We are all familiar with negative temperatures, those chilly winter days when the thermostat falls below zero. For my metric system readers this is a pretty frequent occurrence in winter in the Northern latitudes. For my American readers, a below zero temperature is a sign that going outside is not going to be fun. Celcius and Fahrenheit are both temperature scales that have negative value because they arbitrarily choose a place to put the zero temperature so that temperatures are convenient for day to day use.
The Kelvin scale, on the other hand, places zero not at some arbitrary point but at the point below which no temperature can be: Absolute Zero. The absolute zero temperature is the coldest possible temperature. How do we know this? In fact, absolute zero temperature has never been directly observed on Earth. The coldest temperatures ever achieved on Earth or in space are in experiments with dilute Bose-Einstein condensates, those mysterious states of matter that occur at millionths of a degree above absolute zero.
We know, however, that absolute zero exists because we know how much energy a substance has at a particular temperature and how much it must lose when it loses a degree of temperature Kelvin. If you extrapolate at what temperature a gas or substance would have zero internal energy, you will arrive at zero Kelvin.
At zero Kelvin, all motion of atoms would cease and they would become, theoretically, still.
Of course, such considerations did not stop physicists from asking: what happens if you have negative absolute temperature?
To even ask such as question, however, you need to understand temperature as more than just a measure of how much energy a substance has because nothing can have less than zero energy (at least classically). Instead, you have to look at temperature as a measure of the relationship between entropy, a measure of the disorder of a system, and energy.
This interpretation of temperature comes from two greats of the 19th century, Gibbs and Boltzmann, who developed the mechanical underpinnings of thermodynamics: statistical mechanics. Statistical mechanics describes precisely the relationship between systems of particles and their thermodynamic states such as temperature, pressure, energy, and entropy. With this tool, you can then talk about temperature on a more sophisticated level than just the energy of molecules.
It is generally true that with gases and other ordinary substances when you increase their energy, by, for example, heating them, you also increase their entropy. And from statistical mechanics, this is a rule: temperature is proportional to the change in entropy with energy. Since increasing one increases the other, you can assume that temperature is usually positive.
When it comes to the formation of certain types of crystals, however, this may not be the case. Crystals are solids that have a regular structure such as a triangular or square lattice. Since they are highly ordered, they have very low entropy.
When a crystalline lattice is under a great deal of tension, i.e., its constituent elements are all pulling away from one another and the force of that pull increases the closer the elements are to one another, it turns out that increasing its energy actually causes it to become more ordered. The more energy you add to the system, the tighter the crystal becomes. A crystal with smaller lattice spacing is more ordered than one with larger spacing, all other things being equal. Therefore, as I add energy to the system, the order increases.
The temperature of such systems is negative.
In 1972, Joyce and Montgomery demonstrated negative temperature theoretically in a 2D lattice with a Coulomb interaction (such as you have between electrons or vortex filaments). This kind of crystal arises from filaments of electron or ion plasmas which interact with one another like a two dimensional crystal. (The fancy term for the study of these charged fluids is magnetohydrodynamics.) The crystal is stable as long as the system is isolated from its environment, but, if it is allowed to exchange energy with its environment at all, it will fall apart.
My Ph.D. thesis involved studying a variant of these crystals. The crystals look something like this:
Each filament is repelling every other filament, but, because the system is isolated, energy has to be conserved, so they cannot move further apart without losing energy. Over time they will lose energy to the environment and expand, becoming increasingly disordered.
One of the great unknowns of physics, and the greatest question of classical physics is about turbulence. And I don’t mean quite what you experience on airplanes. Rather, I mean the random transition of fluids from a smooth flow to a chaotic one.
Structures like these are thought to play a role in turbulence where sudden transitions from positive to negative temperature states create bizarre effects. For example, in a negative temperature state, “cooling” causes disorder to increase rather than decrease, meaning that as these structures form and lose energy, they make the fluid disordered.
Experimental studies of negative temperatures go back at least to the early 1950’s in studies of nuclear spin states by Purcell and Pound. Negative temperature states were observed in Lithium-Fluoride crystals where the the loss of energy of the system caused it to gain entropy (disorder). These experiments were designed to study how the crystal reacts when it is placed in magnetic fields of differing directions.
While negative temperatures don’t apply to the molecules we are used to, when it comes to more complex structures such as magnetized solids, vortices, electron filaments, and other more exotic structures, they are an important tool for understand their behavior.
It turns out that negative temperatures, instead of being much colder than zero temperature, are actually hotter than all temperatures. That is, the flow of energy is always from a negative temperature system to a positive temperature system. You can think of negative temperatures as being hotter than infinite temperature! Temperature, in that sense, works like a ring starting at absolute zero, going up to infinity, and then wrapping back around from negative infinity to zero again.
So, while negative temperatures exist, absolute zero is still the coldest temperature.
Joyce, Glenn, and David Montgomery. Negative temperature states for the two-dimensional guiding-centre plasma. No. COO-2059–15. Iowa Univ., Iowa City. Dept. of Physics and Astronomy, 1972.
Montgomery, David, and Glenn Joyce. “Statistical mechanics of “negative temperature” states.” The Physics of Fluids 17.6 (1974): 1139–1145.
Eyink, G. L., and H. Spohn. “Negative-temperature states and large-scale, long-lived vortices in two-dimensional turbulence.” Journal of statistical physics 70.3–4 (1993): 833–886.
Purcell, Edward M., and Robert V. Pound. “A nuclear spin system at negative temperature.” Physical Review 81.2 (1951): 279.
Ramsey, Norman F. “Thermodynamics and statistical mechanics at negative absolute temperatures.” Physical Review 103.1 (1956): 20.