A GREAT PHYSICIST

(On the 50th Anniversary of Ludwig Boltzmann’s Death)

B. I. Davydov

September 6, 1956, marked 50 years since the death of Ludwig Boltzmann—one of the most brilliant figures in theoretical physics of the nineteenth century.

Boltzmann was born on February 20, 1844, in Vienna. A graduate of the University of Vienna, he completed his studies there in 1866. In 1869 Boltzmann moved to Graz. There he held the chair of physics and also headed a physics institute that was quite substantial for that time. Boltzmann lived in Graz, with a short interruption, for 20 years, and all his principal works belong to this period.

In 1889 Boltzmann moved to Munich, then to Vienna, and later to Leipzig. Boltzmann was a deeply convinced supporter of the atomic hypothesis, and he regarded the development of molecular conceptions as the work of his life. Alongside concrete problems of physics, he devoted much attention to general philosophical problems. Taking part in scientific debates with great polemical fervor, Boltzmann sharpened relations with the representatives of Machism, who at that time dominated the universities of Austria and Germany. This forced him to move from place to place. Only after Mach’s retirement in 1902 did Boltzmann return to Vienna with the intention of not leaving it again. At the same time he took up both his former chair of theoretical physics and the chair of natural philosophy.

Impressionable by nature, Boltzmann was keenly interested in art. He furnished his works with epigraphs from Goethe and other poets, and sometimes of his own composition. The article “The Journey of a German Professor to Eldorado,” in which Boltzmann appeared as a popularizer, he dedicated to the memory of Schiller. Boltzmann often played music in his family circle. His lack of equilibrium led to a tragic end: at the age of 62 Boltzmann took his own life during a summer holiday in Duino, near Abbazia (a resort on the Adriatic Sea), where he often spent the summer with his loved ones.

Speaking of Boltzmann’s scientific activity today, 50 years after his death, one usually has in mind exclusively his fundamental works on the foundations of the kinetic theory of gases and statistical physics. However, Boltzmann’s scientific interests were much broader. Thus, he was a zealous propagandist of Maxwell’s electromagnetic theory.

In the nineteenth century there was not yet the complete division of physicists into theorists and experimentalists that is characteristic of the twentieth century. Boltzmann was responsible for a whole series of experimental works intended to verify and vividly demonstrate the validity of Maxwellian electrodynamics. Among them one may mention measurements of the dielectric constant of gases and solids and the establishment of its connection with the optical refractive index. Boltzmann also developed the theory of the Hall effect and indicated that it makes it possible to measure the average velocity of translational motion of the carriers of current. All these

Boltzmann’s works, however, are completely blurred together with his studies in the molecular theory of gases and statistical physics.

A long series of Boltzmann’s works originally appeared in scientific journals, chiefly in the Proceedings of the Vienna Academy of Sciences. In 1909 Boltzmann’s pupil Hasenöhrl published a three-volume collection of his writings, which included all his scientific works, with the exception of popular articles, a collection of which had appeared earlier. In addition, Boltzmann published three lecture courses: the widely known lectures on the theory of gases, then lectures on mechanics, original in their manner of presentation, and lectures on the theory of electricity.

Boltzmann’s works on the kinetic theory of gases naturally at once attracted the attention of scientific circles. They were, however, “more wondered at than acknowledged.” Earlier than in Germany, Boltzmann’s scientific works received full recognition in England, and above all from Maxwell, who himself had devoted so much work to kinetic theory.

In 1894, already after Maxwell’s death, Boltzmann was received with great honor at a scientific congress in Cambridge. In 1899 he was elected a corresponding member of the Royal Society of London. He had become a corresponding member of the Vienna Academy of Sciences as early as 1875, and ten years later he was elected its full member.

Yielding to the persistent urgings of friends, in 1896 and 1898 Boltzmann summarized his fundamental works on kinetic theory and statistical physics by publishing the two-volume course Lectures on the Theory of Gases. This course became widely known, and after its appearance Boltzmann’s scientific discoveries received universal recognition.

After the great successes of the kinetic theory of matter, at the end of the last century, especially in Germany, there came a reaction associated with the names of Mach and Ostwald. A purely descriptive phenomenology became dominant. This reaction was explained primarily by the fact that physicists at that time still could not obtain sufficiently convincing evidence of the existence of molecules and of their thermal motion. Moreover, in electrodynamics Maxwell’s theory likewise had a phenomenological character. Attempts to apply atomic conceptions to it at first failed. They became fruitful only later, with the development of Lorentz’s electron theory.

Atomic conceptions were the foundation of Boltzmann’s entire scientific worldview. To “understand” any phenomenon meant for him to understand its atomic mechanism, to explain it by the interaction of atoms, which to some extent obey the laws of classical mechanics. Here, for example, is a phrase characteristic of Boltzmann: “The mechanical grounding of (phenomenological) differential equations by means of mean values connected with the idea of the arrival and departure of the smallest particles greatly increases their vividness; and to this day, apart from atomistics, no other mechanical explanation of the phenomena of nature has been found”*).

Clearly recognizing the wholly hypothetical character of the ideas then available about molecules and their physical properties, Boltzmann cautiously speaks only of “mechanical analogies” and “models” consisting of the individual molecules of macroscopic bodies. Boltzmann tries to apply atomic conceptions everywhere: in the theory of gases, and there they lead him to fundamental discoveries that laid the foundation of statistical physics; in macroscopic electrodynamics, where at that time they remain sterile. Nevertheless, even there Boltzmann does not want to give up atomic conceptions, and in his course on the theory of electricity he introduces them at least in order to “better picture to oneself the mechanism of the phenomenon.” In doing so, already not pretend—

) All quotations are from Boltzmann’s Lectures on the Theory of Gases*.

referring to any scientific depth, he did not shrink from introducing various, mutually contradictory mechanical analogies in explaining various electrodynamic phenomena.

Boltzmann, a man of broad outlook and exceptional scientific intuition, was, however, wholly alien to narrow dogmatism. Speaking out against Machism, he exclaimed: “Away with all dogmatism, both in the atomistic and in the anti-atomistic sense.” In the eighties and nineties Boltzmann, at scientific congresses, stubbornly continued to speak against the then fashionable “anti-atomistic dogmatics.” It is strange now to hear that he was regarded as a reactionary. Only with the advent of the twentieth century did a general turning point occur.

In Boltzmann’s fundamental works on the kinetic theory one may point to four principal results whose significance goes far beyond the framework of the theory of ideal gases, to which they originally belonged. These are, first, the establishment of the basic integro-differential equation of the theory of gases, now known as the Boltzmann kinetic equation; then the discovery of the “Boltzmann distribution”; further, the proof of the “\(H\)-theorem,” which was the statistical analogue of the law of increase of entropy; and, finally, the establishment of the statistical meaning of the concept of entropy as the logarithm of the a priori probability of a state.

Boltzmann’s works on the kinetic theory of gases were sometimes so closely intertwined with Maxwell’s works that it is even difficult to separate them. As early as 1859 Maxwell moved from considering the mean values of various physical quantities to studying their distribution functions and, following his initial heuristic derivation, gave in 1867 a proof of the stationarity of the Maxwellian distribution of velocities in a gas.

Soon after this Boltzmann obtained his kinetic equation already for a gas in an external force field. This equation and its generalizations are now the foundation of all physical kinetics.

The kinetic equation immediately gave Boltzmann a proof of the stationarity of the Boltzmann distribution. This proof, however, did not satisfy Boltzmann, and in 1872 there appeared his work containing the \(H\)-theorem, probably the deepest of Boltzmann’s works. This theorem proved the uniqueness of the Boltzmann distribution. At the same time it introduced the functional \(H\). Having proved the monotonicity of its increase, Boltzmann obtained the right to regard it (with the opposite sign) as a statistical analogue of thermodynamic entropy. Finally, in 1877 Boltzmann showed that entropy defined in this way coincides with the logarithm of the a priori probability of the given distribution of molecules over small phase cells.

If the derivation of the kinetic equation is a logical development of Maxwell’s ideas, and if the Boltzmann distribution may be regarded as a formal generalization of the Maxwell distribution, then the statistical interpretation of the concept of entropy and of the second law of thermodynamics is entirely Boltzmann’s achievement. It is interesting that Boltzmann’s early, still immature work, which he wrote at the age of 21, was devoted to this question.

Thermodynamic irreversibility stands in qualitative contradiction to the complete reversibility of the laws of classical mechanics. It was therefore necessary to have an unshakable conviction that all observable properties of macroscopic bodies follow from the mechanical laws of interaction of the atoms of which these bodies consist, in order to set oneself the task of revealing the mechanical meaning of the concept of entropy. Boltzmann had such a conviction, founded on direct, visual representations. In this he could proceed from the picture of such simplest irreversible processes as, for example, diffusion.

After the appearance of the \(H\)-theorem, a lively discussion flared up about the statistical interpretation of thermodynamic irreversibility. This discussion was at times carried on with great passion. It was fruitful. Its noise has long since died down, and now one can formulate with complete clarity what, in fact, was proved and what is the physical meaning of the assumptions made in doing so.

At the basis of Boltzmann’s kinetic equation, and consequently at the basis of the \(H\)-theorem as well, lies the notion of “molecular disorder,” i.e., the assumption of the statistical independence of molecules colliding with one another. From this assumption there already follows the law of the frequency of collisions, which underlies the kinetic equation.

Boltzmann quite reasonably points out that “without such an assumption it is in general impossible to prove a single theorem of the theory of gases. It is made in calculating friction, thermal conductivity, etc.” This assumption is analogous to all the assumptions that have to be made in any application of probability theory: assumptions about the equal probability of certain quantities whose exact values are regarded as a priori unknown, or about some simple law of distribution of their probabilities. The application of the mathematical theory of probability becomes possible only after such assumptions have been made. All that we are entitled to demand in this connection is the internal consistency of the theory thus created and its agreement with observations.

It is noteworthy that precisely the agreement of all conclusions of the kinetic theory with experiment never raised any doubts in anyone; they were irrefutably justified by an enormous body of experimental material. The entire discussion that unfolded around the \(H\)-theorem was connected with the question of the internal consistency of the theory and of its fundamental foundations.

So long as the discussion concerns thermodynamically equilibrium states, i.e., distributions satisfying the principle of detailed balance, no internal contradictions arise. On the contrary, in the study of nonequilibrium states any attempt to attach universal significance to the conclusions of the kinetic theory comes into irreconcilable conflict with the complete reversibility of classical mechanics.

The assumption of the complete statistical independence of molecules is perfectly symmetric in time. Therefore, if it is satisfied for some distribution of the coordinates of the molecules and their velocities, then for the same distribution but with the velocities changed to their exact opposites it will also be satisfied. Consequently, if for a given nonequilibrium distribution of molecules the functional \(H\), by Boltzmann’s theorem, must decrease, then for the same distribution but with reversed velocities it must increase. This is the well-known objection of Loschmidt. It shows that the moment for which the assumption of the complete statistical independence of molecules is satisfied must be a point of maximum for the function \(H(t)\). Thus, for nonequilibrium distributions this assumption can be satisfied only at exceptional moments.

The assumption of “molecular disorder” prevailing in a gas at any moment of time thus leads to an internal contradiction, and in order to save the \(H\)-theorem it is necessary to modify it. The way to do this is obvious. In the proof of the \(H\)-theorem it is assumed, in essence, that the colliding molecules are statistically independent before their collision. By virtue of the equations of mechanics, after the collision they will no longer be statistically independent in the same sense, as is easy to see from simple examples. Therefore, without coming into conflict with the laws of mechanics, one may assume either the statistical independence of the colliding molecules before their collision, or after it. These two possibilities are a priori equally—

... valuable. Experience shows that one should stop at the first of them: it is precisely this that leads to the monotonic increase of entropy, whereas the second possibility leads to its decrease.

With this weakened formulation of the initial assumption, Loschmidt’s objection falls away. If all molecular velocities are changed to the directly opposite ones, then the molecules that were approaching one another before a collision will turn into molecules moving apart after it. The assumption of the statistical independence of molecules before their collision will turn into the corresponding assumption concerning molecules that have just undergone a collision. Thus our assumption, which is a kind of initial condition, is no longer symmetric in time; and it is precisely this asymmetry of the initial state, with the complete symmetry of the equations of mechanics, that leads to the monotonic decrease of Boltzmann’s functional $H$.

Initially, in proving the $H$-theorem, Boltzmann proceeded from intuitive notions analogous to those that had already been used before in solving the simplest problems of the kinetic theory of gases. That these assumptions reduce to the hypothesis of the statistical independence of colliding molecules became clear only in the course of the ensuing discussion. Even in the Lectures on Gas Theory, Boltzmann’s formulations are not yet beyond reproach. Boltzmann nowhere explicitly stipulates the necessary weakening of the assumption of statistical independence of colliding molecules in passing to the study of nonequilibrium states, of which we spoke above. At first he defines a “molecularly disordered distribution” as one for which the usual expression for the number of colliding molecules is obtained. Since nonequilibrium states are at issue, such a distribution, upon reversal of the velocities of all molecules, should pass into a “molecularly ordered” one in Boltzmann’s terminology. Nevertheless, Boltzmann writes, for example, in the second part of the book, that “a state opposite to an ordered one is always again an ordered state.”

The wish to reconcile the observed thermodynamic irreversibility with the unlimited applicability of the laws of classical mechanics led Boltzmann to his “fluctuation hypothesis.” According to this hypothesis, the irreversibility of the phenomena surrounding us is of a limited character. It is connected with the fact that we are present at the decay of a fluctuation of monstrous, supercosmic dimensions.

The fluctuation hypothesis cannot be denied grandeur. In its scale, however, it goes far beyond the bounds of any phenomena of nature that have been at all investigated. The complete absence of any direct confirmations compels one to treat it with caution.

We have dwelt in detail on the questions of the statistical foundation of the second law of thermodynamics connected with the proof of the $H$-theorem, since it is probably the most significant of Boltzmann’s works. It appeared in 1872, when Boltzmann was 28 years old. Five years later his second fundamental work in the same direction appeared, establishing the connection of the functional $H$ with the a priori probability of the distribution of molecules in six-dimensional phase space, a connection expressed by the concise formula $H=-\ln W$. Thereby the law of increasing entropy acquired a clear statistical interpretation as an expression of the tendency of the evolution of molecular systems toward “more probable” distributions.

Boltzmann’s theoretical works dealt directly with the simplest object of statistical physics—the ideal gas. The application of statistical methods to more complex macroscopic bodies became possible after the transition from the investigation of statistical distributions in the six-dimensional phase space of one molecule to distributions in the many-dimensional phase space of all the molecules of the macroscopic body under consideration. The trans-

This approach is usually associated with the name of Gibbs, who gave an exposition, astonishing in its completeness, of the method that underlies all of statistical thermodynamics. It must be noted, however, that the idea of a statistical distribution in a multidimensional phase space belongs to Boltzmann, who spoke of such distributions as early as 1871, when he considered his “ergodic distribution.”

In his investigations of kinetic theory, Boltzmann proceeded from the basic conception of the atomic structure of macroscopic bodies. In his time this inevitably required the assumption that the motion of atoms and molecules obeys the laws of classical mechanics. Boltzmann was fully aware of the riskiness of such an assumption. He even pointed to the possibility that, in reality, the elementary laws of motion of individual molecules already have a statistical character. The development of modern quantum mechanics justified such an assumption in an extremely distinctive way.

Boltzmann’s name resounds for us as the name of one of the creators of classical theoretical physics. At the same time, it was precisely the application of the statistical methods he developed to the theory of radiation that led to the creation of quantum theory. The first work on the theory of thermal radiation—the theoretical explanation of the law discovered experimentally by Stefan—belongs to Boltzmann himself.