What is entropy? Why things fall apart, in plain words

Entropy is a measure of how many different ways the tiny parts of a system, such as its atoms, can be arranged while the system looks the same from the outside. A state that can happen in more ways has higher entropy and is more likely. Because of that, the total entropy of an isolated system almost never decreases, a rule called the second law of thermodynamics. It explains why heat spreads, why ice melts in a warm room and why the past differs from the future.

In short

• Entropy counts arrangements: high entropy means many hidden arrangements give the same visible result.
• Things drift toward higher entropy not because of a force, but because those states vastly outnumber the rest.
• "Disorder" is a loose shorthand; the precise idea is about energy spread over more possibilities.

What does entropy actually measure?

Start with something small enough to count. Toss 4 coins. The result you notice, how many came up heads, is called the macrostate. The exact pattern, which coin is which, is the microstate. Some macrostates can happen in more ways than others:

Heads out of 4	Ways it can happen	Chance
0	1	6%
1	4	25%
2	6	38%
3	4	25%
4	1	6%

There are 16 patterns in all, and 6 of them give two heads, so "half heads" is the most likely result. That count of ways is what entropy measures. In 1877 the Austrian physicist Ludwig Boltzmann tied entropy to this count, in a formula later written as S = k log W, where W is the number of microstates and k is a tiny number now called the Boltzmann constant, exactly 1.380649 × 10⁻²³ joules per kelvin by NIST's listing since 2019. The formula is carved on his tombstone in Vienna.

Now scale up. With 100 coins, there are about 1.3 × 10³⁰ patterns. Exactly 50 heads can happen in about 10²⁹ of them. All 100 heads can happen in just one. The chance of tossing 100 heads at once is about 1 in 10³⁰, and between 40 and 60 heads turn up about 96% of the time. A glass of water holds around 10²⁵ molecules, not 100 coins, so the most likely states outnumber the unlikely ones by margins too vast to picture. That is why the second law looks like an iron rule, although it is really a matter of overwhelming odds.

What is the second law of thermodynamics?

The second law says that the total entropy of an isolated system, one that swaps neither energy nor matter with anything else, tends to increase until it reaches a maximum, and essentially never goes down. The word entropy itself was coined in 1865 by the German physicist Rudolf Clausius, from the Greek for transformation, as the Stanford Encyclopedia of Philosophy recounts. Clausius defined a change in entropy from heat: the heat a body gains, divided by its temperature in kelvins.

That definition lets you check the law with a kitchen example. Drop a 10-gram ice cube at 0 °C into a room at 20 °C and let it melt. Melting ice takes about 334 joules per gram, so the cube draws 3,340 joules from the room.

1. The ice gains entropy: 3,340 J ÷ 273.15 K = +12.23 joules per kelvin.
2. The room loses entropy: it gives up the same heat at a higher temperature, 3,340 J ÷ 293.15 K = −11.39 joules per kelvin.
3. The total rises: 12.23 − 11.39 = +0.84 joules per kelvin.

Heat flowing from warm to cold always comes out positive like this, because the same heat counts for more at the lower temperature. Run it backward, with the meltwater refreezing by pushing its heat into the warmer room, and the total would fall by 0.84. Nothing forbids it outright; it's just so improbable it never happens. A freezer can make ice only by using energy and dumping even more heat out the back, raising entropy overall.

The first law of thermodynamics says energy is never created or destroyed. The second adds that energy, left to itself, spreads out, and spread-out energy can do less work. That is why no engine can turn heat into work with perfect efficiency, and why the laws of motion in Newton's laws run equally well forward and backward while the everyday world doesn't.

Is entropy the same as disorder?

Not quite, and the shorthand misleads. A shuffled deck of 52 cards can be in about 8 × 10⁶⁷ orders, and only one is new-deck order, so "a shuffled deck is more disordered" sounds like entropy. But the order of the cards isn't thermodynamic entropy at all; their atoms have essentially the same entropy in any order. The card example is a good picture of counting, not of heat.

Real cases often run against the word "disorder." Oil and water separate into two neat layers, yet the separation raises entropy, because water molecules have more freedom when they aren't forced to arrange themselves around oil. Crystals form from a cooling liquid, an obvious increase in order, while the heat they release raises the entropy of the surroundings by more. Living things build ordered bodies by taking in energy and releasing heat and waste, so the total still goes up. The chemist Frank Lambert argued in 2002 that chemistry teaching should drop "disorder" for "energy spreading," and a number of chemistry textbooks have since changed their wording. The precise version is the count of arrangements, or how widely energy is shared among them.

How does entropy give time a direction?

Most basic laws of physics work the same if you run time backward. A film of two billiard balls colliding looks fine in reverse. A film of an egg smashing doesn't, and entropy is the reason: the smashed egg can be arranged in vastly more ways than the whole one, so the forward direction is overwhelmingly likely. This is called the thermodynamic arrow of time, and memories, causes and aging all point the same way.

It raises a deeper question. If entropy rises toward the future, it must have been lower in the past, and lowest of all at the start of the universe. Cosmologists trace that start to the Big Bang, which seems to have begun in an extraordinarily low-entropy state. Taken far into the future, the same reasoning predicts a "heat death," a universe spread so thin and even that nothing more can happen.

Still argued

Why the early universe had such low entropy is unresolved. The philosopher David Albert named the assumption the Past Hypothesis in 2000 and argued that it explains the arrow of time, together with the laws of physics. Others, such as the philosopher Huw Price, argue that the low-entropy start itself needs an explanation; some, such as Craig Callender, doubt it needs one at all; and some physicists propose models in which it arises naturally. The Stanford Encyclopedia of Philosophy sets out these positions. The evidence fits a low-entropy start; it doesn't say why.

What does entropy mean in information theory?

In 1948, the engineer Claude Shannon defined a quantity for messages with the same mathematical form as Boltzmann's, and called it entropy. In his paper, A Mathematical Theory of Communication, it measures how unpredictable a message is, in bits. A fair coin toss carries 1 bit. A fair die roll carries log₂ 6, about 2.58 bits. A coin that lands heads 90% of the time carries only about 0.47 bits, because you can mostly guess it. Compression works by squeezing out predictable parts, so a file can't be shrunk below its Shannon entropy without losing information.

The two meanings connect: both count possibilities. Thermodynamic entropy is, in a sense, the information you'd need to pin down the exact microstate when you know only the macrostate. Wiping a bit of memory has a minimum heat cost for that reason, a result known as Landauer's principle. At the smallest scales, counting states properly needs quantum mechanics, which is what makes the count finite in the first place. Entropy is one of the few ideas in science that links steam engines, the shape of time and the bits in a phone.

Check yourself

• Without looking back, explain why 100 tossed coins almost never all land heads, using the idea of counting ways.
• Why does the melting ice cube raise the total entropy, even though the room's entropy falls?
• Why is "entropy is disorder" only a rough description?

Sources

• Thermodynamic asymmetry in time — Stanford Encyclopedia of Philosophy
• Philosophy of statistical mechanics — Stanford Encyclopedia of Philosophy
• Boltzmann constant, CODATA value — National Institute of Standards and Technology
• A Mathematical Theory of Communication — Claude E. Shannon, Bell System Technical Journal, 1948
• Entropy — Wikipedia