# An Introduction to Negative Temperature

*d.w.rowlands [at] gmail.com*

*It's probably due to my history of working with NMR, but I have a bit of an obsession with the concept of negative temperature, one of the consequences of thermodynamics that physicists and physical chemists tend to be vaguely aware of but not think about. An article in Science by a group that figured out a way to achieve negative temperature in the mechanical vibrational energy of a material brought the topic to my attention again and persuaded me to teach a Spark class for high school students on the topic. Well, secretly it's an introduction to statistical mechanics and the concepts of entropy and temperature, but I thought negative temperature would be a good way to make the material sound interesting.*

*Actually, I taught two classes: an introductory version with minimal math and no calculus for grades 8-10 and an advanced one with calculus for grades 10-12. The more more basic course is summarized here, while the notes for the advanced class are posted as a PDF file here.*

## What is entropy?

Before we can talk about the nature of temperature, we have to understand the concepts that temperature is defined in terms of: energy and entropy. Energy seems like a fairly intuitive concept. The main difficulty is in keeping track of all the different forms it can take. The First Law of Thermodynamics says that energy is “conserved”: the total amount of energy in the universe never changes. However, energy can change forms, and not all of these changes are reversible. For example, if you throw a ball of clay, it has kinetic energy because it is in motion: *KE = mv*. Ignoring air resistance, it will keep going at the same speed forward, but it will also get downward velocity—and so more kinetic energy—while losing gravitational potential energy, *mgh* at exactly the same rate.

That change of energy is reversible: if you throw a ball upward, gravity slows it down as it converts kinetic energy to potential energy. However, in either case, it will eventually hit the ground and smash, at which point it will stop moving and have lost all its kinetic and potential energy. It turns out that the energy has been completely converted to vibrations in the molecules that make up the clay and the ground. This sort of energy is called “thermal energy”. This type of energy change is “irreversible”: one never sees a lump of clay on the ground suddenly convert its thermal energy into kinetic energy and spontaneously jump into the air.

The reason that some changes between types of energy can only occur in one direction is the Second Law of Thermodynamics, which says that the “entropy” of the universe always increases. Entropy is a much slipperier concept than energy. You might have heard it referred to as “disorder”. However, disorder and order are confusing ideas, because whether something looks ordered can depend on your point of view. For example, a toddler might think a collection of books were “in order” because they’d been sorted by color and size. An elementary school student would correct them by saying they should be alphabetized. But their school librarian would want them sorted according to the Dewey Decimal System, and a college librarian would want to use a different system called the Library of Congress Catalog System.

A better way to think of entropy is as a measure of how many different ways you could arrange the particles something is made of, given its overall properties. An analogy that may be useful in understanding this is a very tall hill that a large group of people are planning to climb to get a better view of a fireworks display. The higher they go, the better the view, but they’ll need more energy to climb higher.

If the crowd of people are all very tired and in bad shape, there aren’t that many possible arrangements for them, because none of them will have the energy to get very high and they’ll all cluster at the bottom. However, if they’re all in good shape and have a lot of energy, they’ll spread out over more of the hill: some may stay near the bottom, but others will climb to the top, and it will be much harder to predict where to find any given person.

## What is temperature?

At first glance, it might seem like energy and entropy are the same thing in this system: providing more energy makes the people spread out more and the entropy goes up. However, they don’t actually vary at the same rate.

Imagine two arrangements of people on the hill. In one case, they don’t have much energy and are all within the bottom twenty feet of the hill. In the other, they have much more energy and are spread over the bottom half-mile of the hill. If, in each case, you increase their energy such that everyone can move twenty feet higher, you’ll have increased the energy of both by the same amount. However, in the first case, you’ll have doubled the area that people are spread over, and greatly increased the entropy (the amount of different places you might find a given person). In the second case, you’ve made only a very small increase in the area the people are spread over, and the entropy hasn’t changed much.

For any system, we can define the quantity β as the increase in entropy produced by a standard, small increase in energy. This quantity is important because of the fact that the Second Law tells us that the entropy of the universe always increases, while the First Law tells us that the energy of the universe always stays the same. That means that how energy can flow between two systems is a function of their values of β.

Suppose two systems with different values of β come into contact. If an amount of energy E flows from system 1 to system 2, then the entropy of system 1 decreases by E × β1 while the entropy of system 2 increases by E × β2 . Since the entropy of the universe must always increase, this can only happen if β2 > β1 . Thus, thermal energy always flows from systems with low values of β to systems with high values of β.

This is exactly the opposite of what you’d expect for temperature and, in fact, it turns out that temperature as measured in Kelvin—what’s called “absolute temperature” since it is zero at absolute zero—is proportional to β. Thus, temperature is just the relationship between the thermal energy and entropy of a given system.

## So, how can temperature be negative?

In the previous section, we assumed that the hill was tall enough that no one was close to the very top. As a result, adding more energy allowed people to spread out more and increased the entropy of the system. The systems we’re most used to measuring the temperature of do behave this way. In a gas, thermal energy is just the kinetic energy of the gas molecules: since there’s no limit to how much kinetic energy they can have, it’s as though the hill has no top for the people to reach. Molecules in a solid are more like masses oscillating on a spring, and their thermal energy is a combination of kinetic and spring potential energy. But still, there’s no upper bound on the energy: if you add enough energy to break the springs, you’ve melted the solid, but there’s nothing preventing you from doing so and thus increasing the thermal energy of the system.

However, what if the hill does have a top and you provide enough energy for people to reach it? As we start increasing the people’s energy from zero, we initially get an increase in entropy as people start to spread out over the hill. As we add more energy, someone will eventually reach the top, but more people will still be near the bottom, and increasing energy will spread them out more. Eventually, we’ll get to a point where any given person has an equal probability of being anywhere on the hill. At this point, β is zero: the system has its maximum entropy and we can’t increase it by adding more energy. In fact, if we add even more energy, the entropy of the system will go down as people start to cluster at the top of the hill. This means that β is negative, and so is β, so the absolute temperature of the system is negative.

This doesn’t conflict with the observation that absolute zero is the coldest possible temperature, though. Remember that energy always flows from low-β systems to high-β systems. Absolute zero corresponds to a β of ∞, and thus it will flow to a system at absolute zero from one at any other temperature. For that matter, since a negative temperature corresponds to a negative value of β, energy will always flow from a negative temperature system to a positive temperature system, meaning that negative temperatures are hotter than positive temperatures!

## Can you make a negative temperature in real life?

As we noted previously, the temperature of particles in a gas or solid normally corresponds to people on a hill with no top. This means the temperature of gas molecule kinetic energy or vibrations in a solid can’t be excited to a negative temperature.. However, there are physical systems with maximum possible energies, and in which one can produce a negative temperature. The most common of these is in MRI experiments. You’re probably aware of MRI as a medical imaging technique, but you may not know how it works. MRI depends on the fact that atomic nuclei can act as bar magnets. An MRI imager applies a strong, constant magnetic field that forces these bar magnets to align either with or against the field. While both alignments can be stable, an energy aligned along the field has less energy than one aligned against it, which means that there are two permitted “energy levels” for each nucleus to be in. At infinite temperature—β = 0—this system has maximum entropy with the two energy levels equally populated. However, it’s possible to use a second magnetic field to flip more bar magnets into the against-the-field orientation, producing a greater population of against-the-field magnets and a negative temperature.