Sadi’s pamphlet finds its way into the hands of a fierce-eyed, austere Prussian professor called Rudolf Clausius. It is he who grasps the fundamental issue at stake, formulating a law that was destined to become famous: if nothing else around it changes, heat cannot pass from a cold body to a hot one.
The crucial point here is the difference from what happens with falling bodies: a ball may fall, but it can also come back up, by rebounding, for instance. Heat cannot.
This is the only basic law of physics that distinguishes the past from the future.
None of the others do so. Not Newton’s laws governing the mechanics of the world; not the equations for electricity and magnetism formulated by Maxwell. Not Einstein’s on relativistic gravity, nor those of quantum mechanics devised by Heisenberg, Schrödinger, and Dirac. Not those for elementary particles formulated by twentieth-century physicists. . . . Not one of these equations distinguishes the past from the future.11 If a sequence of events is allowed by these equations, so is the same sequence run backward in time.12 In the elementary equations of the world,13 the arrow of time appears only where there is heat.* The link between time and heat is therefore fundamentaclass="underline" every time a difference is manifested between the past and the future, heat is involved. In every sequence of events that becomes absurd if projected backward, there is something that is heating up.
If I watch a film that shows a ball rolling, I cannot tell if the film is being projected correctly or in reverse. But if the ball stops, I know that it is being run properly; run backward, it would show an implausible event: a ball starting to move by itself. The ball’s slowing down and coming to rest are due to friction, and friction produces heat. Only where there is heat is there a distinction between past and future. Thoughts, for instance, unfold from the past to the future, not vice versa—and, in fact, thinking produces heat in our heads. . . .
Clausius introduces a quantity that measures this irreversible progress of heat in only one direction and, since he was a cultivated German, he gives it a name taken from ancient Greek—entropy:
I prefer to take the names of important scientific quantities from ancient languages, so that they may be the same in all the living languages. I therefore propose to call entropy the quantity (S) of a body, from the Greek word for transformation: ἡ τροπὴ.14
The page of the article by Clausius in which he introduces for the first time the concept and the word “entropy.” The equation provides the mathematical definition of the variation of entropy (S—S0) of a body: the sum (integral) of the quantity of heat dQ leaving the body at the temperature T.
Clausius’s entropy, indicated by the letter S, is a measurable and calculable quantity 15 that increases or remains the same but never decreases, in an isolated process. In order to indicate that it never decreases, we write:
ΔS ≥ 0
This reads: “Delta S is always greater than or equal to zero,” and we call this “the second principle of thermodynamics” (the first being the conservation of energy). Its nub is the fact that heat passes only from hot bodies to cold, never the other way around.
Forgive me for the equation—it’s the only one in the book. It is the equation for time’s arrow, and I could hardly refrain from including it in my book about time.
It is the only equation of fundamental physics that knows any difference between past and future. The only one that speaks of the flowing of time. Behind this unusual equation, an entire world lies hidden.
Revealing it will fall to an unfortunate and engaging Austrian, the grandson of a watchmaker, a tragic and romantic figure, Ludwig Boltzmann.
BLUR
It is Boltzmann who begins to see what lies behind the equation ΔS ≥ 0, throwing us into one of our most dizzying dives toward understanding the intimate grammar of our world.
Boltzmann works in Graz, Heidelberg, Berlin, Vienna, and then in Graz again. He liked to attribute his restlessness to the fact that he was born during Mardi Gras. He was only partly joking, since the instability of his character was real enough, oscillating as it did between elation and depression. He was short and stout, with dark, curly hair and the beard of a Taliban; his girlfriend called him “my dear sweet chubby one.” It was he, this Ludwig, who was the luckless hero of time’s directionality.
Sadi Carnot thought that heat was a substance, a fluid. He was wrong. Heat is the microscopic agitation of molecules. Hot tea is tea in which the molecules are very agitated. Cold tea is tea in which the molecules are only a little agitated. In an ice cube, warming up and melting molecules become increasingly agitated and lose their strict connections.
At the end of the nineteenth century, there were many who still did not believe in the existence of molecules and atoms: Ludwig was convinced of their reality and entered the fray on behalf of his belief. His diatribes against those who doubted the reality of atoms became legendary. “Our generation were at heart all on his side,” remarked one of the young lions of quantum mechanics years later.16 In one of these fiery polemics, at a conference in Vienna, a noted scientist17 maintained against him that scientific materialism was dead because the laws of matter are not subject to the directionality of time. Scientists are not immune from talking nonsense.
Looking at the sun going down, the eyes of Copernicus had seen the world turning. Looking at a glass of still water, the eyes of Boltzmann saw atoms and molecules frenziedly moving.
We see the water in a glass like the astronauts saw the Earth from the moon: calm, gleaming, blue. From the moon, they could see nothing of the exuberant agitation of life on Earth, its plants and animals, desires and despairs. Only a veined blue ball. Within the reflections in a glass of water, there is an analogous tumultuous life, made up of the activities of myriads of molecules—many more than there are living beings on Earth.
This tumult stirs up everything. If one section of the molecules is still, it becomes stirred up by the frenzy of neighboring ones that set them in motion, too: the agitation spreads, the molecules bump into and shove each other. In this way, cold things are heated in contact with hot ones: their molecules become jostled by hot ones and pushed into ferment. That is, they heat up.
Thermal agitation is like a continual shuffling of a pack of cards: if the cards are in order, the shuffling disorders them. In this way, heat passes from hot to cold, and not vice versa: by shuffling, by the natural disordering of everything. The growth of entropy is nothing other than the ubiquitous and familiar natural increase of disorder.
This is what Boltzmann understood. The difference between past and future does not lie in the elementary laws of motion; it does not reside in the deep grammar of nature. It is the natural disordering that leads to gradually less particular, less special situations.