What Is Entropy? Why Things Fall Apart and Time Runs One Way

Entropy measures how many ways a system's parts could be arranged while leaving it looking the same from outside. A tidy desk has few arrangements that count as tidy and enormously many that count as messy, so if the papers are moved at random, the desk becomes messy, not because mess is a force but because there are more ways to be messy. Ludwig Boltzmann put the idea into an equation in the 1870s: entropy is a constant times the logarithm of the number of microscopic arrangements consistent with the macroscopic state, and it is carved on his gravestone in Vienna.

The ink in water is the same case. There are a few arrangements of the ink molecules in which they are all in one corner and vastly more in which they are spread throughout the glass, so random jostling takes them from the first to the second and effectively never back. Nothing pulls the ink apart. It is simply that of all the places each molecule could wander to, almost all are somewhere other than the drop.

The second law of thermodynamics says that the entropy of an isolated system never decreases. It was found before anyone knew what entropy counted, by engineers in the 1820s to 1860s studying steam engines, who noticed that heat always flows from hot to cold and that no engine could turn heat entirely into work. Rudolf Clausius coined the word in 1865 from the Greek for transformation, and stated the law in its classic form: the energy of the universe is constant, and the entropy of the universe tends to a maximum. Boltzmann's counting later explained why. Heat flowing from hot to cold spreads energy over more molecules in more ways; the reverse would concentrate it, which is possible but so improbable that it will not happen in the age of the universe.

The law is statistical, not absolute. A few molecules can, and do, briefly gather in a corner. For the trillions of trillions in a glass the odds against a visible fluctuation are so long that the law is, in practice, never broken, and it is the only fundamental law of physics with a direction built into it: every other equation works the same run backwards. The arrow of time, the fact that we remember the past and not the future, is the second law seen from inside.

The law does not forbid order; it forbids order for free. A refrigerator makes its inside colder, lowering the entropy there, and pays for it by dumping more heat into the kitchen, so that the total rises. Living things build the most ordered structures known by taking in low-entropy energy as food or sunlight and giving out high-entropy heat and waste; the Earth as a whole runs on sunlight arriving as a few high-energy photons and leaving as many low-energy ones, and the difference is what drives weather, life and everything else that happens here. Some consequences of the second law:

• •No engine can convert heat entirely into work, and the maximum efficiency depends only on the temperatures of the hot and cold sides
• •A perpetual motion machine is impossible, not because energy is lost but because usable energy is
• •Every real process, from a bouncing ball to a computation, dissipates some energy as heat and cannot be exactly undone
• •Information is physical: erasing a bit of memory must release a minimum amount of heat, a result proved in 1961 and confirmed in the laboratory

In 1948 Claude Shannon, working out how much a telegraph line could carry, arrived at an expression for the uncertainty in a message that had exactly Boltzmann's form, and on the advice of John von Neumann called it entropy, partly because nobody understood what entropy was and so he would win every argument. The connection is real. A message that could be many things has high entropy and needs many bits; a gas whose molecules could be in many arrangements has high entropy and would need many bits to describe. The compression that shrinks a photo, the impossibility of a perfect data channel and the heat that limits how small a computer chip can go are all the same principle.

Taken to its end, the second law predicts a universe in which all energy has spread out evenly, no temperature differences remain, and nothing further can happen, a state the Victorians called the heat death. It is a very long way off, and the picture has been complicated by the discovery that the universe is expanding ever faster and by unresolved questions about the entropy of gravity and black holes, which turn out to hold more entropy than anything else. But the direction is not in doubt. The stars are burning down, and the reason anything is happening at all is that the universe began in a state of very low entropy and has been running downhill from it ever since; why it began that way is one of the deepest open questions in physics.

Entropy is often summed up as disorder, and that shortcut causes confusion, because plenty of things become more orderly without anyone pushing them. Shake oil and water and they separate into neat layers. Let a warm salt solution cool and cubic crystals grow out of it. Water freezes into a regular lattice in the freezer. None of these breaks the second law, because the law is about the total entropy of the system and everything it exchanges energy with, not about how tidy one part looks to your eye.

In each case the visible order is paid for elsewhere. When crystals form or water freezes, heat is released into the surroundings, and spreading that energy through the air adds more arrangements than the crystal takes away. Oil and water separate for a subtler reason: oil molecules force the water around them into constrained cage-like arrangements, and pushing the oil together frees many water molecules to move more randomly. The layers look neater, but the water has gained far more freedom than the oil has lost. Counting the microscopic arrangements, not judging appearance, is the only reliable test.

You can watch the arithmetic of entropy with a handful of coins. Flip four and there are sixteen possible outcomes. Only one of them is all heads, while six give two heads and two tails, so a mixed result is six times as likely as the tidy one. Yet all heads still turns up about once in every sixteen throws. With so few parts, the ordered state is not rare at all, and a small system will often wander into it and out again. That is exactly what it means to say the second law is statistical rather than absolute.

Now flip a hundred coins. There are about 1.3 million trillion trillion possible outcomes, roughly 8 percent of them are an exact fifty-fifty split, and about 96 percent land somewhere between forty and sixty heads. All heads is still just one outcome. If you had flipped a hundred coins every second since the Big Bang, the chance that you would ever have seen all heads would be less than one in a trillion. A glass of water holds around ten trillion trillion molecules, so the odds against them gathering on one side of the glass by chance are beyond any meaningful comparison.

In the early 1970s the physicist Jacob Bekenstein argued that black holes must have entropy, because otherwise throwing a hot, disordered object into one would make its entropy simply vanish from the universe, breaking the second law. He proposed that a black hole's entropy is proportional to the area of its event horizon, the boundary from which nothing escapes.

Stephen Hawking at first resisted the idea, then showed in 1974 that black holes should give off a faint glow of radiation, giving them a temperature and confirming that the entropy was real. A black hole turns out to hold an enormous amount of entropy for its size. The result connected gravity, heat and quantum theory, and it remains one of the central clues in the search for a deeper theory of physics.

Entropy counts the number of microscopic arrangements that look the same from outside, and the second law of thermodynamics says it never falls in an isolated system, because there are always far more ways to be disordered than ordered. That is why heat flows from hot to cold, why no engine is perfectly efficient, why order must be paid for with waste heat, why information has a physical cost, and why time has a direction at all.