# Boltzmann, Ludwig

# Boltzmann, Ludwig

(*b*. Vienna, Austria, 20 February 1844; *d*. Duino, near Trieste, 5 September 1906)

*physics*.

Boltzmann’s father. Ludwig, was a civil servant (*Kaiserlich-Königlich Cameral-Concipist*), his mother was Katherina Pauernfeind. He was educated at Linz and Vienna, receiving his doctorate in 1867 from the University of Vienna, where he had studied with Josef Stefan. Boltzmann held professorships at the universities of Graz, Vienna, Munich, and Leipzig. In 1876 he married Henrietta Von Aeigenlter, who bore him four children.

*Distribution Law*. The first stimulus for Boltzmann’ researches came from teachers and colleagues at the University of Vienna, especially Stefan and Josef Loschmidt, in a lecture Stefan suggested the problem in electrical theory whose solution constituted Boltzmann’ first published paper (1865);^{1} he also published a few papers on kinetic theory and did important experimental work on gases and radiation that provided the basis for some of Boltzmann’ theories. Loschmidt (also in 1865) accomplished the first reliable estimate of molecular sizes with the help of the Clausius-Maxwell kinetic theory. Although Loschmidt was later to dispute Boltzmann’ interpretation of the second law of thermodynamics, the problem of finding quantitative relations between atomic magnitudes and observable physical quantities was a common interest of both men.

Boltzmann began his lifelong study of the atomic theory of matter by seeking to establish a direct connection between the second law of thermodynamics and the mechanical principle of least action (1866). Although Clausius, Szily, and others later worked along similar lines, and Boltzmann himself returned to the subject in his elaboration of Helmholtz’ theory of monocyclic systems (1884), the analogy with purely mechanical principles seemed insufficient for a complete interpretation of the second law. The missing element was the statistical approach to atomic motion that had already been introduced by the British physicist James Clerk Maxwell.^{2} Boltzmann’s first acquaintance with Maxwell’s writings on kinetic theory is indicated by his paper on thermal equilibrium (1868). In this paper, he extend Maxwell’ theory of the distribution of energy among colliding gas molecules, trading the case when enternal forces are present. The result was a new exponential formula for molecular distribution, now known as the “Boltzmann factor” and basic to all modern calculations in statistical mechanics. To understand how Boltzmann arrived at this result, we must first review the work of Maxwell on which it is based.

Maxwell, in his first paper on kinetic theory (1859), had pointed out that the collisions of molecules would not simply tend to equalize all their speeds but, on the contrary, would produce a range of different speeds. Most of the observable properties of a gas could be calculated if one knew, instead of the positions and velocities of all the molecules at any given time, only the average number of molecules having various positions and velocities. In many cases it seems reasonable to assume that the gas is spatially uniform, that is, the average number of molecules is the same at different places in the gas. The problem is then to determine the velocity distribution function *f(v)*, defined so that *f(v) dv* is the average number of molecules having speeds between v and *v + dv*.

Maxwell argued that *f(v)* should be a function that depends only on the magnitude of v, and that the velocity components resolved along the three coordinate axes should be statistically independent. Hence, he inferred that

where *N* is the total number of molecules, and α^{2} is inversely proportional to the absolute temperature.

In his long memoir of 1866, Maxwell admitted that the assumptions used in his previous derivation of the distribution law “may appear precarious” he offered another derivation in which the velocities of two colliding molecules rather than the velocity components of a single molecule, were assumed to be statistically independent. This means that one can express the joint distribution function for the probability that molecule 1 has velocity vsub(2 as the product of the probabilities of these two separate events:

To derive the distribution function itself, Maxwell argued that the equilibrium state would be reached when the number of collisions in which two molecules with initial velocities (v_{1}, v_{2}) rebound with final velocities (v_{1}, v_{2}) is equal to the number of collisions in which two molecules with in vsub(1)vsub(2)) rebound with final velocities (v_{1}′, v_{2}′); from this condition it follows that

By combining this equation with that for the conservation of energy (in the case when no forces act),

Maxwell deduced (as before) that

This type of reasoning about velocity distribution functions was repeatedly used and generalized by Boltzmann in his own works on kinetic theory. He began, in his 1868 paper, by considering the case in which one of the particles of a system is acted on by a force with a corresponding potential function,

*V*(x). The condition of conservation of energy would then be

and Boltzmann could then apply Maxwell’s procedure to deduce the distribution function

The constant factor *h* could be related to the absolute temperature of the gas, as Maxwell and Clausius had done, by comparing the theoretical pressure of the gas with the experimental relation between pressure and temperature (Gay-Lussac’s law). In modern notation, *h* is equivalent to 1/kT, where *k* is a constant, now called Boltzmann’s constant, and *T* is the absolute temperature on the Kelvin scale.

The physical meaning of the Maxwell-Boltzmann distribution law is that the energy (*E = mv*^{2}/2 + *V*[x]) of a molecule is most likely to be roughly equal to *kT*; much larger or much smaller energies occur with small but finite probability.

In the same paper of 1868, Boltzmann presented another derivation of the Maxwell distribution law that was independent of any assumptions about collisions between molecules. He simply assumed that there is a fixed total amount of energy to be distributed among a finite number of molecules, in such a way that all combinations of energies are equally probable. (More precisely, he assumed uniform distribution in momentum space.) By regarding the total energy as being divided into small but finite quanta, he could treat this as a problem of combinatorial analysis. He obtained a rather complicated formula that reduced to the Maxwell velocity-distribution law in the limit of an infinite number of molecules and infinitesimal energy quanta.

The device of starting with finite energy quanta and then letting them become infinitesimal is not essential to such a derivation, but it reveals an interesting feature of Boltzmann’s mathematical approach. Boltzmann asserted on several occasions that a derivation based on infinite or infinitesimal quantities is not really rigorous unless it can also be carried through with finite quantities. While this prejudice kept him from appreciating and using some of the developments in pure mathematics that appeared toward the end of the nineteenth century, it also had the curious effect of making some of his equations for energy distribution and transfer look similar to those of modern quantum theory. (This is perhaps not quite accidental, since Planck and other early quantum theories were familiar with Boltzmann’s works and used many of his techniques.)

*Transport Equation and* H-*theorem*. Although Maxwell and Boltzmann had succeeded in finding the correct distribution laws by assuming that the gas is in an equilibrium state, they thought that the kinetic theory should also be able to show that a gas will actually tend toward an equilibrium state if it is not there already. Maxwell had made only fragmentary attempts to solve this problem; Boltzmann devoted several long papers to establishing a general solution.

Approach to equilibrium is a special case of a general phenomenon: dissipation of energy and increase of entropy. It was Boltzmann’s achievement to show in detail how thermodynamic entropy is related to the statistical distribution of molecular configurations, and how increasing entropy corresponds to increasing randomness on the molecular level. This was a peculiar and unexpected relationship, for macroscopic irreversibility seemed to contradict the fundamental reversibility of Newtonian mechanics, which was still assumed to apply to molecular collisions. Boltzmann’s attempts to resolve this contradiction formed part of the debate on the validity of the atomic theory in the 1890’s. Seen in this context, the proof of the distribution law has even more significance than the law itself.

Boltzmann’s major work on the approach to equilibrium (and on transport processes in gases in general) was published in 1872. This paper, like that of 1868, took Maxwell’s theory as the starting point. Boltzmann first derived an equation for the rate of change in the number of molecules having a given energy, *x*, resulting from collisions between molecules. He considered a typical collision between two molecules with energies x and x’s - after the collision. Such a collision reduces by one the number of molecules with energy *x*; the number of such collisions is assumed to be proportional to the number of molecules with energy *x*’s. Boltzmann used here, without any comment, Maxwell’s assumption of statistical independence of the velocities of two colliding molecules (eq. 2); later it was recognized that there might be valid grounds for objecting to this assumption.^{3} With this assumption, the decrease in *f(x)* (x’s), multiplied by an appropriate factor for the collision probability and integrated over all values of x’s. Similarly, the increase in *f(x)* may be attributed to inverse collisions in which the molecules have energies – and x + x’s – before the collision, and x and x’s after the collision. By such arguments Boltzmann arrived at the equation

(This is a special case of the general Boltzmann transport equation [eq.9] terms describing the effect of external forces and nonuniformities on the change of *f* are here omitted. The square root expressions in the denominators, which do not appear in the form of the equation generally used, result from the fact that energy rather than velocity is the variable.)

One additional assumption involved in this derivation should be mentioned: the collision probability function, ψ(*x*, *x*’ξ), is the same for both the direct and inverse collisions; that is, the collision is perfectly reversible.

Following Maxwell’s 1866 development of the transport equations, Boltzmann should how the diffusion, viscosity, and heat conduction coefficients of a gas could be calculated by solving the general transport equation

where (ξ,η,ζ) are components of the velocity of a particle and (*x,y,z*) are components of the force acting on it, and *V*, φ, b, and ω_{1} are variables characterizing of the relative motion of the two molecules during a collision. (Values of the function *f* for velocities of the two molecules before and after the collision are indicated by *f*, *f*^{1}, *f*′, and *f*_{1}′.)

It is difficult to obtain exact solutions of Boltzmann’s transport equation except when the molecules interact with inverse fifth-power forces, a case for which Maxwell had found an important simplification.^{4} Boltzmann made several attempts to develop accurate approximations for other forces laws, but this problem was not satisfactorily solved until the work of S. Chapman and D. Enskog in 1916–1917. Boltzmann’s equation is now frequently used in modern research on fluids, plasmas, and neutron transport.

If the velocity distribution function is Maxwellian, then the integral on the right-hand side of eq. 8 vanished identically for all values of the variables, and we find ∂*f*/∂*t*=0. In other words, once the Maxwellian state has been reached, no further change in the velocity distribution function can occur.

So far this is simply an elaboration of the previous arguments of Maxwell and of Boltzmann himself, but now, with an explicit formula for ∂*f*/∂*t*, Boltzmann was able to go further and show that *f(x)* probably tends toward the Maxwell form. He did this by introducing a function, *E*, depending on *f(x)*

and showing that *E* always decreases unless *f* has the Maxwellian form:

(The proof is straightforward and relies simply on the fact that the quantity (a-b) log *b/a* is always negative if a and b are real positive numbers.) Boltzmann also noted that in the Maxwellian state *E* is essentially the same as the thermodynamic entropy (aside from a constant factor). Thus Boltzmann’s “*H*-function”(the notation was changed from *E* to *H* in the 1890’s) provides an extension of the definition of entropy to nonequilibrium states not covered by the thermodynamic definition.

The theorem that *H* always decreases for nonequilibrium systems was called “Bltzmann’s minimum theorem” in the nineteenth century, and now goes by the name “Boltzmann’s *H*-theorem.” (It has not yet been proved rigorously except with certain specializing assumptions.)

*Reversibility and Recurrence Paradoxes*. The *H*-theorem raised some difficult questions about the nature of irreversibility in physical systems, in particular the so-called “reversibility paradox” and “recurrence paradox.”(The modern terminology goes back only to the Ehrenfests’ 1911 article, in which the words *Umkehreinwand and Wiederkehreinwand* were introduced.) The reversibility paradox First discussed by Lord Kelvin (1874) and brought to Boltzmann’s attention by Loschmidt, is based on the apparent contradiction between one of the basic premises of Boltzamann’s derivation—the reversibility of individual collisions—and the irreversibility predicted by the theorem itself for system of many molecules. Of coures there must be such a contradiction between any molecular theory based on Newtonian mechanics and the general principle of dissipation of energy, but Boltzmann’s work was the first to reveal this inconsistency explicitly.

Boltzmann’ initial response (1877) to the reversibility paradox was the suggestion that the irreversibility of processes in the real world is not form of the intermolecular force law but, rather, seems to be a result of the initial conditions. For some unusual initial conditions the system might in fact decrease its entropy (increase the value of *H*) as time progresses: such initial conditions could be constructed simply by reversing all the velocities of the molecules in an equilibrium state known to have evolved from a nonequilibrium state. But, Boltzmann asserts, there are infinitely many more initial states that evolve with increasing entropy, simply because the great majority of all possible states are equilibrium states. Moreover, the entropy would also be almost certain to increase if one picked an initial state at random and followed it backward in time instead of forward.

The recurrence paradox arises from a theorem in mechanics first published by Poincare in 1890. According to this theorem, any mechanical system constrained to move in a finite volume with fixed total energy must eventually return to any specified initial configuration. If a certain value of the entropy is associated with every configuration of the system (a disputable assumption), then the entropy cannot continually increase with time, but must eventually decrease in order to return to its initial value. Therefore the *H*-theorem cannot always be valid.

Poincaré, and later Zermelo (1896), argued that the recurrence theorem makes any mechanical model, such as the kinetic theory, incompatible with the second law of thermodynamics; and since, it was asserted, the second law is a strictly valid induction from experience, one must reject the mechanistic viewpoint.

Boltzmann replied that the recurrence theorem does not contradict the *H*-theorem, but is completely in harmony with it. The equilibrium state is not a single configuration but, rather, a collection of the overwhelming majority of possible configurations, characterized by the Maxwell-Boltzmann distribution. From the statistical viewpoint, the recurrence of some particular initial state is a fluctuation that is almost certain to occur if one waits long enough; the point is that the probability of such a fluctuation is so small that one would have to wait an immensely long time before observing a recurrence of the initial state. Thus the mechanical viewpoint does not lead to any consequences that are actually in disagreement with experience. For those who are concerned about the cosmological consequences of the second law—the so-called “heat death” corresponding to the final attainment of a state of maximum disorder when all irreversible processes have run their course—Boltzmann suggested the following idea. The universe as a whole is in a state of thermal equilibrium, and there is no distinction between forward and backward directions of time. However, within small regions, such as individual galaxies, there will be noticeable fluctuations that include ordered states corresponding to the existence of life. A living being in such a galaxy will distinguish the direction of time for which entropy increases (processes going from ordered to disordered states) from the opposite direction; in other words, the concept of “direction of time” is statistical or even subjective, and is determined by the direction in which entropy happens to be increasing. Thus, the statement “Entropy increases with time” is a tautology, and yet the subjective time directions in different parts of the universe may be different. In this way local irreversible processes would be compatible with cosmic reversibility and recurrence. (Boltzmann’s concept of alternating time directions has recently been revived in connection with theories of oscillating universes.)

*Statistical Mechanics and Ergodic Hypothesis*. Having followed Boltzmann’s work on irreversible processes into some of the controversies of the 1890’s, let us now return to his contributions to the theory of systems in thermal equilibrium (for which the term “statistical mechanics” was introduced by J. Willard Gibbs).

It would be possible (as is in fact done in many modern texts) to take the Maxwell-Boltzmann distribution function (eq. 7) as the basic postulate for calculating all the equilibrium properties of a system. Boltzmann, however, preferred another approach that seemed to rest on more general grounds than the dynamics of bimolecular collisions in low-density gases. The new method was in part a by-product of his discussion of the reversibility paradox, and is first hinted at in connection with the relative frequency of equilibrium, as opposed to nonequilibrium, configurations of molecules: “One could even calculate, from the relative numbers of the different distributions, their probabilities, which might lead to an interesting method for the calculation of thermal equilibrium.”^{5} This remark was quickly followed up in the same year (1877) in a paper in which the famous relation between entropy and probability,

S =

klogW,

was developed and applied. In this equation, *W* is the number of possible molecular configurations (“microstates,” in modern terminology) corresponding to a given macroscopic state of the system.^{6} (To make this expression meaningful, microstates have to be defined with respect to finite cells in phase space: the size of these cells introduces an arbitrary additive constant in S which can be determined from quantum theory.)

The new formula for entropy—from which formulas for all other thermodynamic quantities could be deduced—was based on the assumption of equal a priori probability of all microstates of the system (that is, all microstates that have the same total energy). As noted above, Boltzmann had already proved in 1868 that such an assumption implies the Maxwell velocity distribution for an ideal gas of noninteracting particles; it also implies the Maxwell-Boltzmann distribution for certain special cases in which external forces are present. But the assumption itself demanded some justification beyond its inherent plausibility. For this purpose, Boltzmann and Maxwell introduced what is now called the “ergodic hypothesis,” the assumption that a single system will eventually pass through all possible microstates.

There has been considerable confusion about what Maxwell and Boltzmann really meant by ergodic systems. It appears that they did not have in mind completely deterministic mechanical systems following a single trajectory unaffected by external conditions; the ergodic property was to be attributed to some random element, or at least to collisions with a boundary. In fact, when Boltzmann first introduced the words *Ergoden and ergodhchc*, he used them not for single systems but for collections of similar systems with the same energy but different initial conditions. In these papers of 1884 and 1887, Boltzmann was continuing his earlier analysis of mechanical analogies for the second law of thermodynamics and also developing what is now (since Gibbs) known as ensemble theory. Here again, Boltzmann was following a trail blazed by Maxwell, who had introduced the ensemble concept in his 1879 paper. But while Maxwell never got past the restriction that all systems in the ensemble must have the same energy, Boltzmann suggested more general possibilities and Gibbs ultimately showed that it is most useful to consider ensembles in which not only the energy but also the number of particles can have any value, with a specified probability.

The Maxwell-Boltzmann ergodic hypothesis led to considerable controversy on the mathematical question of the possible existence of dynamical systems that pass through all possible configurations. The controversy came to a head with the publication of the Ehrenfests’ article in 1911, in which it was suggested that while ergodic systems are probably nonexistent, “quasi-ergodic” systems that pass “as close as one likes” to every possible state might still be found. Shortly after this, two mathematicians, Rosenthal and Plancherel, used some recent results of Cantor and Brouwer on the dimensionality of sets of points to prove that strictly ergodic systems are indeed impossible. Since then, there have been many attempts to discover whether physical systems can be ergodic; “ergodic theory” has become a lively branch of modern mathematics, although it now seems to be of little interest to physicists.

After expending a large amount of effort in the 1880’s on elaborate but mostly fruitless attempts to determine transport properties of gases, Boltzmann returned to the calculation of equilibrium properties in the I890’s. He was encouraged by the progress made by Dutch researchers—J. D. van der Waals, H. A. Lorentz, J. H. van’t Hoff, and others—in applying kinetic methods to dense gases and osmotic solutions. He felt obliged to correct and extend their calculations, as in the case of virial coefficients of gases composed of elastic spheres. The success of these applications of kinetic theory also gave him more ammunition for his battle with the energeticists (see below).

*Other Scientific Work*. Although Boltzmann’s contributions to kinetic theory were the fruits of an effort sustained over a period of forty years, and are mainly responsible for his reputation as a theoretical physicist, they account, numerically, for only about half of his publications. The rest are so diverse in nature—ranging over the fields of physics, chemistry, mathematics, and philosophy—that it would be useless to try to describe or even list them here. Only one common characteristic seems evident: most of what Boltzmann wrote in science represents some kind of interaction with other scientists or with his students. All of his books originated as lecture notes; in attempting to explain a subject on the elementary level, Boltzmann frequently developed valuable new insights, although he often succumbed to unnecessary verbosity. He scrutinized the major physics journals and frequently found articles that inspired him to dash off a correction, design a new experiment, or rework a theoretical calculation to account for new data.

Soon after he started to follow Maxwell’s work on kinetic theory. Boltzmann began to study the electromagnetic theory of his Scottish colleague. In 1872, he published the first report of a comprehensive experimental study of dielectrics, conducted in the laboratories of Helmholtz in Berlin and of Töpler in Graz. A primary aim of this research was to test Maxwell’s prediction that the index of refraction of a substance should be the geometric mean of its dielectric constant and its magnetic permeability . Boltzmann confirmed this prediction for solid insulators and (more accurately) for gases. He also confirmed the further prediction that if the speed of light (and hence the index of refraction) varies with direction in an anisotropic crystal, then the dielectric constant must also vary with direction.

During the next few years, Boltzmann began experimental work in diamagnetism while continuing his theoretical research in kinetic theory. He proposed a new theory of elastic aftereffects, in which the stress on a material at a given time depends on its previous deformation history.

In 1883, as a result of preparing an abstract of H. T. Eddy’s paper (on radiant heat as a possible exception to the second law of thermodynamics) for Wiedemaun’s *Beib1ätter*, Boltzmann learned of a work by the Italian physicist Adolfo Bartoli on radiation pressure. Bartoli’s reasoning stimulated Boltzmann to work out a theoretical derivation, based on the second law of thermodynamics and Maxwell’s electromagnetic theory, of the fourth—power law previously found experimentally by Stefan:

(radiation energy) ∞ (absolute temperature)

^{4}.

Although at the time the “Stefan-Boltzmann law” for radiation seemed to be an isolated result with no further consequences, it did at least show a possible connection between thermodynamics and electromagnetism that was exploited in the later quantum theory. In the 1920’s it was applied by Eddington and others in explaining the equilibrium of stellar atmospheres.

In the 1890’s Boltzmann again revived his interest in electromagnetic theory, perhaps as a result of Hertz’s experiments, which he repeated before a large audience in Graz. He published his *Vorlesungen über Maxwells Theorie…* in 1891 and 1893, along with some papers in which he suggested new mechanical models to illustrate the field equations. In 1895 he published an annotated German edition of Maxewell’s paper “On Faraday’s Lines of Force” in Ostwald’s Klassiker der exakten Wissensehaften. Boltzmann was partly responsible for the eventual acceptance of Maxwell’s theory on the Continent, although he did not advance the theory itself as much as did Lorentz, nor did he grapple with the difficult problems that ultimately led to Einstein’s theory of relativity.

*Defense of the Atomic Viewpoint*. Throughout his career, even in his works on subjects other than kinetic theory Boltzmann was concerned with the mathematical problems arising from the atomic nature of matter. Thus, an early paper with the title “Über die Integrale linearer Differentialgleichungen mit periodischcn Koeffizienten” (1868) turned out to be an investigation of the validity of Cauchy’s theorem on this subject, which is needed to justify the application of the equations for an elastic continuum to a crystalline solid in which the local properties vary periodically from one atom to the next. Every time someone published new data on the specific heats of gases, Boltzmann felt obliged to worry again about the distribution of energy among the internal motions of polyatomic molecules.

Until the 1890’s, it seemed to be generally agreed among physicists that matter *is* composed of atoms, and Boltzmann’s concern about the consistency of atomic theories may have seemed excessive. But toward the end of the century, the various paradoxes—specific heats., reversibility, and recurrence—were taken more seriously as defects of atomism, and Boltzmann found himself cast in the role of principal defender of the kinetic theory and of the atomistic-mechanical viewpoint in general, Previously he had not been much involved in controversy—with the exception of a short dispute with O.E. Meyer, who, ironically, had accused Boltzmann of proposing a theory of elasticity that was inconsistent with the atomic nature of matter. But now Boltzmann found himself almost completely deserted by Continental scientists; his principal supporters were in England.

In retrospect it seems that the criticisms of kinetic theory in this period were motivated not primarily by technical problems, such as specific heats of polyatomic molecules but, rather, by a general philosophical reaction against mechanistic or “materialistic” science and a preference for empirical or phenomenoiogical theories, as opposed to atomic models. The leaders of this reaction, in the physical sciences, were Ernst Mach. Wilhelm Ostwald, Pierre Duhem, and Georg Helm. Mach recognized that atomic hypotheses could be useful in science but insisted, even as late as 1912, that atoms must not be considered to have a real existence. Ostwald, Duhem, and Helm, on the other hand, wanted to replace atomic theories by “energetics” (a generalized thermodynamics); they denied that kinetic theories had any value at all, even as hypotheses.

In the first volume of his *Vorlesungen uber Gastheorie* (1896) Boltimann presented a vigorous argument for the kinetic theory:

Experience teaches that one will be led to new discoveries almost exclusively by means of special mechanical models,… Indeed, since the history of science shows how often epistemological generalizations have turned out to be false, may it not turn out that the present “modern” distaste for special representations, as well as the distinction between qualitatively different forms of energy will have been a retrogression? Who sees the future? Let us have free scope for all directions of research: away with all dogmatism, either atomistic or anti-atomistic! In describing the theory of gases as a mechanical analogy, we have already indicated, by the choice of this word, how far removed we are from that viewpoint which would see in visible matter the true properties of the smallest particles of the body [Brush trans., p. 26].

In the foreword to the second volume of this book (1898), Boltzmann seemed rather more conscious of his failure to convert other scientists to acceptance of the kinetic theory. He noted that attacks on the theory had been increasing, but added:

I am convinced that these attacks are merely based on a misunderstanding, and that the role of gas theory in science has not yet been played out. The abundance of results agreeing with experiment which van der Waals has derived from it purely deductively, I have tried to make clear in this book. More recently, gas theory has also provided suggestions that one could not obtain in any other way. From the theory of the ratio of specific heats, Ramsay inferred the atomic weight of argon and thereby its place in the system of chemical elements—which he subsequently proved, by the discovery of neon, was in fact correct….

In my opinion it would be a great tragedy for science if the theory of gases were temporarily thrown into oblivion because of a momentary hostile attitude toward it, as was for example the wave theory because of Newton’s authority.

I am conscious of being only an individual struggling weakly against the stream of time. But it still remains in my power to contribute in such a way that, when the theory of gases is again revived, not too much will have to be rediscovered…[

ibid., pp, 215–216].

Boltzmann and Ostwald, although on good personal terms, engaged in bitter scientific debates during this period; at one point even Mach thought the argument was becoming too violent, and proposed a reconciliation of mechanistic and phenomenological physics.^{7} While teaching at Leipzig with Ostwald during the period 1900–1902, Boltzmann was undergoing periods of mental depression and made one attempt at suicide. He returned to Vienna in 1902, where he succeeded himself as professor of theroetical physics and also lectured on the philosophy of science, replacing Ernst Mach, who had to retire for reasons of health. In 1904 he went to the United States to attend the World’s Fair at St. Louis, where he lectured on applied mathematics, and also visited Berkeley and Stanford. He later described his experiences on this trip in a satirical article, “Reise eines deutschen Professors ins Eldorado.” But despite his travels and discussions with scientific colleagues, he somehow failed to realize that the new discoveries in radiation and atomic physics occurring at the turn of the century were going to vindicate his own theories, even if in somewhat altered form. The real cause of Boltzmann’s suicide in 1906 will never be known; but insofar as despair over the rejection of his lifework by the scientific community may have been a contributing factor (as has sometimes been suggested without much evidence), it is certainly one of the most tragic ironies in the history of science that Boltzmann ended his life just before the existence of atoms was finally established (to the satisfaction of most scientists) by experiments on Brownian motion guided by a kinetic statistical theory of molecular motion.

## NOTES

1. All of Boltzmann’s publications for which only the year is given may be found in his *Wissenschaftliche Abhandlungen*.

2. See *The Scientific Papers of James Clerk Maxwell* (Cambridge, 1890). The 1859 and 1866 papers of Maxwell, together with other papers by Clausius, Boltzmann, Kelvin, Poincarè, and Zermelo (cited by year in this article) may be found in S. G. Brush, ed., *Kinetic Theory*, 2 vols. (Oxford, 1965–1966).

3. See Boltzmann, *Vorlesungen über Gastheorie*, I, §3;n P. and T. Ehrenfest, “Begriffiche Grundlagen der statistischen Aufassung in der Mechanik.”

4. It was in reference to this result of Maxwell’s that Boltzmann wrote his oft-quoted comparison of styles in theoretical physics and styles in music, dramatizing the almost magical disappearance of *V* from the integrand of eq. 9 when the words “let n=5” were pronounced (*Populäre Schriften*, p. 51)

5. Brush, *Kinetic Theory*, II. 192

6. This formula for *S* is clearly related to Boltzmann’s earlier expression for the *H*- function (eq. 9) If we know that the system has probability W, of being in macrostate *i*, with given values of *W*, for all *i*, then the expectation value of the entropy can be calculated from

S =

kΣWlogW

with an appropriate interpretation of the summation or integration.

7.*Die. Principien der Wärmelehre*, *historisch-kritisch entwickelt* (Leipzig, 1896), pp. 362 ff.

## BIBLIOGRAPHY

I. Original Works. The technical papers that originally appeared in various periodicals have been reprinted in Boltzmann’s *Wissenschaftliche Abhandlungen*, F. Hasen öhrl, ed., 3 vols. (Leipzig, 1909). Lectures and articles of general interest are collected in *Populäre Schriften* (Leipzig, 1905). A review article written with J. Nabl, “Kinetische Theorie der Materie,” was published in *Encyklopädie der mathematischen Wissenschaften*, V, pt. 1 (Leipzig, 1905), art V8. Other works are *Vorlesungen über Maxwells Theorie der Elektrizität und des Lichtes*, 2 vols. (Leipzig, 1891–1893); his ed. of Maxwell’s “On Faraday’s Lines of Force,” *Ueber Faraday’s Kraftlinien* (Leipzig, 1895), with 31 pages of notes by Boltzmann; *Vorlesungen uber Gastheorie*, 2vols (Leipzig, 1896–1898), trans by S.G. Brush, with introduction, notes and bibliography, as lectures on Gas Theory (Berkeley, 1964); *Vkorlesungen uber die principe der Mechanik*, 3 vols. (Leipzig, 1897–1920); and *Über die Prinzipen der Mechanik, Zwei akademische Antrittsreden* (Leipzig, 1903). Books based on Boltzmann’s lectures are Charles Emerson Curry, *Theory of Electricity and Magnetism* (London, 1897) with a preface by Boltzmann; and Hugo Buchholz. *Das mechanische Potential*, published with *Die Theorie der Figur der Erde* (Leipzig, 1908).

II. Secondary Literature. Works on Boltzmann are Engelbert Broda, *Ludwig Boltzmann: Mensch, Physiker, Philosoph* (Berlin, 1955); S.G. Brush, “Foundations of Statistical Mechanics 1845–1915,” in *Archive for History of Exact Sciences*, **4** (1967), 145–183; René Dugas, *Lathrotie physique au sens de Boltzmann et ses prolongements moderns* (Neucātel, 1959); P. and T. Ehrenfest, “Begriffliche Grundlagen der statistischen Aufassung in der Mechanik,” in *Encyklopädie der mathematischen Wissenschaften*, IV, pt, 32 (Leipzig, 1911), trans by M.J Moravesik as *The Conceptual Foundations of the Statistical Approach in Mechanics* (Ithace, N.Y., 1959); and G. Jaeger, “Ludwig Boltzmann,” in *Neue Österreichische Biographie 1815–1918*, pt, 1. *Biographien*, II (Vienna, 1925), 117–137. Other articles on Boltzmann are listed in the bibliography of the Brush trans *of Lectures on Gas Theory*.

Stephen G. Brush

# Boltzmann, Ludwig

# Boltzmann, Ludwig

AUSTRIAN PHYSICIST

1844–1906

Ludwig Edward Boltzmann is one of the foremost theoretical physicists of the latter nineteenth century. A vigorous advocate for the existence of atoms, he made monumental contributions to the **kinetic theory** of gases and established the statistical nature of the second law of thermodynamics.

Boltzmann was born in Vienna, Austria, and graduated from high school in Linz. He entered the University of Vienna in 1863, and he received his doctorate in physics three years later. Then, for two years, he served as assistant professor at the university, where he was strongly influenced by the atomistic thinking of physicists Josef Loschmidt and Josef Stefan.

In 1869 Boltzmann became professor of mathematical physics at the University of Graz, the first of his many academic appointments. After leaving Graz in 1873, he held chairs in mathematics at Vienna (1873–1876), experimental physics at Graz (1876–1890), and **theoretical physics** at Munich (1890–1893), Vienna (1893–1900), Leipzig (1900–1902), and finally Vienna again for the remaining four years of his life. According to many of his students, Boltzmann was an outstanding teacher, and his lectures were often filled to overflowing. He displayed a congenial attitude toward students and their learning, something rather rare among Austrian and German professors at that time.

Boltzmann's first significant contribution to physics was the generalization of James Clerk Maxwell's distribution of velocities and energies for a sample of gaseous atoms. Although Maxwell had deduced this distribution, he provided no physical basis for it. Boltzmann showed that as atoms move toward **equilibrium** they assume the Maxwell distribution—later known as the Maxwell-Boltzmann distribution—and further that this is the only statistically possible distribution for a system at equilibrium.

Boltzmann connected his ideas with those of Rudolf Clausius, who had introduced the concept of entropy in 1865. Somehow related to heat, entropy was known to increase during irreversible processes, but its exact nature was unknown. From the distribution of gas atoms, Boltzmann described a quantity—later symbolized by the letter *H* —which is a minimum when atoms assume a Maxwell-Boltzmann distribution. He recognized his *H* function as the negative of entropy, which is a maximum when the atoms reach thermal equilibrium. Thus Boltzmann offered a kinetic explanation for entropy and, more generally, a connection between the behavior of atoms and thermodynamics.

One of the serious problems with Boltzmann's statistical treatment arose from the reversibility of the laws of mechanics, which holds true for a particle moving in one direction just as they do for a particle moving in the opposite direction. How then could a given set of atomic motions cause *H* to tend toward a minimum (or entropy toward a maximum), rather than away from it? This was a vexing question for Boltzmann and a serious criticism of kinetic theory.

In response, Boltzmann considered the number of different ways that a sample of gaseous atoms could achieve a particular distribution. The more ways the atoms can arrange themselves to achieve some distribution, the more likely it becomes for that distribution to occur. This connection between entropy (*S* ) and the number of ways (*W* ) that a given distribution can occur is embodied in the equation *S = k* ln *W* (*k* is now known as Boltzmann's constant, and ln is the natural logarithm). This famous relationship, which is engraved on Boltzmann's gravestone in Vienna, indicates that maximum entropy is associated with the distribution that has the most ways of occurring, that is, with the Maxwell-Boltzmann distribution. Although it is possible for a system of atoms to move away from a Maxwell-Boltzmann distribution, it is not probable, since it is statistically unlikely for the system to move from a distribution with more ways of achieving it to one with fewer ways. Much of Boltzmann's work in this area was formalized somewhat differently under the name of statistical mechanics by Josiah Willard Gibbs, an American physicist who was well known and well respected in Europe, but not in his own country.

Boltzmann's achievements in theoretical physics are all the more remarkable in view of the considerable opposition to his ideas and in view of his own declining health. He had increasingly severe bouts of mental depression, and he tried to commit suicide several times during his life. In 1906 he succeeded in hanging himself while vacationing with his wife and family at Duino, near Trieste, on the Adriatic Sea.

see also Gibbs, Josiah Willard; Maxwell, James Clerk.

*Richard E. Rice*

## Bibliography

Boltzmann, Ludwig (1905). "On the Trip of a German Professor into El Dorado." In *Ludwig Boltzmann: His Later Life and Philosophy, 1900–1906*, ed. John Blackmore (1995). Boston: Kluwer Academic Publishers.

Cercignani, Carlo (1998). *Ludwig Boltzmann: The Man Who Trusted Atoms.* New York: Oxford University Press.

Cropper, William H. (2001). *Great Physicists: The Life and Times of Leading Physicists from Galileo to Hawking.* New York: Oxford University Press.

Lindley, David (2001). *Boltzmann's Atom: The Great Debate That Launched a Revolution in Physics.* New York: Free Press.

### Internet Resources

O'Connor, J. J., and Robertson, E. F. "Ludwig Boltzmann." September 1998. Available from <http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Boltzmann.html>.

# Ludwig Boltzmann

# Ludwig Boltzmann

The lasting fame of the Austrian physicist Ludwig Boltzmann (1844-1906) rests on the statistical inter pretation which he gave to classical thermodynamics.

Ludwig Boltzmann was born on Feb. 20, 1844, in Vienna, the son of Ludwig and Katharina Pauernfeind Boltzmann. In 1863 he entered the University of Vienna, where one of his teachers was Joseph Stefan, author of the law of radiation named after him.

## Thermodynamics and Kinetic Theory

Evidence of the creativity of Boltzmann's intellect was the fusion of thermodynamics and molecular theory that characterized an early paper on the mechanistic meaning of the second law of thermodynamics. In it Clausius's famous formula for entropy, ∫(*dQ/T*) ≥ 0, was given for the first time in terms of the kinetic theory of gases. The paper was written toward the end of Boltzmann's studies in 1866, and shortly afterward he received his doctor's degree and became a privatdozent at the university. He was also chosen as Stefan's assistant at the university's Institute of Physics, named "Erdberg" after the locality. Stefan urged his assistant to become familiar with James Clerk Maxwell's work on electromagnetism. He also supplied Boltzmann with an English grammar when the latter disclosed his total ignorance of English. Such was the starting point of Bolztmann's lifelong interest in the English-speaking countries and their cultures, but especially in the thought of Maxwell, perhaps the foremost of 19th-century physicists.

Boltzmann left the institute in 1869, when he received an invitation from the University of Graz. During his 3 years there, he twice visited Heidelberg and Berlin and established contacts with Helmholtz, Kirchhoff, Bunsen, and the mathematician Koenigsberger. But he soon realized that his Austrian *Gemütlichkeitclashed* with the stiff formalism prevailing in German academic circles.

Between 1873 and 1876 Boltzmann served as professor of mathematics at the University of Vienna. The titles of articles which he published while in Graz and Vienna indicate that his mind was increasingly absorbed in the interpretation of various aspects of thermodynamics, especially of its second law, on the basis of the kinetic theory of gases. Clearly, he was developing some extraordinary insight which had to surface before long.

Boltzmann was, however, already back in Graz, as professor of experimental physics, when this momentous breakthrough came in the history of physics. In 1877 he submitted two memoirs to the Academy of Sciences in Vienna. In the first of these, presented on Jan. 11, 1877, "Remarks on Some Problems of the Mechanical Theory of Heat," Boltzmann made an incisive analysis of the formula ∫(*dQ/T*) ≥ 0 and argued that its validity was not based on the inherent laws of nature but rather on the choice of the initial conditions. According to him, one had to assume on the basis of the kinetic theory of gases that the perfectly elastic balls representing the molecules always tended to change their actual positions. Consequently, any configuration, however improbable, could conceivably occur as time went on: "The calculus of probabilities teaches us precisely this: any non-uniform distribution, unlikely as it may be, is not strictly speaking impossible."

It now remained to give this conclusion a rigorous derivation and quantitative applicability, which Boltzmann did in his memoir submitted to the academy on Oct. 11, 1877, "On the Relation between the Second Law of the Mechanical Theory of Heat and the Probability Calculus with respect to the Propositions about Heat-Equivalence." He concluded that the entropy of a state is proportional to the probability of the configuration of its component particles. His formula, ∫(*dq/T*) = 2Ω/3, is, however, better known in the form *S* = *k* log *W,* which Max Planck gave it in 1901.

## Later Years

His great memoirs of 1877 represented the culmination of his insights into thermodynamics and kinetic theory. It was therefore natural for him to turn to a new field, electromagnetism, mainly under the impact of his reading James Clerk Maxwell's papers and books. When the two volumes of Boltzmann's *Vorlesungen über die Maxwellsche Theorie der Elektricität und des Lichtes* (Lectures on Maxwell's Theory on Electricity and Light) were published in 1891-1893, he was at the University of Munich as professor of theoretical physics.

In 1894 he succeeded his revered teacher, Stefan, as professor of physics at the University of Vienna and stayed there for 6 years. During that period he published in book form his researches on the kinetic theory of gases, *Vorlesungen über Gastheorie* (1896-1898). When the publication of Boltzmann's *Vorlesungen über Mechanik* was completed in 1904, he had already behind him a 2-year stay at the University of Leipzig (1900-1902). It was there that his sensitivity to criticism began to show itself in spells of depression and grave anxiety. He felt deeply disturbed by the ascendency of energeticism, whose chief proponent, Wilhelm Ostwald, was also at Leipzig. Partly because of this, Boltzmann left Leipzig; but at the same time, he was flattered that Mach's chair of *Naturphilosophie* was offered to him at the University of Vienna. But Mach was still very much alive, and his virulent attacks on atomism also seemed to carry the day. Boltzmann came to be referred to as the last bastion of atomism.

Unfortunately, Boltzmann took the view that he was indeed deserted, although radioactivity had already been for several years a firmly established field of investigation and a powerful pointer toward the existence of atoms. But Boltzmann was brooding over what he considered the weakening of his creative powers. In a fit of despondency he took his life on Sept. 5, 1906, at the summer resort of Duino, near Trieste.

## Further Reading

The most important writing on Boltzmann is in German. A specific study in English is in Edward A. Guggenheim, *Boltzmann's Distribution Law* (1955). Volume 1 of Henry A. Boorse and Lloyd Motz, eds., *The World of the Atom* (2 vols., 1966), touches upon aspects of Boltzmann's work. □

# Boltzmann, Ludwig

**Boltzmann, Ludwig** (1844–1906) Austrian physicist, acclaimed for his contribution to statistical mechanics and to the kinetic theory of gases. His research extended the ideas of James Maxwell. Boltzmann's general law asserts that a system will approach a state of thermodynamic equilibrium because that is the most probable state. He introduced the ‘Boltzmann equation’ (1877) relating the kinetic energy of a gas atom or molecule to temperature. Symbol K in the formula, the gas constant per molecule, is called the ‘Boltzmann constant’. In 1884 he derived a law, often termed the ‘Stefan-Boltzmann law’, for black body radiation discovered by his teacher, Josef Stefan (1835–93). Attacked for his belief in the atomic theory of matter, Boltzmann committed suicide.

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