The count that decides which entropy is right

Hotter than any temperature there is took a collection of two-level units and pushed more than half of them into their upper state. Adding energy beyond that point reduces the number of ways of arranging it, so the entropy falls as the energy rises, and a temperature defined as the inverse of the slope of entropy against energy becomes negative. Negative temperatures, on that account, sit above every positive temperature: a system at a negative temperature gives heat to any system at a positive one.

Nuclear spins were put into such states in 1951, and in 2013 a gas of atoms in an optical lattice was prepared with a negative temperature for their motion. The following year the idea was attacked at its root. The entropy in that argument is Boltzmann’s, the logarithm of the number of states at the system’s energy. There is another, older definition, Gibbs’s, the logarithm of the number of states at or below that energy, and with Gibbs’s entropy the temperature is never negative. If Gibbs’s is the correct entropy, the argument said, negative temperatures were never real.

Both definitions are counts, and neither is obviously wrong. What can be asked is what a temperature is for, and whether the count of states itself, without any definition of entropy imposed on it, agrees with one of them.

Counting at an energy, and counting up to it #

The system is the simplest one with a bounded energy: $N$ independent units, each with two levels a fixed energy apart. The energy is set by how many units are excited, and the number of arrangements with $n$ excited is the binomial coefficient, the height of each bar in the figure. Entropy is a count takes the logarithm of that number, and that is Boltzmann’s entropy.

Gibbs’s entropy takes the logarithm of the running total — every arrangement with $n$ or fewer excited — the line in the figure. For a system whose energy has no ceiling, a gas or a solid, the states pile up so steeply towards higher energy that almost all the states below a given energy lie just below it, and the running total is dominated by its last term. The two logarithms then differ by a small amount that becomes negligible as the system grows.

A bounded spectrum breaks that dominance exactly at half filling. The bars start to fall there, but a running total cannot, so above half filling the line keeps rising while the bars shrink. By half filling the line already holds nearly six-tenths of all the states there are, and above it the line creeps towards all of them. The two definitions of entropy part company exactly where the bars turn over.

Two slopes, two signs #

A temperature is defined from an entropy by its slope: $1/T = dS/dE$. For a thousand units the two slopes agree to within a hundredth everywhere below forty per cent filling, and the agreement improves as the system grows. Above half filling they disagree about the sign. Boltzmann’s entropy is falling, so its inverse temperature is negative: at three-quarters filling it is $-1.097$ in units of the level spacing. Gibbs’s entropy is still rising, but only because the running total is gaining the last few states it has not yet counted, which are exponentially few; its slope is $3.0 \times 10^{-59}$, a temperature so high it has no physical meaning except that it is positive.

So at the same energy of the same system one definition says the temperature is negative, above infinity, and the other says it is a finite positive number, merely astronomically large. Both are well-defined. The question is which, if either, does the work a temperature is supposed to do.

What a temperature is for #

The zeroth law of thermodynamics is the statement that gives temperature its meaning: two systems at the same temperature, put in contact, exchange no heat on average, and if their temperatures differ, heat flows from the hotter to the colder. Any definition that deserves the name has to get that right. It is also the one thing that can be tested without choosing a definition first, because what happens when two systems are put in contact is decided by counting.

Two systems that can exchange energy but not particles share a fixed total. The combined system’s states are all equally likely, so the most probable division of the energy is the one with the most arrangements of the pair: the product of the two systems’ counts, maximised over how the energy is split. For large systems that maximum is so sharp that the energy goes there and stays, fluctuating by a negligible amount. The second law with a probability attached is the same statement from the other end: the direction energy flows is the direction that increases the number of arrangements, overwhelmingly.

Maximising a product of counts is maximising the sum of their logarithms, and the maximum of the sum is where the slopes of the two logarithms are equal. The logarithm of each count is Boltzmann’s entropy. So at the most probable split, the Boltzmann temperatures are equal — not as a definition but as a consequence of counting. Whether that settles the matter is what the next figure tests.

Equal Gibbs temperatures, and heat flows anyway #

The test is to prepare two systems with equal Gibbs temperatures, put them in contact, and see whether energy moves. The figure takes a thousand units with sixty per cent excited and searches for the number of excited units in a system of two hundred that gives the same Gibbs temperature: 145, which is seventy-two per cent. Both are above half filling, so both have enormous positive Gibbs temperatures, and by that definition they are in equilibrium.

The count disagrees. The pair has a few hundred times as many arrangements with twenty-one quanta moved from the smaller system to the larger as it has at the starting split. The starting split is not where the energy stays; it flows. It flows from the system whose Boltzmann inverse temperature is more negative, $-0.964$, to the one whose is less negative, $-0.405$ — from hotter to colder on Boltzmann’s ordering, in which negative temperatures lie above infinity and a more negative inverse temperature is hotter. And it stops where the two Boltzmann temperatures agree, to within 0.006.

The same counting settled where two ordinary finite bodies end up in the work left in two buckets of water, and the answer there needed no argument about definitions, because below half filling there is none to have. What the contact figure adds is the case where the definitions disagree, and the count does not.

For heat flow between large systems, the count of states sides with Boltzmann. Two systems with equal Gibbs temperatures are not in equilibrium; two with equal Boltzmann temperatures are. A definition of temperature that assigns equal temperatures to systems that exchange heat does not do what the zeroth law requires of a temperature, and Gibbs’s does not, above half filling. Below half filling the two definitions agree and so does the test.

How a negative temperature is made #

A system cannot be cooled or warmed into a negative temperature by contact with anything at a positive one, because contact always drives it towards the positive temperature. The upper half of a bounded spectrum has to be reached by a trick, and the trick has always been to invert the system faster than it can exchange energy with its surroundings.

In 1951 Edward Purcell and Robert Pound did it with the nuclear spins of lithium fluoride. The spins came to equilibrium with each other in a fraction of a second but with the crystal lattice only over minutes, so after aligning them in a magnetic field the experimenters reversed the field faster than the spins could follow. The spins that had been in the lower energy state were now in the upper one, more than half the population was excited, and for a few minutes the spin system radiated energy instead of absorbing it, before relaxing back through infinite temperature to positive ones. It is the same spin system the staircase that never reaches the floor uses to reach very low temperatures, run in the other direction.

The 2013 experiment did it for motion, which is harder because kinetic energy normally has no ceiling. Potassium atoms in an optical lattice were confined to its lowest band, whose width caps the kinetic energy, and their interactions and the trapping potential were switched to be bounded above as well; the atoms then gathered in the highest-energy states of the band, the signature of a negative temperature for their motion.

A laser runs on the same idea without calling it a temperature. Lasing needs more atoms in the upper of two levels than in the lower, a population inversion, and the Boltzmann factor that sets the ratio of the two populations at equilibrium can give that ratio only if the temperature in it is negative. The inverted atoms in a laser’s gain medium are, for that pair of levels, a system at a negative temperature, kept there by the pump against the relaxation that would restore it; the coherent light a laser produces is energy flowing out of a system hotter than infinity.

Where the choice stops mattering #

Outside the upper half of a bounded spectrum the argument is academic for any large system. The figure computes the difference between the two inverse temperatures for systems from ten to three thousand units, both for two-level units a quarter excited and for harmonic oscillators, whose energy has no ceiling at all. In both cases the difference falls in proportion to one over the number of units. A gram of anything has around $10^{22}$ of them, and the difference is a part in $10^{22}$.

The dispute is therefore confined to two places: very small systems, where one over the size is not small, and bounded spectra above half filling, where the definitions disagree in sign whatever the size. The second is exactly where negative temperatures live, which is why the argument about them is also an argument about which entropy is correct.

A scale that runs through infinity #

The ordering of temperatures that the contact test confirms is strange only when written in the usual way. From coldest to hottest it runs up from just above zero through room temperature to arbitrarily large positive values, jumps to arbitrarily large negative values, and continues up towards just below zero, which is the hottest of all. In the inverse temperature, which is what the slope of entropy actually gives, there is no jump: it runs smoothly from large positive values, through zero, to large negative ones. Zero inverse temperature, infinite temperature, is simply the point where the entropy is at its maximum and adding energy changes it not at all. Temperature is the awkward variable; its inverse is the natural one, and in it a negative temperature is nothing more than a slope that points down.

That reading also says what goes wrong with an engine. The ceiling on every engine gives the efficiency as one minus the ratio of the two reservoirs’ temperatures, and with the hot reservoir at a negative temperature that formula exceeds one. On Boltzmann’s reading it does so because heat can be drawn from both reservoirs at once and all of it turned into work, with the entropy lost by the inverted reservoir paying for the entropy the cold one would otherwise have to receive. On Gibbs’s reading the formula has been applied outside its domain and the number means nothing. The disagreement over how to count efficiency in that case is one of the places the argument is still live.

What Gibbs’s entropy does better #

The figures do not show that Gibbs’s definition is simply wrong, and the case for it is not frivolous. It satisfies several relations exactly that Boltzmann’s satisfies only approximately, for systems of any size.

The first is equipartition. For a classical system, the Gibbs temperature is exactly related to the average of each quadratic contribution to the energy, the familiar half $kT$ per degree of freedom, for any number of particles; Boltzmann’s gives the same only in the limit of large systems. The second is invariance under slow changes. If a system’s parameters are altered slowly enough, the volume of phase space it occupies below its energy stays fixed — a result going back to Paul Hertz in 1910 — so Gibbs’s entropy is exactly unchanged by a reversible adiabatic process, as thermodynamics says an entropy should be, while Boltzmann’s is unchanged only approximately. For a small system these are real advantages, and a thermodynamics of small systems, which the engine a fluctuation cannot run and molecular machines need, has to decide which properties to keep.

What the contact test shows is narrower and harder to escape: the property Gibbs’s entropy gives up above half filling is the one that makes temperature a guide to heat flow. Different authors have weighed those trade-offs differently since 2014, and the weighing is not a calculation.

Where the model stops #

The units do not interact. Real spin systems interact, and the energy of a system with interactions is not simply a count of excitations; the qualitative picture survives, and the details of where the entropy peaks do not.

The contact is weak and slow. The counting argument assumes the two systems exchange energy through a coupling too weak to change either’s spectrum and slowly enough that each explores its states. A strong coupling makes the pair a single system with its own spectrum, and the question of two temperatures stops being well posed.

The systems are isolated. Everything here is microcanonical: a fixed energy, every state at that energy equally likely. A system in contact with a large reservoir at a positive temperature cannot sit above half filling at all, which is why negative temperatures are only ever prepared by isolating a system and inverting it quickly, and why they do not last.

And the energy comes in whole quanta. For a discrete spectrum the slope of an entropy is a finite difference, and for very small systems the choice of how to take it is itself a small ambiguity, of the same order as the difference between the definitions.

What the pictures cannot show #

The contact figure draws the logarithm of the number of arrangements, and a difference of under six in that logarithm looks modest on the page. It means the energy is found at the new split a few hundred times more often than at the old one, and for systems a hundred times larger the same shift in the logarithm would be a factor far too large to write down. The drawing cannot convey how sharp the maximum becomes for anything macroscopic.

Nor can any figure show what a temperature above infinity feels like, because no thermometer at a positive temperature can come to equilibrium with it. A probe placed in contact with a system at a negative temperature is heated, whatever its own temperature, and the reading it reaches depends on the probe as much as on the system.

Still open: which entropy a small or strange system should use #

The contact test decides heat flow between large systems, and there the physics is settled: energy flows as the count of states dictates, and Boltzmann’s temperature predicts it. The dispute that remains is about definitions for the cases where the test is weak — small systems, systems with few accessible levels, and engines that use a reservoir at negative temperature, for which a naive Carnot formula gives efficiencies above one and nobody agrees on what, if anything, that number means.

Proposals continue: entropies built to satisfy both sets of properties, restrictions to the conditions under which each definition is appropriate, and the view that thermodynamics simply has more than one consistent extension beyond its usual range. A law about spectra, not about heat found the third law to be a statement about the bottom of a spectrum; the negative-temperature argument is the corresponding question about the top, and it is not closed.

The habit worth keeping is the one the contact figure teaches. When two definitions disagree, find the job the quantity was invented to do, and see which definition does it. A temperature was invented to say which way heat flows, and on that job the count of states gives an answer that neither definition had to be assumed to reach.