Illustrated Physics
physics, drawn
Concept

Statistical temperature — where it appears

Temperature defined as the reciprocal of the slope of entropy against energy, rather than as what a thermometer reads. It is what decides which way heat flows, it can be negative for a system whose energy has a ceiling, and negative values sit above every positive one on that scale rather than below zero.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

The temperature that runs off the top of the scale. On the left, the entropy of a collection of two-level systems against how many of them are in the upper level. It rises, reaches 0.69314 per spin — which is ln 2, and is where half of them are up — and then falls, because a system with every spin up is as orderly as one with every spin down. On the right, the slope of that curve, which is one over the temperature. Below the maximum it is positive and ordinary: adding energy adds entropy, and the temperature is what everybody expects. At the maximum it is zero, which means the temperature is infinite. Past it the slope is negative — adding energy now removes entropy — and the temperature is negative. Such a system is not cold. It is hotter than any positive temperature whatever: put it in contact with anything at all and energy flows out of it, because that raises the total entropy. The quantity that orders systems by which way heat flows is not the temperature but its reciprocal, which runs smoothly from large and positive through zero to negative, and it is the temperature that has the discontinuity. None of this is possible unless the energy has a ceiling, which is why it happens in a spin system and not in anything that can move: a gas has no upper bound on its kinetic energy, so its entropy never turns over and its temperature is never negative.

Hotter than any temperature there is

A system whose energy has a ceiling can be pushed past the point where adding energy adds entropy. Its temperature is then negative — and negative temperatures are not cold. They sit above every positive temperature on the only scale that decides which way heat flows, and a working laser is at one.

thermodynamics · Third law The entropy each degree of freedom has not yet given up. The entropy carried by each kind of degree of freedom, drawn against the temperature at which it orders and hands that entropy over. Lattice vibrations freeze out around room temperature; electron spins in a paramagnetic salt order in the millikelvin range, which is what makes adiabatic demagnetisation work; and nuclear spins hold R ln(2I+1) — 11.5 joules per kelvin per mole for copper — down to some tens of nanokelvin, where their own dipolar interactions finally sort them out. A copper sample at a microkelvin therefore has a large entropy and violates nothing: its nuclear spin system has not reached its ground state, and the third law is a statement about ground states rather than about thermometers.

A law about spectra, not about heat

The third law is usually met as a statement about cooling. Its statistical form is a statement about a spectrum: the entropy of a system in its ground state is k ln g, and it vanishes only when the ground state is unique. Every apparent exception is a degeneracy or a system that never reached its ground state — and copper nuclei carry eleven joules per kelvin per mole down to a hundred nanokelvin without violating anything.

thermodynamics · Third law Two entropies that agree until half filling. The entropy per unit of 100 two-level units against the fraction excited, by Boltzmann's definition, the logarithm of the number of arrangements at that energy, and by Gibbs's, the logarithm of the number at or below it. Below half filling they nearly coincide: at a quarter excited they are 0.538 and 0.542 per unit, and both approach the dashed large-system curve. Boltzmann's entropy then turns over and falls back to zero when every unit is excited; at three-quarters it is 0.538. Gibbs's cannot fall, because a running total cannot, and it levels off at ln 2 = 0.693, reaching 0.693 at three-quarters. The slope of each is one over its temperature.

The count that decides which entropy is right

There are two ways to count the states of an isolated system: the states at its energy, which is Boltzmann's entropy, and the states at or below it, which is Gibbs's. For large systems in ordinary conditions they agree to the last measurable digit. For a system whose energy has a ceiling, past the halfway point, one gives negative temperatures and the other forbids them. Definitions cannot settle which is right, but a temperature is for something — saying which way heat will flow — and putting two such systems in contact lets the count of states answer.

thermodynamics · Third law

Named alongside it

The objects these essays reach for when they reach for this one.

EntropyBounded spectrumNegative temperatureBoltzmann entropyThe Boltzmann factorDegeneracyEnergy levelsEquilibriumGibbs entropyGround stateHeat flowLaser

All concepts