Standfirst: Ludwig Boltzmann explained why heat flows one way by counting arrangements of things too small to see. The counting works. What it leaves unexplained is still an open question.
Toss a hundred coins. There is exactly one way for all hundred to land heads. There are about a hundred billion billion billion ways, a number thirty digits long, for exactly fifty to land heads, because any fifty of the hundred can be the ones that do. No coin aims at fifty. Each lands heads or tails at even odds and has no connection to the others. The fifty-fifty result turns up because nearly every way the coins can fall looks like it. The steadiness is a property of the count, and no single coin has it. The count also needs numbers to work with. Toss four coins and all four come up heads about one time in sixteen. Toss a hundred once a second, and the wait for all heads would run about three trillion times longer than the universe has existed. [Observed · arithmetic: 100 choose 50 ≈ 1.01 × 10²⁹; 2¹⁰⁰ seconds ≈ 4 × 10²² years; universe ≈ 1.38 × 10¹⁰ years]
Ludwig Boltzmann applied that kind of counting to air. A room holds something like a billion billion billion molecules, each moving and colliding too fast and too often to follow. What can be asked is how many arrangements of all those positions and speeds fit what you can actually measure, the temperature and the pressure. Air spread evenly through the room at one temperature fits almost all of them. Air gathered into one corner fits almost none. So air spreads and does not gather, and heat runs from a hot cup into a cool room and not back. In a paper of 1877 Boltzmann made the count into a definition. Entropy, the quantity that the second law of thermodynamics says never falls in a sealed system, became a measure of how many hidden arrangements fit the state you can see. The formula carved on his headstone in Vienna, S = k log W, says this in symbols: S is entropy, W the number of arrangements, log a way of taming an enormous count, and k a constant that converts the result into the units of heat. He never wrote it in that form himself. [Sourced · Uffink, "Boltzmann's Work in Statistical Physics," Stanford Encyclopedia of Philosophy (SEP); molecule count is the editor's estimate for a room of about 40 cubic meters]
He was born in Vienna on 20 February 1844, took his doctorate there in 1866 under Josef Stefan, and held chairs in Graz, Munich, Vienna and Leipzig before returning to Vienna for good. In 1872 he published what he took to be a proof that collisions alone must drive any gas toward equilibrium, the state in which nothing more changes on average. He called the result the H-theorem. It leaned on an assumption about how molecules collide, one he did not stress, and in the 1890s critics showed that the assumption already had a direction of time built into it. [Sourced · New World Encyclopedia, "Ludwig Boltzmann" (dates, posts); SEP (1872 paper; Burbury and Bryan on the collision assumption, 1894–95)]
The sharper objection had come earlier, from his Vienna colleague Josef Loschmidt in 1876. The laws of motion run equally well backward and forward. Reverse the velocity of every molecule in a gas that has mixed, and it will retrace its path and unmix. No argument from motion alone, then, can show that a gas must mix. Boltzmann's reply in 1877 conceded the point and changed the claim. Unmixing is possible, he wrote, but the starting states that lead to it are so rare that they can be set aside. The one-way flow of heat became a matter of overwhelming probability rather than strict law. The SEP entry is not fully satisfied with either of his 1877 answers, and notes that the counting paper does not actually prove the tendency it describes. [Sourced · SEP]
In 1896 the mathematician Ernst Zermelo pressed from another side. A theorem of Henri Poincaré's says that a closed mechanical system will, given enough time, return as close as you like to where it started. A gas that mixes must eventually unmix, so thermodynamics and the theory of gases could not both be literally true. Boltzmann answered with sarcasm, and with the point that the wait would dwarf any span of time anyone could observe. [Sourced · SEP (Zermelo's claim; the sarcastic reply) · Poincaré's theorem and the content of the reply to verify, see notes]
Behind these exchanges sat a larger quarrel about whether atoms exist at all. Ernst Mach held that science should deal only in what can be observed, and atoms then could not be. At a meeting in Lübeck in 1895, Wilhelm Ostwald and Georg Helm argued that energy, not matter, was the basic reality, and Boltzmann defended atoms against them. After a lecture Boltzmann gave at the Imperial Academy of Sciences in Vienna in 1897, Mach is reported to have said, "I don't believe that atoms exist!" The question was settled by others. Albert Einstein's 1905 account of Brownian motion, the ceaseless jitter of small particles suspended in water, explained it as the result of molecules striking them, and Jean Perrin's experiments from 1908 confirmed the account closely enough to persuade the doubters. Perrin received the Nobel Prize for the work in 1926. [Sourced · New World Encyclopedia (Lübeck, Mach); APS News, "This Month in Physics History" (Einstein, Perrin) · Mach quote reported on Wikipedia without a named source, to verify]
On 5 September 1906, on a summer holiday at Duino, near Trieste, with his wife and youngest daughter, Boltzmann took his own life. He was sixty-two. The version usually told makes him a martyr to atoms, worn down by his opponents just before experiment proved him right. It is a better story than it is history. The SEP entry notes that his death is often linked to how colleagues treated him and attributes it instead to declining mental and physical health. Lise Meitner, who studied under him, said that those close to him knew of his long struggle with depression. The tidy version turns a man's illness into a parable about ideas. [Sourced · SEP; New World Encyclopedia (Duino, family, Meitner)]
The strongest objection to his account is one he saw himself. Counting explains why a gas that starts unmixed will mix. It does not explain why the gas started unmixed. Run the same reasoning backward: given a half-mixed room now, the overwhelmingly likely past is a more mixed room, because almost every arrangement is a mixed one. Counting alone predicts entropy rising toward the past as well as toward the future, which is not the world anyone remembers.
In a letter to Nature in 1895, Boltzmann floated an answer he credited to his former assistant, Ignaz Schütz. Suppose the universe as a whole rests in equilibrium forever, and our region is a rare fluctuation within it, "so far from thermal equilibrium that we cannot imagine the improbability of such a state." The idea fails on its own arithmetic. Small fluctuations are vastly more likely than large ones. If order arises by chance out of equilibrium, the likeliest fluctuation that contains anything like you is a single brain, assembled for a moment, holding false memories of a universe. A whole ordered universe is far less likely. In 2004 the physicists Andreas Albrecht and Lorenzo Sorbo named these hypothetical minds Boltzmann brains. The philosopher of science John Norton calls the scenario self-defeating: if it were true, the evidence and the physics that predict it would be fabrications as well. [Sourced · Boltzmann, Nature 51:413–415 (1895), as quoted in Bettini, "Anthropic Reasoning in Cosmology: A Historical Perspective"; Norton, "You are not a Boltzmann brain"]
The usual answer now is to add an assumption instead of deriving one. The universe began in a state of extremely low entropy, and everything since is the long climb away from it. The philosopher David Albert, in Time and Chance (2000), calls this the past hypothesis. It works, and it leaves the original puzzle where Boltzmann found it, since nothing in the counting explains why the beginning was so special. [Sourced · SEP, "Chance versus Randomness," supplement on initial conditions (Albert 2000)]
What Boltzmann left is a way of seeing regularity that belongs to a crowd and to no member of it. The even warmth of a room, the cooling of a cup and the one-way flow of heat are facts about counts, and counts need large numbers to hold: four coins swing, and a room of air does not. His statistics explain why the future of a mixed room stays mixed. They do not explain why its past was not. The term to search is the past hypothesis.
Sources- Uffink, "Boltzmann's Work in Statistical Physics," Stanford Encyclopedia of Philosophy — https://plato.stanford.edu/entries/statphys-Boltzmann/- "Ludwig Boltzmann," New World Encyclopedia — https://www.newworldencyclopedia.org/entry/Ludwig_Boltzmann- Bettini, "Anthropic Reasoning in Cosmology: A Historical Perspective" (quotes Boltzmann, Nature 51, 1895) — https://arxiv.org/pdf/physics/0410144- Boltzmann, "On Certain Questions of the Theory of Gases," Nature 51, 413–415 (1895) — https://www.nature.com/articles/051413b0- Norton, "You are not a Boltzmann brain" — https://sites.pitt.edu/~jdnorton/Goodies/Boltzmann_Brain/Boltzmann_Brain.html- SEP, "Chance versus Randomness," supplement on initial conditions — https://plato.stanford.edu/ENTRIES/chance-randomness/initial-conditions.html- APS News, "This Month in Physics History: Einstein and Brownian motion" — https://www.aps.org/publications/apsnews/200502/history.cfm- Ernst Mach (Wikipedia, via Wikiwand) — https://wikiwand.com/en/Ernst_Mach