Boltzmann. At this point the process of explanation must naturally come to an end. But this natural limit of explanation is not a manifestation of the limitation of our intellect, just as it is not a defect of our vision that we cannot see through opaque objects. Boltzmann himself writes on this subject: “Some problems recall the question put to a certain painter: ‘What picture has he hidden behind the curtain?’—‘The curtain is the picture,’ replied the painter. He had been invited to deceive the connoisseurs by his art, and for this purpose he painted a picture depicting a curtain. Usually the limitation of our intellect is seen in the circumstance that if we were to succeed in discovering the simplest, basic laws, we would not be able to explain them further, i.e., to decompose them into still simpler ones. Are we not faced here with the aforementioned painted curtain?”

Boltzmann was a consistent atomist and devoted much effort to defending the right of the atomic theory to exist: “... All observations unanimously prove that there exist bodies of such negligible dimensions that only by cohering in millions can they excite our sense organs. We call them atoms and molecules ... We shall know nothing about the structure of the atom until, on the basis of observations, we manage to formulate some hypothesis ... Spectral analysis gives hope of success. The existence of atoms and molecules is, of course, only a hypothesis. Perhaps the atomic hypothesis will be displaced by some other hypothesis. Perhaps, but it is unlikely.” It is unlikely, in Boltzmann’s opinion, because on the basis of the atomic hypothesis predictions were made which he places alongside Leverrier’s prediction of the planet Neptune. And further: “Closely adjoining atomistics is the hypothesis that the elements of the bodily world do not remain at rest while forming matter, but are in continuous motion. And this hypothesis, which is called the mechanical theory of heat, likewise represents a view firmly resting on facts” (article “The Second Law of the Mechanical Theory of Heat”).

“Neither logic, nor philosophy, nor metaphysics decides in the last instance whether something is true or false, but practice does. That which leads us to correct action is truth. Therefore I regard the achievements of technology not as by-products of the natural sciences, but as logical proofs. If we had not attained these practical results, we would not know how to reason.”

One could cite many more of Boltzmann’s statements on questions of the methodology of physics and even on purely philosophical questions. Boltzmann often appeared with popular lectures and reports, which were published as a separate book, Populäre Schriften; moreover, he did not hesitate to spoil the relations by which he had gained for himself the reputation of an intractable man with a restless character. This, apparently, explains his frequent moves and changes of professorships.

Boltzmann’s struggle against Machism, which he conducted with great energy, is noted by V. I. Lenin. In the book Materialism and Empirio-Criticism we read: “Of the German physicists, Ludwig Boltzmann, who died in 1906, systematically fought against the Machist current. The enthusiasm for new gnoseological

to dogmas he opposed a simple and clear reduction of Machism to solipsism... Boltzmann, of course, is afraid to call himself a materialist and even makes a special reservation that he is by no means opposed to the existence of God. But his theory of knowledge is in essence materialistic, and it expresses the opinion of the majority of natural scientists.”

Let us turn to a characterization of Boltzmann’s physical views, and first of all of his mechanism. He himself writes on this subject: “If you ask me, concerning my conviction, whether the present century will be called the iron century or the century of steam and electricity, I shall answer without hesitation that our century will be called the century of the mechanical worldview, the century of Darwin.” In these words a profound meaning is hidden: Boltzmann is not only a mechanist in the narrow sense of the word. In the language of the natural scientist of that time, the word mechanism often meant simply a worldview based on recognition of causality and law-governed regularity in nature.

According to Boltzmann, mechanics serves as the foundation of general natural science, for the simplest phenomenon is a change of place. “For the explanation of natural phenomena we choose an aggregate of a very large number of very small particles, continuously moving and obeying the laws of mechanics” (“Principles of Mechanics”). Thus Boltzmann’s atomistics is a mechanical atomistics, differing from our contemporary conceptions. The cautious natural scientist Boltzmann was alien to stagnation and conservatism. Interesting, for example, is his attitude toward the assertion of the indivisibility of atoms. On this point he wrote: “At present not a single physicist believes in the indivisibility of the atom.” Boltzmann had to defend atomistics chiefly in disputes with energetics; moreover, what interested him was basically the problem of the structure of matter. He proceeded from the discreteness of matter and from it went on to continua and differential equations, repeatedly discussing this limiting transition in the articles “On the Necessity of Atomistics in the Natural Sciences,” “Once Again on Atomistics,” and others. To the attempts of the energetists to hide behind differential equations Boltzmann objected, saying that he who does not see atomistics behind differential equations does not see the forest for the trees.

Boltzmann did not deny the possibility of a nonmechanical explanation of nature; he did not absolutize mechanism, but simply thought that in his time nothing better existed.

“No one asserts that there exists proof that the totality of natural phenomena can be explained mechanically. I myself once broke a lance for the mechanical view of nature, but only in the sense that it is a colossal advance in comparison with the former, purely mystical one.” And further: “This conception is for us only an experience that can be improved, and in time, perhaps, even completely abandoned” (“Mathematics on Energetics”). At the same time Boltzmann believed that every new nonmechanical worldview would have to include within itself elements of the former mechanical one. But he fiercely repelled attempts to reject his theory only because it was mechanical and defended atomistics, although his opponents could not propose anything better. “Not from energetics, not from phenomenology came the ray of hope for a mechanical explanation of nature, but from the atomistic theory. Needless to say, I have in mind the modern electron theory,”—thus Boltzmann greeted the appearance of the latter.

Boltzmann was a researcher of the highest degree of originality and, alongside outstanding theoretical works, was the author of a number of subtle experimental investigations. The high level of his mathematical culture, rich imagination, intuition, and experimental abilities characterize his scientific personality. Boltzmann does not go by the beaten path toward the solution of complex problems. “In natural science, less than anywhere else, is the proposition justified that the direct

“the path is the shortest,” he says. But even when dealing with problems that seem to him solvable at a given moment, he does not lose sight of larger problems of a general character.

Boltzmann was not only an outstanding scientist, but also an outstanding teacher. One is struck by his ability to set forth the most complex questions of theoretical physics with crystalline clarity. In general, however, the striving for clarity, rigor, and completeness distinguishes all his work.

Boltzmann was a man of very broad range. Among Boltzmann’s works there are works in mathematics, mechanics, hydrodynamics, the theory of elasticity, the theory of the electromagnetic field and optics, thermodynamics, and the kinetic theory of gases. The significance of these works is not uniform. Among them, beyond doubt, the works on the kinetic theory of gases and on the statistical foundation of thermodynamics stand out.

Let us try to trace the appearance of Boltzmann’s most important works in this field.

In 1871–1872 three of Boltzmann’s most important works appeared: “On the Thermal Equilibrium of Polyatomic Gases,” “Analytical Proof of the Second Law of the Mechanical Theory of Heat,” and “Further Investigation of the Thermal Equilibrium of Gas Molecules.” In these works the establishment of thermal equilibrium in gases is analyzed, and Maxwell’s distribution is generalized first to the case of polyatomic molecules and then to the case of a mixture of gases. In doing so, the famous Boltzmann distribution is obtained, which is encountered everywhere classical statistics is applied.

In the 1875 work “On Thermal Equilibrium in a Gas upon Which External Forces Act,” it is shown that the results obtained earlier retain their validity for the case of a gas situated in a field of external forces. In this work Boltzmann formulates the general gas-kinetic equation precisely in the form in which it is used today, although in his earlier works it had already occurred for various special cases (stationary, without the presence of external forces, etc.). He carries out all proofs proceeding from his equation. Introducing the function \(H\) (the mean logarithm of the distribution function), he proves his famous \(H\)-theorem in its most general form (the proof for the special case in the absence of external forces had been given by him already in 1872). The \(H\)-theorem states: the function \(H\) cannot increase with the passage of time, i.e., its behavior is, up to sign, analogous to the behavior of entropy. The function \(H\) remains constant for the case of thermodynamic equilibrium. In the same work it is shown that only the Boltzmann distribution satisfies the conditions of statistical equilibrium.

Two years later, in 1877, in the work “On the Relation between the Second Law of the Mechanical Theory of Heat and Probability Theory,” Boltzmann indicates the connection of the \(H\)-function with the statistical weight of a given state and thereby shows that the most probable state and the state of thermal equilibrium are identical. These results are then generalized to the case of polyatomic molecules and to the case of the presence of a field of external forces. Calculating the function \(H\) for the case of an ideal monatomic gas, Boltzmann showed the proportionality of the entropy and the function \(H\); thus entropy was connected with the probability of a given macroscopic state. By these works a statistical basis was laid under thermodynamics, which is one of the greatest discoveries of physics, making it possible to connect the thermodynamic properties of matter with its molecular structure. This conception is the foundation of all statistical physics.

In a series of subsequent works Boltzmann develops methods for the approximate solution of his equation, derives hydrodynamic equations from it, investigates the possibility of extending the kinetic theory of gases to the case of attractive forces, and so on. He applies his method to transport phenomena—

in gases, considers the phenomena of diffusion, internal friction, and others. His works on thermodynamics and the theory of gases are, of course, not limited to this.

The interpretation of entropy as a quantity proportional to the logarithm of the probability of a state is a brilliant achievement of Boltzmann. This interpretation has played a major role in our time as well, serving, for example, as a fundamental concept of modern information theory.

Today, as never before, the meaning of Boltzmann’s great formula

\[ S = k \ln W, \]

which, as Sommerfeld said, “carved on Boltzmann’s monument in the Vienna cemetery, hovers against the background of clouds floating above the grave of the great Boltzmann,” is understood.

Boltzmann’s \(H\)-theorem provoked an enormous and very fruitful discussion, thanks to which a number of new scientific directions arose, for example the so-called ergodic theory.

Opponents of the \(H\)-theorem pointed out that the monotonic change of the quantity \(H\) contradicts the complete reversibility of mechanics (Loschmidt) and the so-called recurrence theorem (Poincaré—Zermelo), according to which the states of dynamical systems must, after some time, approximately repeat themselves.

Having undertaken a more careful analysis of the premises of the \(H\)-theorem, Boltzmann arrives at a clearer formulation of the statistical character of the second law of thermodynamics: increase is only the most probable change of entropy under certain conditions imposed on the initial state of the molecular system under consideration. Namely, considering a system which at the initial moment \(t_0\) was in an improbable state and which, with the passage of time, passes into more probable states, we arrive at an overwhelmingly probable increase of entropy, although its decrease is also possible. On the other hand, from the reversibility of mechanics it follows that before the moment \(t_0\) the entropy must have decreased with just as great a probability. Thus we have obtained a reverse fluctuation at the moment \(t_0\).

To understand the irreversibility of statistical processes, one need only assume that we are present at the damping of a cosmically grandiose fluctuation. What has been set forth expresses precisely the brief meaning of Boltzmann’s fluctuation hypothesis, which sought to reconcile the observed thermodynamic irreversibility with the materialistic conception of the unlimited existence of the universe. This hypothesis played its role in the struggle against the antiscientific theory of the “heat death” of the universe. The notion of heat death was based on applying thermodynamic laws to the whole universe as a whole. However, this is an unfounded extrapolation. Such an objection, of course, also applies to Boltzmann’s fluctuation hypothesis.

The problem of creating a statistical theory of irreversible processes, the first solution of which was given by Boltzmann, proved to be very fruitful. His pupil, the great Polish physicist Marian Smoluchowski, devoted his entire life to this problem.

Smoluchowski’s investigations led to the creation of the statistical theory of Brownian motion, which played an enormous role in science. Suffice it to say that on the basis of these works there has grown up today an extensive branch of mathematics—the theory of Markov random processes.

The study of Brownian motion led to the discovery of the Wiener measure, which at the present time has fundamental significance after the work of Feynman and in such an entirely new field as quantum field theory.

The second large group of Boltzmann’s works is devoted to Maxwell’s theory of electromagnetism. Boltzmann belongs among the admirers and propagandists of Maxwell’s theory, which at that time ran counter to customary views and seemed to many mathematically extremely complicated.

Boltzmann’s first major experimental work was devoted to proving the validity of the Maxwellian relation between a substance’s refractive index and its dielectric constant.

At first Boltzmann investigated solid insulators. Using the condenser method, he determined the dielectric constants of sulfur and paraffin. The determination of the dielectric constants of gases presented considerably greater difficulties. He investigated air, hydrogen, oxygen, carbon dioxide, and others. The great difficulties that were overcome in the process testify to Boltzmann’s experimental skill. Boltzmann overcame still greater difficulties in determining the dependence of the dielectric constant on direction in the case of anisotropy of the medium. He was the first to draw this conclusion from Maxwell’s theory, and he himself verified it in experiments with a sphere cut from a crystal of rhombic sulfur. Thus Boltzmann was among the first to provide experimental proof of Maxwell’s theory.

The second part of his works on the theory of the electromagnetic field, belonging to these same years (1872–1874), was devoted to the study of the polarization of dielectrics in an electric field. In particular, the aim of one of his works was to show that the change in the capacitance of a condenser is explained precisely by the polarization of the dielectric, and not by its weak conductivity. Along the way, by another method, he determined the dielectric constants of a number of substances.

In addition, one may point to some of his theoretical works, for example, on the motion of electricity on curved surfaces, on the interaction of the parts of an electric circuit of arbitrary form, and so on.

One cannot fail to note his works on the theory of thermoelectricity, diamagnetism, the Hall effect, where he indicated the possibility of determining the sign of charge carriers, the theory of electro- and magnetostriction, studies of the action of a magnetic field on an electric discharge in rarefied gases, and several works on capillary phenomena and the heat capacity of gases.

Carrying out Boltzmann’s idea, Toepler constructed an apparatus that made it possible to analyze acoustic oscillations by an optical method. By this method, periods, amplitudes, phase shifts, and other characteristics of oscillations in air resonators were investigated. Several of Boltzmann’s works are devoted to the theory of elastic aftereffect; here he defended the point of view that the elastic force depends not only on the instantaneous deformation, but also on the preceding deformation. Boltzmann tried to confirm his theory by experiments on the torsional oscillations of metal rods.

In 1884 Boltzmann theoretically derived Stefan’s law of thermal radiation. From general considerations he obtained the well-known formula relating the pressure and the energy density of black radiation, and then, applying thermodynamics to thermal radiation, obtained his famous formula.

Lorentz called this work “a pearl of physics.” The works listed do not exhaust the entire list of Boltzmann’s writings, but they give an idea of the breadth of his scientific interests and the scope of his activity. Of course, however, his principal results belong to the kinetic theory of gases and to the statistical foundation of thermodynamics. Boltzmann himself said: “An individual person can attain great significance only through complete devotion to some idea.” With these words one would like to conclude the survey of his activity.