THE ONLY MAN BESIDES BOLTZMANN to grasp the growing importance of statistics and probability in physics was James Clerk Maxwell in England. At the age of 19 he had come across a book by the Belgian mathematician Quetelet in which statistical analysis—for example, of the range of heights of a group of soldiers—was expounded in a distinctly modern way. Quetelet also hinted that statistics could usefully adopt the rigorous methods employed in physics, while Maxwell in turn saw that physics might benefit from the introduction of statistics. “The true Logic for this world is the Calculus of Probabilities,” he observed in a letter; “This branch of Math., which is generally thought to favour gambling, dicing, and wagering, and therefore highly immoral, is the only ‘Mathematics for Practical Men.’ ”
He was quick to capitalize on this insight, publishing in 1859 a groundbreaking analysis in which he used a new style of argument to prove that the rings of Saturn must be composed of numerous tiny particles. Galileo, with his first telescope, was surprised to find that Saturn appeared to have “handles”; Christiaan Huygens, using a better instrument, had in 1656 concluded that a ring encircled the planet, and later observations revealed that the ring was multiple, consisting of bands and gaps. By the middle of the 19th century, the nature of these rings remained unknown, and in 1855 the structure and stability of Saturn’s rings was set as the topic for Cambridge University’s Adams Prize. Maxwell, adept at mathematics and confident of his knowledge in mechanics, tackled the problem.
It proved more difficult than he had expected, and he wrestled with the analysis for two or three years before he had sorted it out to his satisfaction. Solid rings were impossible, as others had already concluded; the planet’s gravity would try to make different parts of the ring rotate at different speeds, creating stresses that no physical material could withstand. Maxwell tried instead fluid rings, and what he called “dusky” rings composed of countless tiny particles, like dust grains. Modeling the latter kind of structure demanded a novel mathematical technique. He could not literally track the actual motion of every such particle; instead, he set up an essentially statistical description of the rings, which allowed for prescribed numbers of particles to follow certain classes of orbit, just as a population census might divide up people according to age, weight, and height classes.
Applying Newtonian mechanics to this dusky ring, Maxwell showed that the particles could move in collective modes corresponding to waves in the ring’s density. Only if these waves remained limited in magnitude would the ring itself remain stable, and this entailed certain conditions on the size and number of particles in the ring. He thus proved that Saturn’s rings could exist indefinitely, retaining their shape and density, if they were composed of particles of a suitable size. For this demonstration Maxwell won the Adams Prize.
With this considerable achievement in hand, Maxwell was ideally placed to apply the same sort of analysis to gases, in which, the atomists now asserted, broad properties such as pressure, temperature, and so on were to be understood as the gross manifestations of all the tiny and incalculable motions of countless atoms. Maxwell’s proposed distribution of the velocities of atoms in a gas was, in mathematical terms, a straightforward variation on his description of the particles in Saturn’s rings. And Maxwell’s investigation of the motions and stability of the rings made it an obvious next step for him to think similarly about the stability of atomic velocity distributions. In 1866, he published a long analysis, “On the Dynamical Theory of Gases,” setting out everything he had learned about the kinetic model of gases, showing how to obtain from the underlying distribution of velocities all manner of physical properties of the gas itself.
Boltzmann was well aware of Maxwell’s 1866 paper. Indeed, on one occasion he flew off into a romantic fancy about its brilliance, rhapsodizing over Maxwell’s ability to orchestrate a mathematical argument with symphonic coherence. “First the variations in velocity develop majestically, then the equations of state enter on one side, the equations of motion on the other; ever higher surges the chaos of formulas. Suddenly, four words sound out: ‘Put N = 5.’ The evil demon V vanishes, just as in music a disruptive figure in the bass abruptly falls silent . . .” Even in physics and mathematics, Boltzmann could find the melodrama he so much admired in music and the theater.
But Boltzmann’s sometimes extravagant passion also gave him the courage of his convictions. It was he, not Maxwell, who pushed kinetic theory further. Maxwell hesitated; he saw a problem looming that did not become clear to Boltzmann until some years later. Aware of the work of Loschmidt, Stefan, and Boltzmann, Maxwell had written praising the efforts of the Viennese. But just a few years later, in a letter of December 1873 to the Scottish physicist P. G. Tait, whom he knew from his school days in Edinburgh, Maxwell was mocking the continentals. It is “rare sport,” he said, to see “those learned Germans” tangled in confusion.
The cause of Maxwell’s aloof amusement was ostensibly another priority dispute that Boltzmann had gotten himself into, this time involving Clausius, the man who had first explained how atomic motion manifested itself as heat. But there were deeper issues here that would dog Boltzmann for many years, and in part cause Maxwell to turn his attentions away from the blossoming kinetic theory of gases. The looming problem sprang from a seemingly banal fact: heat always flows from high temperatures to low, so that anything hot inevitably cools down of its own accord. But why is this so, and why doesn’t the reverse ever happen?
In 1865, the same year that Loschmidt had estimated the size of molecules in air, Clausius published an influential work clarifying some formerly elusive notions about the nature of heat, energy, and mechanical work and in the process coining a new word: entropy. The word arose in the context of what is now called the second law of thermodynamics. The first law is the rule that Helmholtz had done so much to enunciate—the law of conservation of energy.
The second law of thermodynamics, like the first, existed in rudimentary form before it found a precise formulation. In 1824, a French engineer named Sadi Carnot came up with a profound but rather mystifying analysis of the efficiency of steam engines. At that time tinkerers and inventors of all sorts were trying to make better steam engines, mainly through inspired guesswork, since there was no theory of steam engines to guide them. Into this breach stepped Carnot, who imagined an idealized engine in which a steam-filled cylinder expands, pushing on a piston and performing mechanical work, then cools and returns to its starting position. By thinking of the transformation of energy and heat involved in this complete cycle of activity, Carnot proved there was a maximum amount of work that such an engine could perform, depending only on the high and low temperatures between which the device cycles.
Carnot’s argument, or variations on it, applies very generally. It is why, for example, you can’t cool your house down by leaving the refrigerator door open: the fridge uses energy to stay cool inside, but only at the expense of expelling more heat into its surroundings than it removes from its interior. It was thought at first that Carnot’s principle was in some way a consequence of the conservation of energy, but investigations over the next several decades, particularly by William Thomson and William Rankine in Britain and by Clausius in Germany, showed that there was a second independent principle at work. These efforts generated the science now called thermodynamics—literally, the dynamics of heat.
Thomson and Clausius particularly used Carnot’s insight to understand the nature of thermodynamic changes. In an idealized, isolated system, the total amount of energy must remain constant; that was the first law of thermodynamics. But within that system, energy could change from one form to another and back again. Physicists distinguished two types of changes: reversible ones, in which the system could be exactly restored to its starting point, and irreversible ones, in which it could not—not, that is, without the application of further external energy. In reversible changes, something stayed the same; in irreversible changes, it did not.
That something, Clausius said in 1865, was entropy. In reversible changes entropy remained constant, but in irreversible changes it grew. Entropy in an isolated system can never decrease, which is why irreversible changes are indeed irreversible. Once the entropy has increased, it can’t fall back to its former level. By the same reasoning, entropy in any isolated system would tend to increase until it reached its maximum possible value. That state of maximum entropy, Clausius said, was the state of perfect thermal equilibrium. The rule that entropy can never decrease, only increase or stay the same, was a new physical principle: the second law of thermodynamics.
Whereas heat and energy are physical quantities that can be fairly directly grasped, entropy has a more abstract character. It represents a sort of potential energy: mechanical work can be extracted from a system as long as there is room for entropy to increase, but in a uniform volume of gas in thermal equilibrium, entropy has attained its maximum possible value, and no further work can be obtained.
Clausius defined entropy in terms of the heat going in or out of a system and the temperature at which such exchanges are happening. Immediately, enthusiasts for the kinetic theory of heat wanted to understand entropy in terms of the underlying atomic constitution of a gas rather than its overt bulk properties. The temperature and pressure of a gas were simply related to the average kinetic energy of its constituent atoms; that was now straightforward. But what was the kinetic definition of entropy? What quality or average property of moving atoms corresponded to this newly minted thermodynamic quantity?
In 1866, the 22-year-old Boltzmann had published an attempt to answer this question—a very sketchy and preliminary attempt, relying on some restrictive and in truth unrealistic restrictions on the way atoms could move. This was the first attempt on a problem that was to keep Boltzmann busy, one way or another, all his life. It was not, however, a great piece of work, and since Boltzmann’s name was unknown at the time, it went largely unnoticed. A few years later, Clausius hit on a similar idea and in 1871 published a brief note saying more or less the same thing. Clausius being who he was, his argument gained a little attention.
Most notably, it drew some attention from Boltzmann, by then in his first appointment as physics professor in Graz. He submitted a lengthy, not to say long-winded, memo to the Viennese Academy of Sciences, in which he reprinted several pages’ worth of his 1866 paper and concluded, in case any reader did not see what he was getting at, “I think I have established my priority.” That was heavy-handed enough, but Boltzmann went further: “Finally I wish to express my pleasure that an authority such as Dr. Clausius contributes to the dissemination of the ideas contained in my papers on the mechanical theory of heat.”
Subtleties of phrasing were never Boltzmann’s strong point. A colleague once commented that “style is the man, as a Frenchman has said. . . . Boltzmann wrote good, flowing German, with occasional Austrianisms, but he did not turn his sentences. Everything came out quite unaffectedly, just as it came into his head.”
Clausius was presumably not pleased at being thanked by someone he didn’t know for acting as the messenger for work he knew nothing about. Still, he published a short, mild reply acknowledging that Boltzmann had indeed had the idea first and apologizing that he had not been sufficiently conversant with the scientific literature to be aware of the younger man’s work. But he finished by saying that he thought his result a little more general than Boltzmann’s.
Boltzmann might have been inclined to snipe back, but as 1871 progressed he became caught up in the ideas that would power his monumental work of 1872; he saw, in fact, that he could solve this fundamental problem in its entirety. His analysis of atomic collisions, derivation of the transport equation, and proof that the Maxwell distribution was the only possible distribution corresponding to thermal equilibrium were (to use a symphonic analogy Boltzmann might have liked) merely the overture to what was the great theme of his 1872 analysis. The culmination was something Boltzmann called his minimum theorem, a result that some years later became known as the H-theorem when an English physicist apparently misread a German script upper-case E in one of Boltzmann’s papers for an H.
What came to be called H, at any rate, was a numerical quantity defined in terms of the velocity distribution of the atoms, whatever form that might take. For any collection of atoms, moving at any assortment of speeds, the value of H came from a formula that Boltzmann devised.
The significance of H was twofold. First, when the atoms fell into a Maxwell-Boltzmann distribution, H assumed its minimum possible value. Second, Boltzmann argued, a collection of atoms whose H-value was greater than this minimum would, through the effect of collisions, transform its distribution of velocities in such a way as to decrease H, moving it toward the minimum value associated with the Maxwell-Boltzmann distribution.
This was a result of astounding power. It implied not only that the Maxwell-Boltzmann distribution was uniquely the correct description of a collection of atoms at equilibrium but also that any other distribution would, because of atomic collisions, inevitably evolve toward the Maxwell-Boltzmann form. In fact, as Boltzmann was eager to believe, his quantity H was by all appearances exactly what he needed as a kinetic definition of the thing Clausius called entropy. All he had to do was put a minus sign in front of it. Then H reached a maximum in thermal equilibrium, was less than that for any other distribution, and starting from any other value naturally evolved toward equilibrium. This was how entropy behaved: whatever its value, it increased until it attained the maximum possible value, which corresponded to thermal equilibrium. H was precisely the kinetic definition of entropy, Boltzmann declared, and his H-theorem showed that the mysterious second law of thermodynamics, stating that entropy must always increase, was itself the consequence of the elementary principles of mechanics applied to the collisions of atoms. The H-theorem seemed to give a simple kinetic explanation for thermodynamics in its entirety. It seemed to be a proof, from first principles, of the inescapable fact that everything in the universe cools down and never spontaneously heats up.
But, as Maxwell was keenly aware, something about this result was fishy. In an 1869 letter to his friend and fellow-physicist Tait, he had come up with a whimsical character that came to be known as Maxwell’s demon. Imagine two adjacent chambers filled with gas, one side hot and the other cold, and with an aperture connecting them. Normally, as atoms pass at random through the aperture, from one side to the other and back again, the gases will become mixed and their temperatures will become equal. Maxwell imagined, however, a tiny creature watching over the atoms passing back and forth and able to operate a shutter in the aperture. This demon’s sole task was to open and close the shutter so as to let only the faster moving atoms into the chamber containing the hot gas and only the slower moving ones go the other way. The result of this monitoring would be to reverse the normal course of affairs: the gas in the hotter chamber would grow hotter still, and the gas on the other side would go colder. Heat would flow the wrong way.
Generations of undergraduates have been taught about Maxwell’s demon, with often confusing consequences. The philosopher Karl Popper came to believe that Maxwell had somehow proved the laws of thermodynamics incorrect. There is, of course, no actual demon.
What Maxwell intended was a more subtle observation, which he made clear to Tait in the form of a catechism “Concerning Demons” of which item 3 read: “What was their chief end? To show that the second law has only a statistical certainty.” The demon’s actions contravened no law of physics. Maxwell’s point, therefore, was that one could imagine atomic motions, no matter that they were somewhat fanciful, which resulted in heat flowing the wrong way.
Of course, the demon is a fictitious creature, a figment of Maxwell’s imagination. But what the demon accomplishes on purpose can also happen by accident, without any demonic intervention. The probability might be exceedingly low, but it is not impossible that atoms would move purely by chance in such a way that heat flowed from cold to hot. Consequently, the second law of thermodynamics could not be an absolute law; there were circumstances in which it might not hold true.
This realization was the cause of Maxwell’s skepticism over what Boltzmann claimed to have done. The H-theorem supposedly proved that any set of atoms, banging around and colliding at random, would move inexorably toward thermal equilibrium. But as Maxwell perceived, there must be physically permissible atomic motions that would correspond to heat moving the wrong way and, therefore, a system moving away from equilibrium. It might happen very rarely and only transiently, but it could nevertheless happen. Boltzmann’s theorem, on the other hand, seemed to say that such a thing could not happen at all.
Maxwell therefore regarded the efforts of Boltzmann and others as basically futile. They were chasing after a mirage. Or, as he more extravagantly put it in his letter to Tait, “the German Icari flap their waxen wings in nephelococcygia amid those cloudy forms which the ignorance and finitude of human science have invested with the incommunicable attributes of the invisible Queen of Heaven.” Nephelococcygia is cloud cuckoo land, the foolishly idealistic city of birds, imagined by Aristophanes, that was meant to exist midway between earth and heaven. In Maxwell's opinion, the German theorists did not understand that they were chasing after an unrealizable fantasy.
Maxwell had, indeed, a peculiar sense of humor and an odd turn of phrase. A joking ironic manner was his habit. He had, like Boltzmann, lost a parent when young. Maxwell’s mother had died, of intestinal or stomach cancer, when the boy was only seven years old. Young James’s reaction was “Oh! I’m so glad. Now she’ll have no more pain!” The boy was thereafter brought up by his father, a provincial lawyer, with the assistance of an aunt and an assortment of tutors. For the first 10 years of his life James Clerk Maxwell, an only child, grew up on the run-down estate of Glenlair, some 16 miles from Dumfries in the southwestern corner of Scotland, which his father had inherited. Here he explored the country and learned the stars, curious from an early age about everything around him. “Show me how it doos,” he would ask at the age of three, his mother reported, and “what’s the go o’ that?” And if he didn’t get what he considered an adequate answer he would insist “but what’s the particular go of it?”
A streak of eccentricity ran in the Maxwell family. A grandfather reportedly saved himself from drowning in the Hooghly River in India by floating to shore on his bagpipes, which he then played to entertain the other members of his party and frighten away the tigers. Maxwell’s father was similarly self-reliant and was endlessly fascinated by the ingenuity and invention displayed in all manner of novel industrial processes. He not only designed new buildings for his Glenlair estate, but had roomy square-toed shoes made up according to his own specification, and likewise had shirts cut to his personal preference. This was all very well when he and his young son were at Glenlair, but when young James was sent away to Edinburgh at the age of 10 to live with his aunt and continue his schooling at the Edinburgh Academy, his unusual garb and rustic accent caused a good deal of merriment among the more sophisticated city boys. They tore at his odd clothes, teased him for his outlandish and stuttering speech, and called him “Dafty.” James showed a good deal of wit and resilience, and gradually earned the respect of his tormentors.
During his school years he wrote frequently to his father in zany letters filled with puns and misspellings, embellished with elaborate doodles, and containing secret messages in different colored inks. One letter begins: “My dear Mr Maxwell, I saw your son today, when he told me that you could not make out his riddles.” And he took to signing himself anagrammatically as Jas. Alex. McMerkwell. His letters betray an affectionate and familiar manner, as well as some sharpness of mind. At the age of 11 he noted that “Ovid prophesies very well when the thing is over.”
The boy showed early aptitude in science. When he was 14, some modest but original ideas of his in geometry were presented to the Edinburgh Royal Society. P. G. Tait, the physicist with whom Maxwell later corresponded, was a schoolfellow, and through his father, young Maxwell knew the Thomsons of Edinburgh, whose son William, later Lord Kelvin, was also to become a great scientist, engineer, and Victorian entrepreneur. He left the Edinburgh Academy in 1847 and moved on to the University of Edinburgh as an undergraduate. He was still only 16 years old.
Where Boltzmann was raised exclusively by a doting and perhaps overprotective mother, living with her until his marriage and only reluctantly separating even then, Maxwell moved between his father’s residence at Glenlair, his aunt’s house in Edinburgh, and the testing environment of the Edinburgh Academy—which, for all that Maxwell endured there from the other boys, gave him a solid education. He developed the habit of writing light verse and humorous doggerel, and commemorated the virtues of his first school:
Let Pedants seek for scraps of Greek,
Their lingo to Macadamize;
Gie me the sense, without pretence,
That comes o’ Scots Academies.
(Macadam was the Scotsman who invented a durable road covering of stone chips and gravel embedded in a bituminous mix.)
Maxwell learned a good deal of self-reliance, bolstered by an ironic and occasionally waspish sense of humor. This attitude spilled over into his studies. As a student at the University of Edinburgh, he gave himself a summer project: “Kant’s Kritik of Pure Reason in German, read with a determination to make it agree with Sir W. Hamilton.” Sir William Hamilton was an Edinburgh professor of logic and metaphysics.
After three years at the University of Edinburgh, he traveled south to study at Cambridge. Here he was a misfit all over again, reading the lesson in Trinity College chapel in his wild Scots accent. He tended to speak, moreover, in a “spasmodic” manner, rushing out bursts of words then abruptly halting before delivering another burst.
But by this time his brilliance was becoming apparent, and Cambridge has traditionally admired eccentricity and idiosyncrasy when it is accompanied by the signs of genius. Maxwell, unable now to run about the Scottish moors for exercise, took to charging up and down the staircases of Trinity College in the small hours of the morning. His fellow undergraduates, recognizing the pattern, began to lay in wait behind their doors and fling shoes and hair-brushes at him as he passed. Maxwell made a number of friends with whom he kept close throughout his life.
He graduated from Cambridge in 1854 and stayed on to teach. After a couple of years he took a position in Aberdeen, then moved on to King’s College London soon after. In 1865, at the age of 34, he withdrew from his academic position and began to spend more time at Glenlair, though he kept up his scientific work and maintained correspondence with his fellow physicists. Six years later, Henry Cavendish, the Duke of Devonshire and a talented physicist himself, endowed the University of Cambridge to build a laboratory for experimental physics, and Maxwell became the first director of what is still known around the world as the Cavendish Laboratory.
Maxwell’s scientific endeavors showed great versatility. There were, moreover, great contrasts in the kind of theorizing he undertook. In his analysis of the rings of Saturn, he studied the mechanics of tiny particles in a gravitational field. In his electromagnetic work, he delved into pure field theory. In his work on the kinetic theory of gases, everything derived from mechanics; there was no field theory at all. This range of interests, compared to Boltzmann’s more singular focus on gas theory, gave Maxwell a more agnostic view of theorizing in general. He found atomic theory fascinating, no doubt, and perceived its many virtues and possibilities. But at the same time he could see imminent difficulties.
BOLTZMANN, YOUNGER by 13 years, was introduced to Maxwell’s work by Stefan, and throughout his life remained a great admirer of his Scottish counterpart. On one occasion, lecturing on Maxwell’s electromagnetic theory, he borrowed a phrase from Goethe’s Faust to ask rhetorically, “Was it a God who wrote these signs?” His respect was, however, only partially reciprocated. Maxwell and Boltzmann never met or even, apparently, corresponded. They might usefully have done so during the late 1870s, when objections to the H-theorem came up, but it perhaps seemed to Boltzmann that Maxwell had more or less withdrawn from the battle. Certainly, he didn’t pursue the subject with Boltzmann’s bulldog tenacity. Maxwell was guided by elegance and brevity and aimed to encapsulate in precise mathematics an idea or theory whose form he could already perceive. Boltzmann, by contrast, plowed on regardless, confident that because an answer must exist, it would be only a matter of time and effort for him to find it. He was never an introspective man, and in these circumstances that was an advantage; it never occurred to him to doubt whether he would succeed.
The differences in style between Maxwell and Boltzmann may have been in part consciously aesthetic, but were more likely the offspring of their individual psychologies. Both have their advantages and disadvantages. Boltzmann had doggedness coupled to passionate belief. Maxwell had a fierce sense of design or logic, which helped him find the powerful and beautiful simplicity of his electromagnetic theory. But, as Boltzmann’s own work demonstrated, science is not always clean and precise, especially in its formative stages. Elegance is for the tailor and the shoemaker. Boltzmann’s style enabled him to keep pushing ahead through the theoretical thorns and tangles of kinetic theory.
These differences also made Boltzmann, on his good days, a memorable lecturer and speaker, while Maxwell could find himself so anxious to convey the subtlety of each and every point that he lapsed into inarticulate stumbling and then silence. Though he complained with increasing frequency, as he got older, of the tedium of lecturing to dull and indifferent undergraduates, Boltzmann was capable of expounding the subjects he cared about with a passion unconstrained by doubt or hesitation.
Maxwell, by contrast, was not a man of overt exuberance. There was in him almost something of the dilettante, a lighter touch that enabled him to hop from one subject to another, as he did throughout his scientific career, but that made him easily distractible when attempting to teach. He had difficulty keeping his mind on a single track and, as one contemporary put it, he had, as a result, “his full share of misfortunes at the blackboard.” At one point both Maxwell and Tait applied for the same position at the University of Edinburgh, and although Maxwell was acknowledged to be the greater scientist, Tait got the job, it seems, because he could teach.
If, in their spoken expositions, Boltzmann was forceful and Maxwell hesitant, their writings show their characters in a different light. Maxwell thought and analyzed a great deal before he committed anything to paper, trying to work his way through every possible byway in advance. What he wrote was therefore clear and complete, carefully and logically guiding the reader to an inescapable conclusion. Boltzmann, characteristically, wrote as he spoke, forging his way ahead without worrying that every possible byway had been inspected and every possible objection assessed and discarded. His bulldozer style made his writings often dense and difficult and—disturbing to his readers and often to himself—not always consistent from one exposition to the next.
Maxwell had the essence of the ironic perspective: the capacity to stand away from his own work and see it as others might see it. He could write persuasively because he could anticipate objections from those with another point of view, and answer them before the detractor had clearly articulated the problem. Boltzmann, throughout his life, in his personal dealings and in his science, was deaf to other sensibilities. “I think I have established my priority,” he would write, not understanding that the reader was already there.
In yet another letter to Tait, Maxwell expressed his opinion of the differences between them: “By the study of Boltzmann I have been unable to understand him. He could not understand me on account of my shortness, and his length was and is an equal stumbling block to me. Hence I am very much inclined to join the glorious company of supplanters and to put the whole business in about six lines.” He was admittedly reading Boltzmann in German, but then Maxwell had read Kant in German as a teenager, so he had the necessary fortitude.
This was written in 1873, the year after Boltzmann published his H-theorem, which Maxwell found impossible to understand because of his own ideas concerning what he called the demon. He could not see how Boltzmann had been able to derive an equation demanding a one-way trend in the behavior of atomic motions, when clearly there must be sets of atomic motions that behaved differently. And so Maxwell concluded that somewhere in the dense reasoning of his 1872 paper Boltzmann must have gone wrong.
Boltzmann did not immediately perceive the acuteness of Maxwell’s criticism. A detailed account of the demon appeared in Maxwell’s Theory of Heat, which was published in 1871 and translated into German in 1877. But by that time the same issue had been raised in a somewhat different guise, and in circumstances such that Boltzmann could hardly evade the question. The source of the objection this time was his friend and colleague Josef Loschmidt.
What Loschmidt articulated, in 1876, became known as the reversibility problem. It hinges on the fact that the laws of mechanics governing atomic motions and collisions are, as physicists like to say, time-reversible; that is, any set of motions and collisions obeying Newton’s laws can be run backward, as if on a videotape, and they will still obey Newton’s laws. This, as Loschmidt explained, leads to a problem with the H-theorem: any set of atomic motions that causes H to decrease has a time-reversed counterpart that must cause H to increase. How then can Boltzmann’s theorem dictate that H must always decrease? Loschmidt, a sympathizer of atomic theory, did not intend his observation as a disproof of Boltzmann’s work in particular or kinetic theory in general. But undoubtedly he had located a problem that demanded an answer.
Loschmidt’s reversibility objection was essentially what Maxwell had in mind with his somewhat cryptic invocation of the demon. But Maxwell was perhaps too cute for his own good. Thomson had published something very close to Loschmidt’s argument a couple of years earlier but seemed to conclude that in the absence of the hypothetical demon, these oddities would not arise.
Faced now with Loschmidt’s specific objection, presented moreover to the Viennese Academy of Sciences, Boltzmann had to come up with an equally specific response. His first answer was simple. He agreed, as he must, that for some atomic distributions the value of H, and therefore the entropy, must go in the wrong direction. But he asserted that such cases would demand an extraordinary degree of order—a sort of conspiracy—among the atoms. As a matter of probability, and because of the enormous number of disorderly arrangements of atoms compared to “special” arrangements, H would almost always do what the H-theorem said it would do.
All kinds of traps and implications lay concealed in this answer. For one thing, Boltzmann had originally said that the H-theorem was exact, that atomic collisions would always lead to an increase of entropy. Now he was saying that in certain cases, rare, unlikely, but physically legitimate, this would not be so. But in that case was the H-theorem a true theorem, a theorem with limited validity, a useful approximation, or what exactly? And if the H-theorem did not always hold true, what was the extent of its validity, and what precisely was the nature of atomic distributions for which it wasn’t true? Boltzmann had derived the theorem in a seemingly general way, using basic elements of Newtonian mechanics and some broad, seemingly plausible arguments about the behavior of atoms. Was there some hidden assumption that was not quite true, or not always true, so that the H-theorem would not invariably follow?
Furthermore, if Boltzmann was now saying that in some odd cases entropy could decrease instead of increasing, was the implication that in nature, in reality, there could be occasional instances of systems behaving counter to the recently minted second law of thermodynamics, or was he implying that the atomic distributions that gave the “wrong” behavior were, for some as-yet-unspecified reason, physically disallowed? Critics seized on the suggestion that kinetic theory seemed to imply that the laws of thermodynamics were not true laws after all, but only approximate laws, true “almost always.” If this was so, it seemed an unhappy development. There had never been any implication that Newton’s laws of mechanics were true only most of the time, or that the refraction of light by lenses almost always went according to plan. What use was there—indeed what meaning—in a supposed law of physics that on closer examination turned out not to be quite a law after all?
Critics of atomic thinking, who were beginning to rally against the seeming triumphs of kinetic theory, now believed they had found a crucial flaw. They held the laws of thermodynamics to be absolute and inviolable, as true physical laws must surely be. Kinetic theory stumbled on this point. Taken literally, it implied that the second law of thermodynamics was inexact and therefore not really a law at all. Or if, as Boltzmann sometimes seemed to hint, kinetic theory must be amended or augmented somehow so as to disallow violations of the laws of thermodynamics, then its pretensions to be a complete explanation of thermodynamics based only on mechanics were demolished. Either way, atomic theory looked shaky.
Maxwell, the first man to perceive the probabilistic nature of the second law, doubted Boltzmann’s theorem because it seemed to offer absolute certainty where there could be none. Thomson, on the other hand, who had originally praised Boltzmann’s work because of the very certainty it promised, now began to have doubts because probability was creeping into the picture.
In Germany and Austria the prevailing opinion was with Thomson. The laws of thermodynamics must be absolute, so kinetic theory must be wrong. And beginning to be influential was the voice of Ernst Mach, still in Prague but making a few inroads with books of historical and philosophical commentary on physics. As a student in Vienna, just a couple of years ahead of Boltzmann, Mach had been swayed by atomism and counted himself for a time a believer in atomic theory. But in Prague he was finding his own voice and was beginning to evolve a philosophy of science according to which observations and data were primary and theorizing was intrinsically suspicious. The goal of science, Mach implied, was to provide logical and rational relationships between facts and phenomena that could be directly observed; the more one invoked the existence of entities whose existence was not immediately apparent, the more one was going astray. Theorizing, in Mach’s view, was a necessary evil at best, and frequently an unnecessary one.
In atomism and kinetic theory, Mach found a natural target. It demanded a belief in unseen and quite possibly unseeable objects, yet its results, which merely confirmed what the laws of thermodynamics already said, were supposed to lend credence to the assumptions on which it was based. Apart from the circular nature of this reasoning, it ran counter to what Mach had decided was the essence of scientific explanation: to find laws, as simple as possible, linking observable phenomena. Classical thermodynamics passed the test; it posited fundamental relationships between the overt properties of gases—their pressure, volume, temperature, and so on. Kinetic theory, on the other hand, sought to replace these perfectly acceptable and straightforward laws with new and mysterious explanations based on unprovable assumptions about the existence and properties of atoms. How was this an advance?
The discovery of flaws, paradoxes even, in the kinetic theory represented not just problems for the theory itself—problems that Boltzmann at least thought he knew how to deal with—but deeper flaws in the very structure and essence of the theory, so far as Mach was concerned. To tinker with the theory so as to bring it into line with established thermodynamics laws was, Mach concluded, an admission of failure. Proponents of kinetic theory had originally proclaimed that armed with nothing but the laws of mechanics, they could explain the properties of gases. Now they discovered that perhaps they could not, so they began to modify their already unfounded theoretical assumptions.
To Mach, the conclusion was simple. Atomic theory had failed in what it professed it could do and must therefore be wrong. His distaste for theorizing was vindicated. His insistence on sticking to simple laws linking observable data was shown to be reliable. The kinetic theory of heat, in the view of Mach and those who were beginning to rally around him, had had its day.