In September 1906, Ludwig Boltzmann traveled with his wife and daughter to Duino, a small village near Trieste on the Adriatic coast. He was sixty-two years old and had been suffering for years from severe depression, headaches, and failing eyesight. He had been fighting a battle with the scientific establishment for three decades and had not, by any honest accounting, won. On September 5th, while his wife and daughter were swimming, he hanged himself.
He was right about everything.
Boltzmann's central achievement was the statistical interpretation of the second law of thermodynamics. The second law, in its classical formulation, says that the entropy of an isolated system never decreases. Heat flows from hot to cold, not the reverse. A drop of dye disperses through water; it does not spontaneously reconcentrate. Gases expand to fill their container; they do not unmix. These are observed facts, and the classical thermodynamics of Clausius and Kelvin captured them in the language of entropy — a quantity that always increases or stays the same, never decreases. But this raised an immediate question that classical thermodynamics could not answer: why? What physical principle prevents the reverse process?
Boltzmann's answer, worked out over the 1870s and 1880s, was probabilistic. There is no physical law preventing a drop of dye from spontaneously reconcentrating. The reverse process is not forbidden by any equation of motion. What prevents it is counting. There are an astronomically larger number of arrangements of dye molecules that are dispersed than concentrated. When the system evolves randomly among its possible states, it overwhelmingly tends toward dispersal simply because dispersed states vastly outnumber concentrated ones. Entropy, in Boltzmann's framework, is a logarithmic measure of the number of microscopic arrangements (microstates) consistent with a given macroscopic state (temperature, pressure, volume). High entropy means many microstates. The second law is not a fundamental prohibition but a statistical near-certainty: entropy increases because high-entropy states are overwhelmingly more probable.
The equation on Boltzmann's tombstone in Vienna, S = k log W, is the compressed form of this. S is entropy, k is the constant that now bears his name, and W is the number of microstates. This formula is not just an empirical relation; it is a definition, a bridge between the statistical counting of microstates and the thermodynamic quantity that Clausius had defined without explanation. Boltzmann built that bridge.
The simplest way to feel the argument is to think about a gas. Imagine 10²³ molecules bouncing around in a box. One particular microstate has all the molecules crammed into the left half. How many microstates look like that? Compared to the number that have molecules roughly evenly distributed throughout the box? The ratio is so extreme — roughly 2 raised to the power of 10²³ — that you could run the universe for a trillion times its current age and never, not once, expect to observe the spontaneous reconcentration of gas from the right half into the left. It is not impossible in principle. It is just so improbable that the distinction between "impossible" and "overwhelmingly unlikely" has no practical content.
This answer is profound and correct. It is also unsettling in ways that Boltzmann's opponents found decisive, and that he spent his career defending against.
The most serious objection was what became known as the reversibility paradox, raised sharply by Josef Loschmidt. Newton's laws of motion are time-reversible: if you take any mechanical trajectory and reverse all the velocities, you get another valid trajectory, one that runs the dynamics backward. If a forward-running trajectory takes a concentrated gas and disperses it, then the time-reversed trajectory takes a dispersed gas and spontaneously concentrates it — and is an equally valid solution to Newton's equations. So, the argument went, if entropy increases in one direction of time because the laws of mechanics say so, then there should be an equal tendency for entropy to decrease in the other direction. But we never observe entropy decreasing. How can time-reversible mechanics underlie a time-asymmetric thermodynamics?
Boltzmann's response was careful and, in hindsight, correct. The reversibility argument shows that individual microstates that decrease entropy exist. But they form a set of measure zero — a vanishingly small fraction of all microstates. The overwhelming majority of microstates, evolved forward in time, lead to entropy increase. The time-asymmetry is not in the laws of motion; it is in the initial condition. We find ourselves in a universe whose initial state — the Big Bang, low entropy and highly ordered — was, by cosmological standards, extraordinarily special. The arrow of time, the felt directionality of entropy increase, is a consequence of that boundary condition rather than any time-asymmetric law.
Ernst Mach and Wilhelm Ostwald, the leaders of a positivist movement in German-speaking science, took a different line. Their objection was not to Boltzmann's reasoning but to his ontology. Mach and Ostwald were deeply skeptical of unobservable entities — including atoms. Their philosophy held that science should confine itself to observable phenomena and relationships between them, and since no one had ever directly observed an atom, the atomic hypothesis was metaphysics, not physics. They preferred to ground thermodynamics in a general concept of energy without committing to any picture of what matter was made of. Boltzmann's entire program rested on atoms. Without them, there was no statistical mechanics, no counting of microstates, no S = k log W.
The battle was not just academic. Ostwald was an influential editor and academic organizer; Mach had enormous philosophical prestige. Boltzmann, who suffered from what appears to have been bipolar disorder, found the sustained professional opposition physically and psychologically costly. He wrote, in an essay near the end of his life, that he was not certain he would live to see his ideas vindicated. He was correct in the most painful way: vindication came quickly after his death. In 1905, Albert Einstein had already calculated the statistical fluctuations in Brownian motion — the visible jiggling of pollen grains in water — in terms of atomic parameters, and Jean Perrin would measure them precisely in 1908, providing direct evidence for atoms and a measurement of Avogadro's number. The entire edifice of objections collapsed within a few years of Boltzmann's death.
What I find most worth sitting with in Boltzmann's story is not the tragedy of a man rejected by his contemporaries, though that is real enough. It is the deeper question his work raises about what the arrow of time actually is. We feel time as directional — the past is fixed, the future is open, entropy increases — but Boltzmann showed that this asymmetry is not written into the laws of physics. It is written into the initial conditions of the universe. This is a strange conclusion. It means that the felt asymmetry between before and after, the irreversibility we experience as the most basic feature of time, is a consequence of the peculiar low-entropy state in which the universe began, 13.8 billion years ago. Physics is symmetric. We are not. The directionality of everything — memory, causation, the fact that we can remember the past but not the future — traces back to that single extraordinary initial condition.
Boltzmann spent his professional life trying to persuade people that matter was made of atoms and that the second law was a probabilistic theorem. Both things are now taught in introductory courses. His equation is on his tombstone. The atoms are not in doubt. He died in September, the summer nearly over, in a village on the Adriatic, on what might have been any morning, while his wife and daughter swam below.