Emergence of temperature

Why does a single atom have no temperature at all when a jar of them plainly does?

A jar of air has a temperature. You can measure it, and everyone agrees on the number. The jar contains nothing but atoms, and the atoms are all there is. So it seems to follow that the temperature must somehow be distributed among them — each atom carrying its own small share, the way each atom carries its own small share of the jar's mass.

Try to collect that share and it is not there. Ask a physicist for the temperature of one argon atom moving at 400 metres per second and you will not be given a small number. You will be told the question is malformed. Mass is additive and temperature is not, even though both are properties of the same collection of the same atoms. What kind of property behaves that way?

Begin with the tempting answer. Kinetic theory says the average translational kinetic energy of a gas particle is three-halves of Boltzmann's constant times the temperature. So can we not just invert it — take one atom's kinetic energy and solve for its temperature?

Look carefully at what the formula actually contains. The word "average" is not decoration. It is the whole content. An average over one thing is just that thing, so the inversion would assign a different "temperature" to every atom in the jar, changing millisecond by millisecond as each collides. Nothing would be shared, nothing would equalise, and a thermometer would read whatever it last happened to be struck by. That is not the quantity we started with.

So the formula does not describe an atom. It describes a distribution of atoms, and reports one number that characterises the spread. Which points at the real definition.

The atoms in the jar do not all move at the same speed. Their speeds are spread out, and at equilibrium that spread takes a particular shape — the Maxwell-Boltzmann distribution — whose entire form is fixed by a single parameter. Temperature is that parameter. It is not the energy of any atom; it is the shape of how energy is shared across all of them. One atom does not have a shape. It has an energy.

This also explains what temperature does. Bring two jars into contact and energy flows until the two distributions take the same shape; that shared parameter is what makes thermal equilibrium meaningful, and what makes the zeroth law true rather than accidental. Nothing about a single atom's energy would give you that.

Now push to the deepest version, because it is the one that shows temperature is not merely an average dressed up. The thermodynamic definition is that the reciprocal of temperature equals the rate of change of entropy with energy. Entropy is a count of how many microscopic arrangements are consistent with what you specified macroscopically. So temperature answers the question: if I add a small amount of energy to this system, by how much does its number of available arrangements grow?

Ask that of a single atom. How many arrangements are consistent with "this atom has exactly this energy"? Essentially one. There is no counting to do, therefore no entropy that varies with energy, therefore no derivative — and therefore no temperature. Not a temperature too small to measure. No such quantity.

That is what makes temperature emergent in the strict sense: it is not a property the parts have in smaller amounts, and not a property that appears because something new was added. It is a property of the statistics of the parts, and statistics need a population before they mean anything at all.

Three qualifications keep this from being too tidy.

First, how many is "many"? There is no threshold at which temperature switches on. What happens is that the relative fluctuation in a system's energy scales roughly as one over the square root of the number of particles, so a jar of a billion billion atoms has a temperature defined to absurd precision, while a cluster of ten atoms has one so noisy that quoting it is nearly meaningless. Temperature does not appear; it sharpens. The concept in between is genuinely fuzzy, and thermodynamics of small systems is an active field rather than a settled one.

Second, there is an important escape hatch. A single trapped ion is routinely described as having a temperature — laser cooling brings ions to microkelvin. Is that a contradiction? No, because the distribution is gathered over time instead of over particles: measure the same ion repeatedly and its energies trace out a distribution with the same characteristic shape, and that distribution has a parameter. The requirement was never "many particles" but "enough samples to define a distribution". Whether a time-based and a population-based average must agree is the ergodic question, and it is not free — there are real systems where they do not.

Third, be careful with the phrase "temperature is average kinetic energy". It is a special case, true for the translational motion of an ideal gas, and it fails elsewhere. In a solid much of the energy is potential, in a molecule it hides in rotation and vibration, and in a system of spins with a bounded energy ceiling you can even reach negative absolute temperature — a state hotter than infinity, where adding energy reduces the number of arrangements. That last case is only intelligible on the entropy-derivative definition; on the average-energy definition it is nonsense. Which of the two is the real definition is settled by the case where they disagree.

• Negative absolute temperature in bounded-energy spin systems, and why it is hotter than infinite temperature rather than colder than zero.
• Thermodynamics of small systems, where fluctuation theorems replace the sharp laws that large numbers supply.
• Whether other familiar quantities — pressure, viscosity, phase — are emergent in the same strict sense.

Every term the collection defines is gathered in the glossary.