1In the city of Vienna, on a night in late winter, a son was born to a keeper of the emperor's tax rolls. It was the night between the last feast of carnival and the first grey morning of the fast. The boy was named Ludwig Boltzmann. When he was a grown man, he would say that this was the reason his spirits went up and down as they did, with never a level day between: he had been born between the dancing and the ashes.
2He was a round, short-sighted, curly-headed child, who loved the piano and collected beetles and came first in his class. When he was fifteen his father died. A year after, his brother died too. His mother spent all she had to keep him at his books. He never forgot what the books had cost.
3At the university he found a teacher named Josef Stefan, who had a rare way with students: he treated them as friends. One day Stefan put some printed pages in his hand. They were the papers of the Scotsman Maxwell, on the swarm of grains and how their speeds are shared. With them he gave the young man an English grammar. For Boltzmann did not know one word of English. He learned it in order to read those pages.
4There was another man in that small, poor laboratory, a quiet older one named Loschmidt. He had done a thing no one had done before. He had reckoned how big the grains must be. They were so small, by his reckoning, that a row of some millions of them would not cross a fingernail. The young student and the old one became friends. They would stand at the window arguing about grains that neither had seen, nor ever would.
5For you must know that this was a matter of belief, in those days. No one had seen a grain of matter. Many of the greatest scholars in the German lands held that there were none. They said the grains were a handy fiction, like the lines on a map, and that a careful man of science speaks only of what he can measure: heat, pressure, energy. Boltzmann believed in the grains as a sailor believes in the sea-bed. He could not see it. Everything he could see was resting on it.
6At twenty-five he was made professor in the mountain town of Graz. There, on an outing in the hills, he met a girl of nineteen named Henriette. She had long fair hair and no parents, and she was training to be a schoolteacher. She told him she wished she could understand what he worked at. He told her that she could, if she learned the mathematics.
7So she went to the university and asked to hear the lectures. No woman had asked before, and there was no rule against it, and for one term she sat at the back and listened. Then the professors met and made a rule. Henriette was shut out.
8She wrote to Boltzmann, who was then away in Vienna. He told her not to accept it, and to send a petition to the minister himself. She did. She won leave for one term more, and then the door was shut again, and again she knocked. It was in the midst of that fight that he wrote and asked her to marry him. It was a strange love-letter. He said he did not think love could last where a wife had no part in her husband's striving and was only his housekeeper and not his companion. She said yes. And all her life, when he had a new thought, she was the first to whom he tried to explain it.
9Now hear the thought, and what drove him to it.
10He had taken up the question that Clausius had left: why does heat fall one way? And he had first tried to answer it by force. He wrote a long proof that a swarm of flying grains, however they begin, must settle toward Maxwell's even sharing of speeds, and can never go back. He thought he had made the law of heat as certain as the law of the falling stone.
11Then his old friend Loschmidt, gently, put a knife into it.
12The grains, said Loschmidt, are like little balls. They fly and strike and bounce by the plain laws of Newton. And those laws do not care about before and after. Watch two balls strike and part; now picture it backward; the backward picture is just as lawful as the forward. So take your cup of tea that has gone cold. At one instant, turn every grain exactly about, so that each flies back the way it came. Every knock will be undone, in order. The heat will climb out of the room and gather itself into the cup again. Nothing in Newton forbids it. So how can you have proved that it must never be?
13Boltzmann saw that his friend was right. His proof could not stand as he had built it. For some months he was a wretched man. He walked up and down. He played the piano badly. Henriette, newly married, watched him and held her peace.
14Then one evening he came into the kitchen, where she was, and asked whether she had any coins in the house.
15She had ten small copper ones in a jar. He tipped them out on the table.
16"Suppose I throw these," he said, "and I tell you only how many fell heads. I do not tell you which. Now. In how many ways can they fall so that all ten are heads?"
17"One way," she said. "Every coin heads."
18"And nine heads, with one tail?"
19She thought. "Ten ways. The tail could be the first coin, or the second, and so on."
20"And five heads, with five tails?" He did not wait; he was already scratching on the back of a bill. "Two hundred and fifty-two. I throw the coins blindly. Each single way of falling is as likely as any other. All heads is no rarer than any one mixed fall you care to name. But there is only one all-heads. And there are two hundred and fifty-two ways to be half and half. So when I look at the table I shall nearly always see something near half and half. Not because the coins are made to mix. Only because there are more ways of being mixed than of being sorted."
21"That is ten coins," she said.
22"Yes. Now take a hundred. All heads: still one way. Half and half—" he worked it, and whistled. "A number with thirty figures. Throw a hundred coins once every second since the world began, and you would not yet have seen them all come down heads. Yet nothing forbids it. It is only one way, lost among more ways than there are drops in the sea."
23She was quick. "And the grains in a cup of tea are more than a hundred."
24"They are more than a hundred as the sea is more than a spoon." He sat down. He was sweating. "Listen. This is the answer to Loschmidt. He is right that the heat could climb back into the cup. Nothing forbids it. But count the ways. For the heat to be all gathered in the cup, the fast grains must all be here, and the slow all there. That can be done in some number of ways. For the heat to be shared out through the whole room, fast and slow all mingled, that can be done in a number of ways so much greater that there is no name for how much greater. The grains knock about blindly. They pass from one arrangement to another, with no aim. And nearly every arrangement there is, is a shared-out one. So that is where we find them."
25"The tea does not cool because it must," said Henriette slowly. "It cools because there are more ways to be cool."
26"More ways for the heat to be spread than for it to be gathered. Yes. And the turning-back that Loschmidt asks for would be like throwing all heads with more coins than there are stars. You may wait for it. You will wait longer than there has been time."
27He took the cup she had been drinking from and held it up. It was a plain white one with a blue rim.
28"In how many ways can these grains of clay be laid together, and still be a cup that holds tea? A great many, no doubt. But in how many ways can they lie on the floor in pieces? In more ways than I could count in a thousand lives. That is why a cup falls and breaks, and the pieces never leap up and join. Each is as lawful as the other. One is the whole table of coins, and the other is the single throw."
29"Do not show me," said his wife, and took the cup away from him.
30That year he wrote it down for the scholars. He showed that the number Clausius had found, the one that only ever grows, the one with the Greek name, was nothing mysterious. It was a count. The entropy of a thing is a measure of in how many ways its grains could be arranged and it would still look the same to us. A thing of low entropy is a rare arrangement: the hot cup in the cold room, the sorted deck, the whole vessel. A thing of high entropy is a common one. And the law that entropy grows is only this: left to themselves, things drift from the rare arrangements toward the common ones, because there are so many more of the common.
31The counts themselves are too long to write. A number with thirty figures is nothing; these have more figures than there are grains of sand. So he did what a sensible man does with such a number. He did not write the number. He wrote how long it was. A child knows that a million is a bigger thing than a hundred because it has more noughts; and it is easier to say "six noughts" than to hold a million in the mind. The mathematicians have a word for the length of a number, taken in that way. They call it its logarithm. The entropy, said Boltzmann, goes as the logarithm of the count.
32That was all. A boy with a jar of coins can follow it. And with it, the last wall between the great and the small fell down. The law of the engine, which Carnot found by thinking of mill-wheels, and Clausius named, and Maxwell teased with his doorkeeper, turned out to rest on nothing but this: there are more ways to be mixed than sorted, and the grains do not care.
33Now see what kind of law that makes it, for this troubled people more than anything.
34Every law before it had said must. The stone must fall. The planet must keep its path. This law said only nearly always. It was a law of chance, as sure as the sunrise but of another kind: sure in the way it is sure that a town of ten thousand will not all sneeze at the same instant. The old scholars disliked it deeply. A law of nature, they said, should not come out of a gambling-house. And it rested on the grains, which no one had seen. So they did not so much refute him as turn their chairs a little away.
35He minded that less, in those years, than he would later. They were his good years. He was back in Graz with a chair of his own. He and Henriette took a house on a hill outside the town, with a meadow. He bought a cow, so that his children should have fresh milk, and led her home through the streets himself, on a rope; then he went to the professor of animals to ask how one milked her. He skated on the pond with his children and came down heavily and often. In the evenings he played the piano and they sang. Students came from far countries to sit at his table.
36But Henriette had not forgotten the kitchen, and one night she asked him a thing he could not answer.
37They were sitting by the stove. Their first son, who was named Ludwig after him, was asleep on her knee. He was a year old, and warm, and every part of him was in its place: ten fingers, a small ear folded like a shell, a pulse going in the soft top of his head.
38"Tell me about your count again," she said. "Things go from the rare arrangements to the common ones."
39"Left to themselves. Yes."
40"Then tell me what this is." She did not lift her hand from the child's back. "This is not a common arrangement. I should think there is no rarer one in the world. And he was not here two years ago. And every day he is more himself, not less. So is a tree. So was that cow of yours, from a calf. You say the pieces of the cup never leap up and join. What have I got on my knee?"
41He opened his mouth to say that the child was not left to himself; that he ate. And he shut it again, because that was a word and not an answer. A stove eats too. What did the eating do? Why should bread go in at one end of a creature and order come out?
42"I do not know," he said. "The count is right. I am sure of the count. And the boy is here. Both are true, and I cannot yet lay one upon the other."
43"Then you are not finished," said Henriette, with some satisfaction. She stood up carefully, so as not to wake him, and carried the rare arrangement off to bed.
44He sat on by the stove a long while. The iron ticked as it cooled. Its heat was going out into the room, and from the room into the night over the hills, the one way, as it must; no, he thought, not as it must. As it nearly always would.
45He had the count. He had not yet seen what feeds on the falling. That came some years after. It came in his best hour, and it was followed hard by his worst.