I. The Tombstone

Vienna Central Cemetery. A tombstone. An equation cut into the stone.

S = k log W

S is entropy. k is the Boltzmann constant. W counts the microscopic arrangements that can give rise to the same macroscopic state. Here, log means the natural logarithm. A few letters. A great deal beneath them.

Look at a bottle of gas. The temperature is steady, the pressure steady, the needle on the gauge hardly moving. Inside, no company of molecules stands at attention. They move, collide, change direction, collide again. Many different things can be happening in the bottle while the instruments give almost the same readings.

The steadiness is real. So is the motion. The neat number is not lying. It simply does not tell you everything that is going on underneath.

Boltzmann connected these two levels. He was not announcing that order did not exist. He was asking how it could exist without every molecule first being made to stand in line.

Planck later gave the relation its familiar form. Later still, others had it carved on the tomb. A lifetime of work comes down, here, to a few letters.

An equation remains in the stone.

The person who did the work is gone.

II. The Man

Ludwig Boltzmann. Born in Vienna on February 20, 1844.

He studied physics at the University of Vienna, where Josef Stefan was his teacher. In 1866, at twenty-two, he earned his doctorate. After that came Graz, Vienna, Munich, Leipzig: sometimes mathematics, sometimes experimental physics, sometimes theoretical physics. Back in Vienna, he also lectured on natural philosophy.

Read the place names in succession: luggage never quite unpacked. Appointments, research, relationships all entered into the decision to stay or leave. Another city, another classroom. A few more years lived at a different address.

He was stocky, with curly hair. Those who wrote about him describe sensitivity and warmth, and periods of profound depression. Very different weather could pass through the same person.

In the lecture room, another side of him appeared.

Lise Meitner attended his lectures. Paul Ehrenfest was a student too. In 1899–1900, Ehrenfest took his course on the mechanical theory of heat. For a young man who had become weary of school, those lectures made physics worth learning again. He later completed his doctorate under Boltzmann's supervision.

A person worn down by arguments can still make someone else want to go on studying. A student may leave with more than a distribution or a derivation. There may also be this discovery: the difficult subject has a way in. Someone has shown you where to begin.

That, too, is work he left behind. It is not carved on the tombstone.

III. Beneath the Floor Is Disorder

Newtonian mechanics gave people a formidable kind of confidence. Know where the particles are, how fast and in what direction they are moving, and how they act on one another; then follow the equations forward.

Laplace carried that confidence to an imagined extreme. An intellect that knew every relevant state and force, and possessed sufficient powers of analysis, could bring the past and the future within a single calculation.

Standing in front of a bottle of gas, we are not that intellect. There are too many particles. We cannot keep a register of what each one has just hit and where it is heading next.

Boltzmann's response was not simply to declare Newton wrong and the world lawless. He asked a different question. Without knowing the path of every molecule, could we still explain what the gas would do? Why heat would pass from one place to another? Why the pressure would settle at a particular value?

Instead of asking where this molecule will be a moment from now, ask how many molecules fall within a given range of speeds. Ask how many internal arrangements could produce the same reading on the instrument outside.

Imagine a thermometer standing on a floor. Twenty-six degrees Celsius.

Lift the floorboard: molecules move and collide below. The thermometer need not follow each one. In the classical ideal-gas model, absolute temperature corresponds to their mean translational kinetic energy. Their continual impacts transfer momentum to the container walls, producing pressure. The steady needle depends on motion that never settles down.

Yet the same temperature can accommodate a vast number of different molecular paths.

Throughout this series, Aristotle has been laying a floor: distinguishing things, finding places from which they can be understood. Boltzmann lifts a plank in the thermodynamics we had thought was already finished. He does not make the floor disappear. He shows us that its reliability need not depend on something underneath remaining forever still.

There is a harder question here. Mechanics allows a molecular collision to be described in reverse. Why, then, do the changes in a whole bottle of gas usually refuse to run backward in ordinary life? Loschmidt's objection about reversibility reached precisely this difficulty. It was not an insult. It was a problem Boltzmann had to answer.

Faraday directed attention toward the field. Boltzmann directed it toward the motion of enormous numbers of particles. They did not discover rival worlds beneath the floor, only one of which could be real. In each case, an explanation that had seemed sufficient revealed something more to ask about.

There is disorder underneath. That does not put it beyond understanding.

IV. Who Believed in Atoms?

Do atoms exist?

At the end of the nineteenth century, not every physicist would have given the same answer.

Boltzmann took molecular motion to be a genuine clue to the nature of matter. Mach worried about something else: the usefulness of a model did not automatically entitle scientists to treat its contents as the world's ultimate reality. He was deeply skeptical about atoms that could not be directly observed. Did a description of experience really have to include them?

Ostwald took another route. He wanted to organize physics around energy and its transformations, rather than make a host of hypothetical little particles bear the weight of explanation. He and Mach were not offering the same argument. Opposition to Boltzmann was not merely a chorus repeating “They don't exist.”

In 1895, at a scientific meeting in Lübeck, Ostwald attacked mechanical explanations. Boltzmann rose to argue against him.

Writing much later, Sommerfeld likened the encounter to a bull confronting an agile fighter, and thought the bull had come off better. He also remembered Felix Klein supporting Boltzmann, with the young mathematicians in the audience on the same side. This was a recollection across many years, not a stenographic record of the meeting. But in that recollection, at least, Boltzmann was anything but alone.

He had opponents. He also had readers, students, and colleagues. In 1899, he was elected a Foreign Member of the Royal Society. For his sixtieth birthday in 1904, physicists from several countries contributed to a volume of papers in his honor.

The arguments were not over. Recognition had already arrived. A birthday volume could not answer Loschmidt's objection for him. Nor could the objections raised at a meeting make that volume cease to exist.

Some people challenged his arguments; others took them further. His voice was not an echo in an empty room. Someone was listening. Listening did not always mean agreeing. Disagreement did not always mean a refusal to listen.

V. Boltzmann and Cantor

Cantor found that infinities could differ in size. The promise of listing every member could not be kept for every infinite set. His diagonal argument brought an apparently complete list up against its own limit.

Boltzmann lifted another floorboard. Beneath a steady reading lay innumerable molecular paths. Cantor tested a list that claimed to include everything; Boltzmann studied a gas whose particles could not be followed one by one. Mathematical proof and physical experiment each had their own work to do.

The two men can still stand beside each other.

In each man's work, a familiar way of understanding things encountered something it had not previously made room for. Kronecker's opposition could not make the mathematical questions Cantor had raised disappear. Nor did uncertainty about atoms oblige Boltzmann to pretend that statistical methods explained nothing.

Neither man had to wait until death for all recognition. Cantor helped found the German Mathematical Society and served as its first president. He received the Sylvester Medal in 1904. That same year, Boltzmann received the collection of papers assembled for his sixtieth birthday.

Recognition was already there. Illness and difficulty were still there too. Both could be true.

Cantor died in a sanatorium in 1918. Boltzmann died at Duino in 1906. These were the ends of two lives. What happened at the end did not erase the society, the classrooms, or the papers their colleagues had written.

At this point we may be waiting for a happy ending: someone finally acknowledges you, and everything becomes all right. But the medal arrives, the tribute arrives, and tomorrow still has to be lived. A person needs more than being told he was right.

A page of mathematics can survive its author. So can an equation.

Neither contains the whole of a life.

VI. Boltzmann and Maxwell

The connection between Maxwell and Boltzmann is more than a line drawn by a later writer. Their names belong to the same distribution.

The Maxwell–Boltzmann distribution describes molecular speeds in a gas under the relevant conditions of classical equilibrium. The molecules do not all move at the same speed. Some are faster, some slower, but under given conditions the proportions in the different speed ranges follow a rule.

Maxwell went first. Boltzmann carried the work further, connecting the distribution more deeply with equilibrium, collisions, and the problems of thermodynamics. Both were learning how to stop chasing molecules one by one without abandoning them altogether.

In this series, Maxwell brings light. He developed the investigation of fields that Faraday had opened into an electromagnetic theory. Boltzmann brings “disorder”: all those movements beneath the steady readings of thermodynamics.

But the theory of light did not meet an unbroken welcome. Maxwell died in 1879. Hertz's important experiments with electromagnetic waves came in the late 1880s. The theory and experimental evidence that now sit together on a textbook page were then separated by work still to be done.

Boltzmann himself was a supporter of Maxwell's theory. He taught it, published lectures on it, and helped it circulate on the European continent. The man who asked questions within “disorder” also helped open a way for the theory of light.

Light makes us look up; molecules make us bend closer. Here I would rather leave the two men side by side than reserve beauty for one discovery and coldness for the other.

Countless molecules go their separate ways, yet a distribution connects them. Even here, in the disorder, there is something we can learn to read.

VII. Entropy

In macroscopic thermodynamics, the second law says that the entropy of an isolated system—one exchanging neither matter nor energy with its surroundings—does not decrease. Reaching equilibrium does not mean that every motion inside has stopped.

“Isolated” matters. As hot water cools in a room, the entropy of the water itself can decrease. Its surroundings receive the heat. Only by accounting for the whole change can we tell whether the second law has been violated. We cannot turn the law into a demand that every place, at every moment, must grow messier.

Imagine a box divided in two. Gas fills one half; the other is a vacuum. Remove the partition.

The gas spreads.

There are enormously many microscopic ways for the gas to occupy roughly the whole box. The arrangements in which every molecule happens to remain in the original half make up an astonishingly small proportion. Equilibrium is common not because each molecule knows where it ought to go, but because the microscopic states compatible with it overwhelmingly outnumber the alternatives.

Yet “more ways” is not a force pushing the particles along. Their motions and collisions matter, as does the state in which the system began. Boltzmann's statistical explanation leaves room for small fluctuations. It also asks us to distinguish the extraordinarily unlikely from the absolutely impossible.

A broken cup will not put itself back together while we wait in ordinary life. That does not mean each fragment has received an instruction to make things worse. Turning entropy increase into “life can only deteriorate” gives a physical law a verdict it never delivered.

Heat death can be misunderstood in the same way. Its unsettling prospect is not that every particle must finally be motionless, but that the differences available to sustain useful work disappear. Without a temperature difference, a heat engine cannot keep working in the same way. The remote future of the universe also depends on its history and the conditions of its evolution. One line on a tombstone cannot write that whole ending by itself.

There is still an echo of Chekhov here.

The sisters in Three Sisters do not reach Moscow. But “they do not reach Moscow” does not finish telling us who they are. In the fourth act, they are still facing departures, still confronting how to go on living.

A temperature reading does not tell us every movement inside the bottle either. The resemblance lies in the distance between a brief account and everything it has not individually described. It does not mean that the characters' regrets are entropy, or that molecules have human hopes left unfulfilled.

Chekhov does not live the sisters' lives for them. An equation cannot decide how its reader is to live.

VIII. Duino

September 5, 1906. Duino, near Trieste.

Boltzmann was on holiday with his wife and daughter. During that holiday, he died by suicide. He was sixty-two.

The scientific disputes were real. So were illness and difficulties in his life. But the records have not preserved everything about that moment. They cannot be arranged into a straight line of cause and effect running from the meeting in 1895 to this day.

In 1905, Einstein had already published his paper on Brownian motion. Starting from molecular motion, he made quantitative predictions about the movement of suspended particles that experiments could test. Publication did not deliver atoms to every reader as directly as a photograph. In 1908–1909, Perrin's measurements supplied powerful experimental support for molecular theory. Boltzmann did not live to see those experiments.

But he had not spent his entire life standing alone before a wall.

In 1904, he attended the scientific congress held during the St. Louis World's Fair. In 1905, he went to Berkeley to lecture and also visited Stanford. Two years, two journeys—not a single trip.

In his account of California, he worried about his English and made a point of mentioning the words “blackboard” and “chalk.” A man who could teach people to understand molecular motion still had to learn how to get his teaching across in another language.

The passages about meals, travel, and lectures belong to Boltzmann too. They do not prove that he was free of pain. They simply keep pain from swallowing the whole person.

In places, the scientific wall was beginning to give way. We do not have enough evidence to say how much of that he saw.

He did not live to see what came after. What came after cannot bring him back.

IX. Disorder

The encounter at the bridge in this section is a work of literary imagination.

S = k log W.

The letters on the stone are orderly. They are not describing a company of molecules standing in formation.

Someone else has arrived at the bridge. He stands there, heavyset, his eyesight poor, holding a stone and catching his breath.

The equation is carved into it. He cradles it in his arms. It is heavy.

Socrates stands in the clearing. Plato crouches over his blueprints. Hume sets out his backgammon board. Schopenhauer looks beneath the bridge. Kierkegaard has made his leap. Turing looks at an apple. Chekhov leans against the railing. Cantor looks up at the sky. Copernicus has put down his book and walked away. Sartre paces. Beauvoir holds a mirror. Quine has said a sentence.

Tesla listens to a hum. Edison holds a light bulb. Heisenberg's position is uncertain. Bohr holds an unsent letter. Tolstoy stands opposite Chekhov with a prescription in his hand. Shakespeare is not there: he is the water below the bridge. Spinoza has glass dust on his hands. Aristotle crouches to lay the floor. Faraday crouches to lift a plank. Maxwell stands writing equations.

Joan of Arc floats above the bridge, carrying fire. Wilde stands with easy elegance, holding that sentence. Ramanujan has emerged halfway through a gap. Oppenheimer walks on, carrying ash. Charlotte Brontë holds a pen. Emily is out on the moor beyond the bridge. The wind comes from where she is.

Boltzmann stands beside Aristotle and looks down.

Aristotle is settling a plank into place. Boltzmann sees more than the plank. Beneath it, the different microscopic arrangements go on changing. Putting a name above them has not made them all stop.

The floor still bears their weight. Boltzmann looks down. The steadiness under his feet and the motion he sees beneath it are both there.

Faraday looks across from another gap. Maxwell keeps writing. Boltzmann looks at the pen, then at the stone in his arms.

He shifts the stone a little. An edge presses into his hand. He stops to catch his breath. The letters sink a little as his arms lower.

Then he looks up.

Kant is standing in the distance.

The figure is so far away that, for a moment, he thinks of a statistical fluctuation. Could it be only a passing shape, something that will appear and then dissolve?

He walks toward it nevertheless. Not quickly. The stone in his arms. The bridge beneath his feet. The wind blowing from Emily's moor.

Order may not remain secure forever. But the movement toward it is real.

Notes

^[1] The equation and gas examples draw on Planck’s Theory of Heat Radiation, §120, and The Feynman Lectures on Physics, volume I, chapters 39, 40, 44, and 46. The temperature relation concerns the classical ideal-gas model used here. The isolated-system condition matters, as do dynamics and initial conditions; counting possibilities alone is not a complete proof.

^[2] Biographical sources include Vienna University, MacTutor, and historical studies by Brush and Flamm. The 1895 encounter reaches us through Sommerfeld’s later recollection. Royal Society obituary pages xi and xiii record the 1899 election and 1904 Festschrift. The 1992 abridged translation of the 1905 travel account distinguishes St. Louis in 1904 from Berkeley in 1905. Einstein’s 1905 predictions and Perrin’s 1908–1909 experiments are kept separate.

^[3] No retrospective clinical diagnosis or single-cause account of the suicide is offered. The bridge, stone, gestures, and Kant in section IX are literary invention. The floor, disorder, and fluctuation imagery is not physical proof of human worth or an ethical direction.

Sources: The tombstone and the history of the formula — photograph of the grave in Vienna's Central Cemetery · The tombstone and the history of the formula — book, PDF page 155 · The tombstone and the history of the formula — paper · Birth, education, and appointments — university record · Appearance and health — University of St Andrews biography · Teaching and students — biography · Teaching and students — “University of Vienna” section of the Lise Meitner article · Laplace, statistical description, temperature, and pressure — entry · Laplace, statistical description, temperature, and pressure — chapter 39 · Laplace, statistical description, temperature, and pressure — §40–4 · Reversibility and the conditions of a statistical explanation — chapter 46 · Mach and Ostwald — entry · Mach and Ostwald — biographical study · Lübeck, 1895, and Sommerfeld's recollection — publication page for Robert Deltete's 1999 study · Recognition during his lifetime — contemporary obituary · Cantor: work, recognition, and later life — biography · Maxwell, molecular speeds, and electromagnetic waves — chronology · Maxwell, molecular speeds, and electromagnetic waves — biography · The second law, local entropy decrease, and available temperature differences — chapter 44 · Chekhov's characters and the literary comparison — text · Einstein's predictions and Perrin's experiments — original paper · Einstein's predictions and Perrin's experiments — French text · The American journeys and the travel writer — DOI · The American journeys and the travel writer — readable PDF