BOLT WOOD BOLTZMANN rocks. If the rate of formation of an inactive decay prospectors, mine owners, speculators, chemical re- product could be determined, the total amount found finers, and wholesalers to analyze samples, devise in a mineral would immediately yield its age . Both separation processes, and find financial backing (from lead and helium (believed by most to be the alpha wealthy Yale alumni) for various projects . These particle) were seen as suitable elements and, indeed, efforts probably helped stimulate the production of served in radioactive dating techniques . The helium radium, in which the United States led the world by method, pioneered in England by R . J. Strutt (later about 1915, although they did not appreciably aid the fourth Baron Rayleigh), could not, however, give the progress of science. more than a minimum age because a variable portion In 1918 Boltwood was appointed director of the of the gas would have escaped from the rock. But Yale College chemical laboratory and presided over the lead method, developed by Boltwood in 1907, the consolidation of the Yale and Sheffield chemistry proved satisfactory and is still in use today . In effect, departments. To cement this union, the new Sterling Boltwood reversed his procedure of confirming the Chemistry Laboratory was proposed, and Boltwood accuracy of lead :uranium ratios by the accepted was placed in charge of its design . He completed it geological ages of the source rocks, and used these successfully, but the strain of this effort caused a ratios to date the rocks. Because most geologists, breakdown in his health from which he never fully under the influence of Lord Kelvin's nineteenth- recovered . Periods of severe depression alternated century pronouncements, inclined toward an age of with his more customary cheerful spirits, and resulted the earth measured in tens of millions of years, in his suicide during the summer of 1927 . Boltwood's claim for a billion-year span was met with Boltwood's influence in radioactivity was wide- some skepticism . However, the subsequent work of spread-through his published papers, correspon- Arthur Holmes, an understanding of isotopes, and the dence, and personal contacts, for he trained surpris- increasing accuracy of decay constants and analyses ingly few research students. Part of his success finally brought widespread acceptance of this method stemmed from his close association with Rutherford, in the 1930's . but like Rutherford's other chemical collaborators, Boltwood's major contributions lay in the under- Soddy and Hahn, he was eminently capable of major standing of the uranium decay series . Still, he was contributions in his own right . able to suggest, with Rutherford in 1905, that actin- ium is genetically related to uranium, though not in the same chain as radium, while in the thorium series BIB1_IOGRAPIIY he almost beat Hahn to the discovery of mesothorium in 1907 . His other significant service to the study of I. ORIGINAL WORKS . A reasonably complete list of radioactivity was to bring greater precision and ad- Boltwood's publications is in Alois F . Kovarik's sketch of him in Biographical Memoirs of the National Academy of vanced techniques into the laboratory, as in his insis- Sciences, 14( 1930), 69-96 . His unpublished correspondence . tence that only by complete dissolution and boiling papers, and laboratory notebooks are preserved in the of the mineral could all the emanation be extracted Manuscript Room, Yale University Library . His extensive from radioactive bodies . correspondence with Rutherford is in the Rutherford Col- Boltwood remained at Yale the rest of his life, lection, Manuscript Room, Cambridge University Library . except for the academic year 1909-1910, when he 11 . SECONDARY LITERATURE . In addition to Kovarik's accepted an invitation to Rutherford's laboratory at the memoir (see above), the following obituary notices offer University of Manchester . Yale, fearing that he would information about Boltwood : Yale Alumni Weekly, 37 (7 remain in England indefinitely, offered Boltwood a Oct. 1927) . 65; Kovarik, in Yale Scientific Magazine, 2 full professorship in radiochemistry . This appoint- (Nov . 1927), 25, 44, 46 : Rutherford, in Nature, 121 (14 Jan . ment brought him back to New Haven, but it also 1928), 64-65 ; Kovarik, in American Journal of Science, 15 1928) . 188-198 . marked the end of his research career. Heavy aca- (Mar. demic duties, including supervision of construction of LAWRENCE BADASH the new Sloane Physics Laboratory and unsuccessful efforts to obtain large quantities of radioactive min- BOLTZMANN, LUDWIG (b. Vienna, Austria, 20 erals for research, seem to have taken all his time February 1844 ; d. Duino, near Trieste, 5 September and energy. His stature as the foremost authority on 1906), physics. radioactivity in the United States brought him mem- Boltzmann's father, Ludwig, was a civil servant bership in the National Academy of Sciences, the (Kaiserlich-Koniglich Cameral-Concipist) ; his mother American Philosophical Society, and other organiza- was Katherina Pauernfeind . He was educated at Linz tions, but it also brought him numerous requests from and Vienna, receiving his doctorate in 1867 from the

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University of Vienna, where he had studied with Josef given time, only the average number of molecules Stefan. Boltzmann held professorships at the uni- having various positions and velocities . In many cases versities of Graz, Vienna, Munich, and Leipzig. In it seems reasonable to assume that the gas is spatially 1876 he married Henrietta von Aigentler, who bore uniform, that is, the average number of molecules is him four children . the same at different places in the gas. The problem Distribution Law. The first stimulus for Boltzmann's is then to determine the velocity distribution function researches came from teachers and colleagues at the f (v), defined so that f (v) dv is the average number of University of Vienna, especially Stefan and Josef molecules having speeds between v and v + dv. Loschmidt. In a lecture Stefan suggested the problem Maxwell argued that f(v) should be a function that in electrical theory whose solution constituted depends only on the magnitude of v, and that the Boltzmann's first published paper (1865) ;' he also velocity components resolved along the three coor- published a few papers on kinetic theory and did dinate axes should be statistically independent . important experimental work on gases and radiation Hence, he inferred that that provided the basis for some of Boltzmann's f(v) = ( a3i3/2)e-(v2/•z) (1) theories . Loschmidt (also in 1865) accomplished the N/ first reliable estimate of molecular sizes with the help where N is the total number of molecules, and a 2 of the Clausius-Maxwell kinetic theory . Although is inversely proportional to the absolute temperature . Loschmidt was later to dispute Boltzmann's interpre- In his long memoir of 1866, Maxwell admitted that tation of the second law of thermodynamics, the the assumptions used in his previous derivation of problem of finding quantitative relations between the distribution law "may appear precarious" ; he atomic magnitudes and observable physical quantities offered another derivation in which the velocities of was a common interest of both men . two colliding molecules, rather than the velocity Boltzmann began his lifelong study of the atomic components of a single molecule, were assumed to theory of matter by seeking to establish a direct be statistically independent. This means that one can connection between the second law of thermo- express the joint distribution function for the prob- dynamics and the mechanical principle of least action ability that molecule 1 has velocity v 1, while at the (1866). Although Clausius, Szily, and others later same time molecule 2 has velocity v 2, as the product worked along similar lines, and Boltzmann himself of the probabilities of these two separate events : returned to the subject in his elaboration of 1 Helmholtz' theory of monocyclic systems (1884), the F(v ,v2) =f(v1)f(v2) . (2) analogy with purely mechanical principles seemed To derive the distribution function itself, Maxwell insufficient for a complete interpretation of the second argued that the equilibrium state would be reached law. The missing element was the statistical approach when the number of collisions in which two molecules to atomic motion that had already been introduced with initial velocities (v 1 ,v2) rebound with final veloci- by the British physicist James Clerk Maxwell .2 ties (v 1 ',v2') is equal to the number of collisions in Boltzmann's first acquaintance with Maxwell's writ- which two molecules with initial velocities (v 1',v2') ings on kinetic theory is indicated by his paper on rebound with final velocities (v 1,v 2) ; from this condi- thermal equilibrium (1868) . In this paper, he ex- tion it follows that tended Maxwell's theory of the distribution of energy F(v 1,v 2) = F(v1',v2') (3) among colliding gas molecules, treating the case when . external forces are present . The result was a new By combining this equation with that for the conserva- exponential formula for molecular distribution, now tion of energy (in the case when no forces act), known as the "Boltzmann factor" and basic to all . To un- 2 M JV1 2 + 2 m2v22 m lv l 2 + m2V2 2, (4) modern calculations in statistical mechanics 2 2 derstand how Boltzmann arrived at this result, we must first review the work of Maxwell on which it Maxwell deduced (as before) that is based . ( a313/2) e-(" /•z). (5) Maxwell, in his first paper on kinetic theory (1859), f(vl) = N/ had pointed out that the collisions of gas molecules This type of reasoning about velocity distribution would not simply tend to equalize all their speeds functions was repeatedly used and generalized by but, on the contrary, would produce a range of differ- Boltzmann in his own works on kinetic theory . He ent speeds . Most of the observable properties of a began, in his 1868 paper, by considering the case in gas could be calculated if one knew, instead of the which one of the particles of a system is acted on positions and velocities of all the molecules at any by a force with a corresponding potential function,

26 1 BOLTZMANN BOLTZMANN

V(x). The condition of conservation of energy would accidental, since Planck and other early quantum then be theorists were familiar with Boltzmann's works and 2 used many of his techniques.) miv1 2 + V(x i) + m2v22 Transport Equation and H-theorem . Although 2 Maxwell and Boltzmann had succeeded in finding the correct distribution laws by assuming that the gas is m rv i'2 + m2v2' 2 , (6) = 2 V(xi') + 2 in an equilibrium state, they thought that the kinetic theory should also be able to show that a gas will and Boltzmann could then apply Maxwell's procedure actually tend toward an equilibrium state if it is not to deduce the distribution function there already. Maxwell had made only fragmentary .2 f(v) = (const .)e-htm1 2+rtrp . (7) attempts to solve this problem ; Boltzmann devoted several long papers to establishing a general solution . The constant factor h could be related to the absolute Approach to equilibrium is a special case of a temperature of the gas, as Maxwell and Clausius had general phenomenon : dissipation of energy and in- done, by comparing the theoretical pressure of the crease of entropy . It was Boltzmann's achievement gas with the experimental relation between pressure to show in detail how thermodynamic entropy is and temperature (Gay-Lussac's law) . In modern nota- related to the statistical distribution of molecular tion, h is equivalent to 1/kT, where k is a constant, configurations, and how increasing entropy corre- now called Boltzmann's constant, and T is the abso- sponds to increasing randomness on the molecular lute temperature on the Kelvin scale. level. This was a peculiar and unexpected relation- The physical meaning of the Maxwell-Boltzmann ship, for macroscopic irreversibility seemed to con- distribution law is that the energy (E = mv2/2 + tradict the fundamental reversibility of Newtonian V[x]) of a molecule is most likely to be roughly mechanics, which was still assumed to apply to molec- equal to kT; much larger or much smaller energies ular collisions . Boltzmann's attempts to resolve this occur with small but finite probability . contradiction formed part of the debate on the validity In the same paper of 1868, Boltzmann presented of the atomic theory in the 1890's. Seen in this con- another derivation of the Maxwell distribution law text, the proof of the distribution law has even more that was independent of any assumptions about colli- significance than the law itself. sions between molecules . He simply assumed that Boltzmann's major work on the approach to equilib- there is a fixed total amount of energy to be distrib- rium (and on transport processes in gases in general) uted among a finite number of molecules, in such a was published in 1872 . This paper, like that of 1868, way that all combinations of energies are equally took Maxwell's theory as the starting point . probable. (More precisely, he assumed uniform dis- Boltzmann first derived an equation for the rate of tribution in momentum space.) By regarding the total change in the number of molecules having a given energy as being divided into small but finite quanta, energy, x, resulting from collisions between molecules . he could treat this as a problem of combinatorial He considered a typical collision between two mole- analysis . He obtained a rather complicated formula cules with energies x and x' before the collision, and that reduced to the Maxwell velocity-distribution law energies ~ and x + x' - after the collision . Such in the limit of an infinite number of molecules and a collision reduces by one the number of molecules infinitesimal energy quanta . with energy x; the number of such collisions is as- The device of starting with finite energy quanta and sumed to be proportional to the number of molecules then letting them become infinitesimal is not essential with energy x, and also to the number of molecules to such a derivation, but it reveals an interesting with energy x' . Boltzmann used here, without any feature of Boltzmann's mathematical approach . comment, Maxwell's assumption of statistical inde- Boltzmann asserted on several occasions that a deriva- pendence of the velocities of two colliding molecules tion based on infinite or infinitesimal quantities is not (eq. 2) ; later it was recognized that there might be really rigorous unless it can also be carried through valid grounds for objecting to this assumption .3 With with finite quantities . While this prejudice kept him this assumption, the decrease in f(x) will be equal from appreciating and using some of the develop- to the product, f(x)f(x'), multiplied by an appro- ments in pure mathematics that appeared toward the priate factor for the collision probability and inte- end of the nineteenth century, it also had the curious grated over all values of x' . Similarly, the increase effect of making some of his equations for energy infix) may be attributed to inverse collisions in which distribution and transfer look similar to those of the molecules have energies ~ and x + x' - t before modern quantum theory . (This is perhaps not quite the collision, and x and x' after the collision . By such

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arguments Boltzmann arrived at the equation now, with an explicit formula for of/at, Boltzmann was able to go further and show that f(x) probably af - f 00f x+x' (x + x' - t) f (x,t)f(x',t) ' - ,/x -\/x' tends toward the Maxwell form . He did this by in- a t - o [f ( ~,t)fvl~-/x + x troducing a function, E, depending on f(x) -\/xx't(x,x',~) dx' d.c. (8) E f(x,t) (log If( -] - 1) A (10) (This is a special case of the general Boltzmann = fo transport equation [eq. 9]; terms describing the effect of external forces and nonuniformities on the change and showing that E always decreases unless f has of f are here omitted. The square root expressions the Maxwellian form : in the denominators, which do not appear in the form -hx, of the equation generally used, result from the fact dE < 0 iff (const .) ~e that energy rather than velocity is the variable .) dE -hx . One additional assumption involved in this deriva- = 0 if f = (const.) \ e tion should be mentioned : the collision probability dt function, y(x, x',~), is the same for both the direct and (The proof is straightforward and relies simply on the inverse collisions ; that is, the collision is perfectly fact that the quantity (a - b) log b/a is always nega- reversible . tive if a and b are real positive numbers .) Boltzmann Following Maxwell's 1866 development of the also noted that in the Maxwellian state E is essen- transport equations, Boltzmann showed how the tially the same as the thermodynamic entropy (aside diffusion, viscosity, and heat conduction coefficients from a constant factor) . Thus Boltzmann's "H-func- of a gas could be calculated by solving the general tion" (the notation was changed from E to H in the transport equation 1890's) provides an extension of the definition of of + of +,of af +X af +Y af entropy to nonequilibrium states not covered by the , +o at ax ay az a~ an thermodynamic definition. The theorem that Halways decreases for nonequilib- + Z + fdwi fb db fd~V(ffi - f'f,') = 0, (9) rium systems was called "Boltzmann's minimum f theorem" in the nineteenth century, and now goes where are components of the velocity of a by the name "Boltzmann's H-theorem ." (It has not particle and (X, Y,Z) are components of the force yet been proved rigorously except with certain acting on it, and V, -~, b, and w i are variables char- specializing assumptions .) acterizing the relative motion of the two molecules Reversibility and Recurrence Paradoxes. The during a collision . (Values of the function f for H-theorem raised some difficult questions about the velocities of the two molecules before and after the nature of irreversibility in physical systems, in particu- collision are indicated by f, f1 , f', and f1 '.) lar the so-called "reversibility paradox" and "recur- It is difficult to obtain exact solutions of Boltzmann's rence paradox." (The modern terminology goes back transport equation except when the molecules inter- only to the Ehrenfests' 1911 article, in which the act with inverse fifth-power forces, a case for which words Umkehreinwand and Wiederkehreinwand were Maxwell had found an important simplification.4 introduced .) The reversibility paradox, first discussed Boltzmann made several attempts to develop accurate by Lord Kelvin (1874) and brought to Boltzmann's approximations for other force laws, but this problem attention by Loschmidt, is based on the apparent was not satisfactorily solved until the work of contradiction between one of the basic premises of S. Chapman and D . Enskog in 1916-1917 . Boltz- Boltzmann's derivation-the reversibility of indi- mann's equation is now frequently used in mod- vidual collisions-and the irreversibility predicted by ern research on fluids, plasmas, and neutron trans- the theorem itself for a system of many molecules . port. Of course there must be such a contradiction between If the velocity distribution function is Maxwellian, any molecular theory based on Newtonian mechanics then the integral on the right-hand side of eq . 8 and the general principle of dissipation of energy, but vanishes identically for all values of the variables, Boltzmann's work was the first to reveal this incon- and we find of/at = 0 . In other words, once the sistency explicitly . Maxwellian state has been reached, no further change Boltzmann's initial response (1877) to the reversi- in the velocity distribution function can occur . bility paradox was the suggestion that the irreversi- So far this is simply an elaboration of the previous bility of processes in the real world is not a conse- arguments of Maxwell and of Boltzmann himself, but quence of the equations of motion and the form of

263 BOLTZMANN BOLTZMANN the intermolecular force law but, rather, seems to be backward directions of time . However, within small a result of the initial conditions . For some unusual regions, such as individual galaxies, there will be initial conditions the system might in fact decrease noticeable fluctuations that include ordered states its entropy (increase the value of H) as time pro- corresponding to the existence of life . A living being gresses; such initial conditions could be constructed in such a galaxy will distinguish the direction of time simply by reversing all the velocities of the molecules for which entropy increases (processes going from in an equilibrium state known to have evolved from ordered to disordered states) from the opposite direc- a nonequilibrium state . But, Boltzmann asserts, there tion ; in other words, the concept of "direction of time" are infinitely many more initial states that evolve with is statistical or even subjective, and is determined by increasing entropy. simply because the great majority the direction in which entropy happens to be increas- of all possible states are equilibrium states . Moreover, ing. Thus, the statement "Entropy increases with the entropy would also be almost certain to increase time" is a tautology, and yet the subjective time if one picked an initial state at random and followed directions in different parts of the universe may be it backward in time instead of forward. different . In this way local irreversible processes The recurrence paradox arises from a theorem in would be compatible with cosmic reversibility and mechanics first published by Poincare in 1890 . Accord- recurrence. (Boltzmann's concept of alternating time ing to this theorem, any mechanical system con- directions has recently been revived in connection strained to move in a finite volume with fixed total with theories of oscillating universes .) energy must eventually return to any specified initial Statistical Mechanics and Ergodic Hypothesis. Hav- configuration . If a certain value of the entropy is ing followed Boltzmann's work on irreversible pro- associated with every configuration of the system (a cesses into some of the controversies of the 1890's, let disputable assumption), then the entropy cannot us now return to his contributions to the theory of continually increase with time, but must eventually systems in thermal equilibrium (for which the term decrease in order to return to its initial value. There- "statistical mechanics" was introduced by J . Willard fore the H-theorem cannot always be valid . Gibbs). Poincare, and later Zermelo (1896) . argued that the It would be possible (as is in fact done in many recurrence theorem makes any mechanical model, modern texts) to take the Maxwell-Boltzmann dis- such as the kinetic theory, incompatible with the tribution function (eq . 7) as the basic postulate for second law of thermodynamics ; and since, it was calculating all the equilibrium properties of a system . asserted, the second law is a strictly valid induction Boltzmann, however, preferred another approach that from experience, one must reject the mechanistic seemed to rest on more general grounds than the viewpoint . dynamics of bimolecular collisions in low-density Boltzmann replied that the recurrence theorem gases . The new method was in part a by-product of does not contradict the H-theorem, but is completely his discussion of the reversibility paradox, and is first in harmony with it . The equilibrium state is not a hinted at in connection with the relative frequency single configuration but, rather, a collection of the of equilibrium, as opposed to nonequilibrium, con- overwhelming majority of possible configurations, figurations of molecules : "One could even calculate, characterized by the Maxwell-Boltzmann distribution . from the relative numbers of the different distribu- From the statistical viewpoint, the recurrence of some tions, their probabilities, which might lead to an particular initial state is a fluctuation that is almost interesting method for the calculation of thermal certain to occur if one waits long enough ; the point equilibrium . This remark was quickly followed up is that the probability of such a fluctuation is so small in the same year (1877) in a paper in which the that one would have to wait an immensely long time famous relation between entropy and probability, before observing a recurrence of the initial state . Thus S = k log W, the mechanical viewpoint does not lead to any conse- quences that are actually in disagreement with experi- was developed and applied . In this equation, W is ence . For those who are concerned about the the number of possible molecular configurations cosmological consequences of the second law the ("microstates," in modern terminology) correspond- so-called "heat death" corresponding to the final ing to a given macroscopic state of the systems (To attainment of a state of maximum disorder when all make this expression meaningful, microstates have to irreversible processes have run their course be defined with respect to finite cells in phase space ; -Boltzmann suggested the following idea. The the size of these cells introduces an arbitrary additive universe as a whole is in a state of thermal equilib- constant in S which can be determined from quantum rium . and there is no distinction between forward and theory.)

2 64 BOLTZMANN BOLTZMANN

The new formula for entropy-from which indeed impossible . Since then, there have been many formulas for all other thermodynamic quantities attempts to discover whether physical systems can be could be deduced was based on the assumption of ergodic ; "ergodic theory" has become a lively branch equal a priori probability of all microstates of the of modern mathematics, although it now seems to be system (that is, all microstates that have the same total of little interest to physicists. energy). As noted above, Boltzmann had already After expending a large amount of effort in the proved in 1868 that such an assumption implies the 1880's on elaborate but mostly fruitless attempts to Maxwell velocity distribution for an ideal gas of determine transport properties of gases, Boltzmann noninteracting particles ; it also implies the Maxwell- returned to the calculation of equilibrium properties Boltzmann distribution for certain special cases in in the 1890's . He was encouraged by the progress which external forces are present . But the assumption made by Dutch researchers-J . D. van der Waals, itself demanded some justification beyond its inher- H . A. Lorentz, J . H. van't Hoff, and others-in apply- ent plausibility . For this purpose, Boltzmann and ing kinetic methods to dense gases and osmotic solu- Maxwell introduced what is now called the "ergodic tions. He felt obliged to correct and extend their hypothesis," the assumption that a single system will calculations, as in the case of virial coefficients of eventually pass through all possible microstates . gases composed of elastic spheres . The success of There has been considerable confusion about what these applications of kinetic theory also gave him Maxwell and Boltzmann really meant by ergodic more ammunition for his battle with the energeticists systems. It appears that they did not have in mind (see below) . completely deterministic mechanical systems follow- Other Scientific Work. Although Boltzmann's ing a single trajectory unaffected by external condi- contributions to kinetic theory were the fruits of an tions; the ergodic property was to be attributed to effort sustained over a period of forty years, and are some random element, or at least to collisions with mainly responsible for his reputation as a theoretical a boundary . In fact, when Boltzmann first introduced physicist, they account, numerically, for only about the words Ergoden and ergodische, he used them not half of his publications . The rest are so diverse in for single systems but for collections of similar systems nature-ranging over the fields of physics, chemistry, with the same energy but different initial conditions . mathematics, and philosophy-that it would be use- In these papers of 1884 and 1887, Boltzmann was less to try to describe or even list them here . Only one continuing his earlier analysis of mechanical analogies common characteristic seems evident : most of what for the second law of thermodynamics and also de- Boltzmann wrote in science represents some kind of veloping what is now (since Gibbs) known as en- interaction with other scientists or with his students . semble theory . Here again, Boltzmann was following All of his books originated as lecture notes ; in at- a trail blazed by Maxwell, who had introduced the tempting to explain a subject on the elementary level, ensemble concept in his 1879 paper . But while Boltzmann frequently developed valuable new in- Maxwell never got past the restriction that all sys- sights, although he often succumbed to unnecessary tems in the ensemble must have the same energy, verbosity. He scrutinized the major physics journals Boltzmann suggested more general possibilities and and frequently found articles that inspired him to Gibbs ultimately showed that it is most useful to dash off a correction, design a new experiment, or consider ensembles in which not only the energy but rework a theoretical calculation to account for new also the number of particles can have any value, with data. a specified probability . Soon after he started to follow Maxwell's work on The Maxwell-Boltzmann ergodic hypothesis led to kinetic theory, Boltzmann began to study the electro- considerable controversy on the mathematical ques- magnetic theory of his Scottish colleague . In 1872, he tion of the possible existence of dynamical systems published the first report of a comprehensive experi- that pass through all possible configurations . The mental study of dielectrics, conducted in the labora- controversy came to a head with the publication of tories of Helmholtz in Berlin and of Topler in Graz . the Ehrenfests' article in 1911, in which it was sug- A primary aim of this research was to test Maxwell's gested that while ergodic systems are probably non- prediction that the index of refraction of a substance existent, "quasi-ergodic" systems that pass "as close should be the geometric mean of its dielectric constant as one likes" to every possible state might still be and its magnetic permeability (i = e°) . Boltzmann found. Shortly after this, two mathematicians, confirmed this prediction for solid insulators and Rosenthal and Plancherel, used some recent results (more accurately) for gases . He also confirmed the of Cantor and Brouwer on the dimensionality of sets further prediction that if the speed of light (and hence of points to prove that strictly ergodic systems are the index of refraction) varies with direction in an

26 5 BOLTZMANN BOLTZMANN anisotropic crystal, then the dielectric constant must periodically from one atom to the next . Every time also vary with direction . someone published new data on the specific heats of During the next few years, Boltzmann began ex- gases, Boltzmann felt obliged to worry again about perimental work in diamagnetism while continuing the distribution of energy among the internal motions his theoretical research in kinetic theory . He proposed of polyatomic molecules . a new theory of elastic aftereffects, in which the stress Until the 1890's, it seemed to be generally agreed on a material at a given time depends on its previous among physicists that matter is composed of atoms, deformation history . and Boltzmann's concern about the consistency of In 1883, as a result of preparing an abstract of H . T. atomic theories may have seemed excessive . But Eddy's paper (on radiant heat as a possible excep- toward the end of the century, the various par- tion to the second law of thermodynamics) for adoxes-specific heats, reversibility, and recurrence Wiedemann's Beibldtter, Boltzmann learned of a work -were taken more seriously as defects of atomism, by the Italian physicist Adolfo Bartoli on radiation and Boltzmann found himself cast in the role of pressure . Bartoli's reasoning stimulated Boltzmann to principal defender of the kinetic theory and of the work out a theoretical derivation, based on the sec- atomistic-mechanical viewpoint in general . Previously ond law of thermodynamics and Maxwell's electro- he had not been much involved in controversy-with magnetic theory, of the fourth-power law previously the exception of a short dispute with O. E. Meyer, found experimentally by Stefan : who, ironically, had accused Boltzmann of proposing a theory of elasticity that was inconsistent with the cc (absolute temperature)'. (radiation energy) atomic nature of matter. But now Boltzmann found Although at the time the "Stefan-Boltzmann law" for himself almost completely deserted by Continental radiation seemed to be an isolated result with no scientists ; his principal supporters were in England . further consequences, it did at least show a possible In retrospect it seems that the criticisms of kinetic connection between thermodynamics and electro- theory in this period were motivated not primarily magnetism that was exploited in the later quantum by technical problems, such as specific heats of poly- theory . In the 1920's it was applied by Eddington and atomic molecules but, rather, by a general philo- others in explaining the equilibrium of stellar atmos- sophical reaction against mechanistic or "mate- pheres . rialistic" science and a preference for empirical or In the 1890's Boltzmann again revived his interest phenomenological theories, as opposed to atomic in electromagnetic theory, perhaps as a result of models. The leaders of this reaction, in the physical Hertz's experiments, which he repeated before a large sciences, were Ernst Mach, Wilhelm Ostwald, Pierre audience in Graz. He published his Vorlesungen fiber Duhem, and Georg Helm . Mach recognized that Maxwells Theorie . .. in 1891 and 1893, along with atomic hypotheses could be useful in science but some papers in which he suggested new mechanical insisted, even as late as 1912, that atoms must not models to illustrate the field equations . In 1895 he be considered to have a real existence . Ostwald, published an annotated German edition of Maxwell's Duhem, and Helm, on the other hand . wanted to paper "On Faraday's Lines of Force" in Ostwald's replace atomic theories by "energetics" (a generalized Klassiker der exakten Wissenschaften . Boltzmann was thermodynamics) ; they denied that kinetic theories partly responsible for the eventual acceptance of had any value at all, even as hypotheses . Maxwell's theory on the Continent, although lie did I n the first volume of his Vorlesungen fiber Gustheorie not advance the theory itself as much as did Lorentz, (1896), Boltzmann presented a vigorous argument for nor did he grapple with the difficult problems that the kinetic theory : ultimately led to Einstein's theory of relativity. Defense of the Atomic Viewpoint . Throughout his Experience teaches that one will be led to new dis- career, even in his works on subjects other than kinetic coveries almost exclusively by means of special mechan- theory, Boltzmann was concerned with the mathe- ical models.... Indeed, since the history of science matical problems arising from the atomic nature of shows how often epistemological generalizations have "U ber turned out to be false, may it not turn out that the matter . Thus, an early paper with the title present "modern" distaste for special representations, die Integrale linearer Differentialgleichungen mit as well as the distinction between qualitatively different periodischen Koeffizienten" (1868) turned out to be forms of energy, will have been a retrogression? Who an investigation of the validity of Cauchy's theorem sees the future? Let us have free scope for all directions on this subject, which is needed to justify the applica- of research ; away with all dogmatism, either atomistic tion of the equations for an elastic continuum to a or anti-atomistic! In describing the theory of gases as crystalline solid in which the local properties vary a mechanical analoet% we have already indicated, by

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the choice of this word, how far removed we are from suicide in 1906 will never be known ; but insofar as that viewpoint which would see in visible matter the despair over the rejection of his lifework by the true properties of the smallest particles of the body scientific community may have been a contributing [Brush trans ., p . 26] . factor (as has sometimes been suggested without much evidence), it is certainly one of the most tragic In the foreword to the second volume of this book ironies in the history of science that Boltzmann ended (1898), Boltzmann seemed rather more conscious of his life just before the existence of atoms was finally his failure to convert other scientists to acceptance established (to the satisfaction of most scientists) by of the kinetic theory . He noted that attacks on the experiments on Brownian motion guided by a kinetic- theory had been increasing, but added : statistical theory of molecular motion . 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 NOTES of results agreeing with experiment which van der Waals has derived from it purely deductively, I have tried to l . All of Boltzmann's publications for which only the year is given make clear in this hook . More recently, gas theory has may be found in his Wi.ssenschaJiliche Abhandlungen . .s Clerk Maxwell (Cambridge, also provided suggestions that one could not obtain in 2. See The Scientific Papers ofJame 1890) . The 1859 and 1866 papers of Maxwell, together with any other way . From the theory of the ratio of specific other papers by Clausius, Boltzmann, Kelvin, Poincare, and heats, Ramsay inferred the atomic weight of argon and Zermelo (cited by year in this article) may he found in S . G . thereby its place in the system of chemical elements- Brush, ed., Kinetic Theory, 2 vols . (Oxford, 1965-1966) . which he subsequently proved, by the discovery of neon, 3. See Boltzmann, Vorlesungen uber Gastheorie, 1, „3 ; P. and T. Ehrenfest, "Begriffliche Grundlagen der statistischen Aufassung was in fact correct .. . . in der Mechanik ." In my opinion it would be a great tragedy for science 4. It was in reference to this result of Maxwell's that Boltzmann if the theory of gases were temporarily thrown into wrote his oft-quoted comparison of styles in theoretical physics oblivion because of a momentary hostile attitude toward and styles in music, dramatizing the almost magical disap- it, as was for example the wave theory because of pearance of V from the integrand of eq. 9 when the words "let n = 5" were pronounced (Popdare Schri_fien, p. 51) . Newton's authority . 5. Brush, Kinetic Theory, It, 192. I am conscious of being only an individual struggling 6. This formula for S is clearly related to Boltzmann's earlier weakly against the stream of time . But it still remains expression for the //-function (eq . 9) . If we know that the system in my power to contribute in such a way that, when has probability W ; of being in macrostate i, with given values of It', fix all i. then the expectation value of the entropy can the theory of gases is again revived, not too much will be calculated from have to be rediscovered ... [1bid., pp . 215-216] . S=kEWlogW Boltzmann and Ostwald, although on good per- with an appropriate interpretation of the summation or integra- sonal terms, engaged in bitter scientific debates during tion . this period ; at one point even Mach thought the 7. Die Principien der Warntelehre, historisch-kritisch entnrickelt argument was becoming too violent, and proposed (Leipzig, 1896), pp . 362 f a reconciliation of mechanistic and phenomenological physics . 7 While teaching at Leipzig with Ostwald during the period 1900-1902, Boltzmann was under- BIBI.IOGRAPII Y going periods of mental depression and made one 1 . ORIGINAL WORKS . The technical papers that origi- attempt at suicide . He returned to Vienna in 1902, nally appeared in various periodicals have been reprinted where he succeeded himself as professor of theoretical in Boltzmann's Wissenschaftliche Abhandlungen, F. Hasen- physics and also lectured on the philosophy of science, ohrl, ed ., 3 vols . (Leipzig, 1909) . Lectures and articles replacing Ernst Mach, who had to retire for reasons of general interest are collected in Populdre Schrifien of health . In 1904 he went to the United States to (Leipzig, 1905) . A review article written with J . Nabl, attend the World's Fair at St . Louis, where he lectured "Kinetische Theorie der Materie," was published in En- on applied mathematics, and also visited Berkeley and ctklopiidie der nathenatischen Wissenschaften, V . pt . I Stanford . He later described his experiences on this (Leipzig, 1905), art . V8. Other works are Vorlesungen fiber Max we//s Theorie der Elektrizitat and des Lichtes, 2 vols . trip in a satirical article, "Reise eines deutschen Pro- (Leipzig, 1891-1893) ; his ed . of Maxwell's "On Faraday's fessors ins Eldorado ." But despite his travels and Lines of Force," Ueber Faraday's Krafilinien (Leipzig, discussions with scientific colleagues, he somehow 1895), with 31 pages of notes by Boltzmann ; Vorlesungen failed to realize that the new discoveries in radiation fiber Gasiheorie, 2 vols . (Leipzig, 1896-1898), trans. by S. G . and atomic physics occurring at the turn of the century Brush, with introduction, notes, and bibliography, as Lec- were going to vindicate his own theories, even if in tures on Gas Theory (Berkeley . 1964) ; Vorlesungen fiber die somewhat altered form . The real cause of Boltzmann's Principe der Mechanik, 3 vols. (Leipzig, 1897-1920) ; and

267 BOLYAI BOLYAI

Uber die Prin_ipien der Mechanik, Zwei akademische An- especially in the so-called Euclidean or parallel triusreclen (Leipzig, 1903) . Books based on Boltzmann's lec- axiom, to which Kastner and Seyffer, as well as Gauss, tures are Charles Emerson Curry, Theory of Electricity and were devoting attention . Bolyai maintained a cor- Magnetism (London . 1897), with a preface by Boltzmann : respondence with Gauss that, with interruptions, and Hugo Buchholz . Das mechanische Potential, published lasted all their lives . with Die Theorie der Figur der Erde (Leipzig. 1908). After his return to Transylvania, Bolyai became a II . Si oNiARY LITERATURE . Works on Boltzmann arc superintendent in the house of the Kemznys in Kolo- Engelbert Broda, Luchnig Boltzmann : Men.sch, Physiker, Philosoph (Berlin, 1955): S . G . Brush . "Foundations of szvar (German, Klausenburg : now Cluj, Rumania) . Statistical Mechanics 1845-1915," in Archive for History In 1801 he married Susanna von Arkos, the daughter of Exact Sciences. 4 (1967), 145-183 : Rene Dugas, La of a surgeon . His wife was talented but sickly and theorie physique au seas de Boltzmann et ses prolongentents nervous, and the marriage was not a happy one . The modernes (Neuchatel . 1959): P. and T. Ehrenfest, couple settled in Domald, where Bolyai farmed from "Begriffiche Grundlagen der statistischen Aufassung in der 1801 to 1804 . In 1802 their son, Janos, was born, at Mechanik," in Encrklopadie (let - nucthematischen Wissen- the von Arkos home in Koloszvar . schaften, IV. pt . 32 (Leipzig. 1911), trans . b y M . J . In 1804, Farkas accepted the position of professor Moravcsik as The Conceptual Foundations of the Statistical of mathematics, physics, and chemistry at the Evan- Approach in Mechanics (Ithaca, N .Y., 1959): and G . Jaeger, gelical-Reformed College at Marosvasarhely, where "Ludwig Boltzmann ." in Neue Osterreichische Biographic he taught until his retirement in 1853 . During this 1815-1918, pt . I . Biographien, II (Vienna, 1925), 117-137 . Other articles on Boltzmann are listed in the bibliography half century he was known as a patient and kind of the Brush trans. of Lectures on Gas Theory . teacher, but one who lacked the faculty of easily transmitting to others his own scientific enthusiasm, STEPHEN G. BRUSH despite the emphasis he placed on correct mathemat- ical education . Meanwhile, he continued his research, BOLYAI, FARKAS (WOLFGANG) (b. 9 February concentrating on the theory of parallels . He sent a 1775, Bolya [German, Bell], near Nagyszeven [Ger- manuscript on this subject, Theoria parallelarum, man . Hermannstadt], Transylvania, Hungary [now with an attempt to prove the Euclidean axiom, to Sibiu, Rumania] : d. 20 November 1856, Marosvasar- Gauss in 1804 . The reasoning, however, satisfied hely, Transylvania . Hungary [now Targu-Murex, neither Gauss nor himself ; and Bolyai continued to Rumania]), mathematics . work on it and on the foundations of mathematics Farkas Bolyai was the son of Gaspar (Kasper) in general . Bolyai and Christina Vajna (von Pava) Bolyai . Bolya The Euclidean axiom, which appears as the fifth was the hereditary estate of the noble family of Bolyai postulate in Book I of Euclid's Elements, is equivalent de Bolya . which was mentioned as early as the thir- to the statement that through a given point outside teenth and fourteenth centuries . By the time of Gas- a given line only one parallel can be drawn to the par it had been reduced to a small holding, but line. It is also equivalent to the statement that there Gaspar added another holding (which belonged to exists a triangle in which the sum of the three angles his wife's family) in Domald, near Marosvasarhely . is equal to two right angles and, hence, that all He enjoyed a reputation as an industrious and intelli- triangles have this property . Attempts to prove this gent landholder of strong character . axiom-that is, to deduce it from other, more obvious, Young Farkas received an education at the Evan- assumptions began in antiquity . These attempts gelical-Reformed College in Nagyszeven, where he were always unsatisfactory, however, and the nature stayed from 1781 to 1796. He excelled in many fields, of the axiom had remained a challenge to mathemati- especially in mathematics, and showed interest in cians . Bolyai, working in almost total scientific isola- theology, painting, and the stage . In 1796, he traveled tion, often despaired while trying to understand it . to Germany, going first to Jena and then, with a During such moments of discouragement, he fellow student at Nagyszeven, Baron Simon Kemeny, sought consolation in poetry, music, and writing for entered the University of Gottingen, where he stud- the stage . In 1817, his Ot Szomorujatek, Irta egy ied until 1799 . Among his teachers were the astrono- Hazafi ("Five Tragedies, Written by a Patriot") was mer Felix Seyffer and the mathematician Abraham entered in a contest . The following year, another play, Gotthelf Kastner . It was at this time that Bolyai began A Parisi Par ("The Paris Process"), appeared . Bolyai's his lifelong friendship with Carl Friedrich Gauss, also wife died in 1821, and in 1824 he married Theresia a student at Gottingen, who already was intensely Nagy, the daughter of an iron merchant in Marosva- engaged in mathematical research . From this period sarhely . They had one son, Gregor . dates Bolyai's interest in the foundations of geometry, Farkas began to interest himself in mathematics

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