## What 'Negative Temperature' Really Means 204

On Friday we discussed news of researchers getting a quantum gas to go below absolute zero. There was confusion about exactly what that meant, and several commenters pointed out that negative temperatures have been achieved before. Now, Rutgers physics grad student Aatish Bhatia has written a comprehensible post for the layman about how negative temperatures work, and why they're not actually "colder" than absolute zero. Quoting:

*"...you first need to engineer a system that has an upper limit to its energy. This is a very rare thing – normal, everyday stuff that we interact with has kinetic energy of motion, and there is no upper bound to how much kinetic energy it can have. Systems with an upper bound in energy don’t want to be in that highest energy state. ...these systems have low entropy in (i.e. low probability of being in) their high energy state. You have to experimentally ‘trick’ the system into getting here. This was first done in an ingenious experiment by Purcell and Pound in 1951, where they managed to trick the spins of nuclei in a crystal of Lithium Fluoride into entering just such an unlikely high energy state. In that experiment, they maintained a negative temperature for a few minutes. Since then, negative temperatures have been realized in many experiments, and most recently established in a completely different realm, of ultracold atoms of a quantum gas trapped in a laser."*
## Purcell and Pound (Score:5, Informative)

It would have been nice for Aatish to go a bit into what Purcell and Pound did in their 1951 experiment, namely "inverting" the orientation of the fluorine nuclei in the presence of an applied magnetic field by application of a radio frequency magnetic pulse, where the frequency is the Larmor frequency of fluorine and the pulse amplitude and length was sufficient to cause a 180 degree nutation. The result is that the nuclei have the same order (entropy) as the rest state, but have higher energy. In NMR, this is referred to as applying a 180 degree or pi pulse.

Aatish's comment about reality being liberal is unconvincing.

## Re:Uhhhh (Score:5, Informative)

The point they're making is that temperature can refer to energy and entropy other than just the kinetic motion kinds.

Unfortunately understanding the definition still doesn't get us very far for those of us without intuitive models of those other kinds of situations, so we're no farther ahead.

## Re:Uhhhh (Score:5, Informative)

## Re:Uhhhh (Score:5, Informative)

The short answer is that physicists throw out the "temperature describes amount of molecular movement" definition and replace it with something more abstract.

The abstract definition of temperature allows negative values, and that's ok because nobody cares anymore about molecular movements in that case.

## Re:Uhhhh (Score:5, Informative)

heating, but higher states can be reached through other mechanisms. Once we realized that, we needed another scale for the higher-energy states at the other side of infinity, so we started using negative numbers for them. So negative temperatures are not at the cold side of 0K, but at the hot side of inifinity K. More complete explanations here [wikipedia.org] and here [slashdot.org].## Re:It wants to get colder (Score:4, Informative)

## Ok i will give it my best shot (Score:5, Informative)

I thought about explaining it, and i will do so *without* mentioning the Dalai Lama.

The Situation is very simple: The definition of Temperature you learned in school, namely that it is only related to the average energy of many equal systems *is right*, but only for *classical systems*.

What does it mean?

If i have a classical gas, e.g. air at room temperature and i have to input to it, i can add this energy in whichever distribution i want. Easy to do that, no matter at which temperature we are.

No lets consider a quantum gas (to be complete: a quantum gas and not consiting of harmonic oscillators), e.g. electrons spins which are aligned to a magnetic field. Each of the electron can either have an Energy of -1/2E or +1/2E, where E depends on the electron spin and the magnetig field, but is constant. This means that if i have N electrons, we wont be able to input more energy than N * E into the system. Moreover if only a single electron in not in the high-energy state, we have to flip exactly this electron to get the system into its highest energy state. That may be pretty hard, statistically speaking.

So now imagine a quantum gas somehow statistically exchanging energy with a classical gas. That means, in our case, to bring the quantum gas to the state of Total energy = N*E (from the state of (N-1)*E) a high energy gas molecule would have the hit the very last of the low-ebergy electrons. If the high-energy molecules bounce from the electron in the excited state, then nothing will happen.

It is intuitive that, even if the two gases are in contact, the avergae energy between the systems will *not* be the same, just because its unlikely to flip *all* or *nearly all*.

The fromal version if this consideration is the textbook definition of the Temperature as a property in statistical physics, which is T=dE/dS, where E is the total energy and S is the Entropy (yes, the very same one as in computational science).

In the case of the two-level systems we find (let n be the numebr of systems in exited state)

S is proportional to -(n*log(n/N) + (N-n)*log((N-n)/n))

E is proprotioanl to n

That means that the sign of the temperature changes, as soon as more systems are excited than not.

## Temperature (Score:5, Informative)

Actually, it's not to hard to intuitively understand negative temperature if you think of it as something hotter than the hottest possible temperature. Classically, that isn't possible, but then you need a bit of quantum weirdness.

In a typical system of normal temperature particles of occupy various quantum energy levels available to them. In thermal equilibrium, statistically, lower energy levels tend to get occupied first and higher energy levels have fewer particles. If somehow you can create a stable system where higher energy states are occupied, but by some quirk (of quantum mechanics), lower ones are not, it turns out that is what a negative temperature system is.

As it turns temporarily creating a system where the higher energy levels are occupied before the lower ones is something that people do all the time to create a pumped laser. But lasers aren't designed to be a stable system (you eventually want the higher energy state to emit light/photons and fall to the lower energy state), so although the population of the energy states are inverted (more in the upper energy states), it's not stable, so it's generally not accurate to call this a negative temperature system.

The reason the "sign" of the temperature is negative is just a problem with the definition of temperature. For most defintions of temperature, if you add energy, you increase entropy, so temperature is a measure of how these relate to each other (the slope). If somehow when you add energy to your system, you decrease entropy of your system (e.g, you pack the upper energy states even tighter reducing entropy instead of just letting particles in all energy states into statistically higher energy states), the slope is negative.

## The article links to a better explanation (Score:5, Informative)

There's a link in the article to Leprechauns and Laser Beams [coffeeshopphysics.com], which IMHO does a much better job of explaining things. As I understand it, negative temperatures don't just come from the entropy-based definition of temperature. You also need to be talking about a system whose energy content is capped. Normal materials don't do this -- you can keep adding energy (speeding up atoms) as long as you want. But if you have a group of atoms with exactly two energy states (high and low), once every atom is in the high-energy state you can't add more energy. Apparently, one example of this is a laser.

From an entropy point of view, the lowest energy and highest energy states have identical entropy (i.e. none -- one possible state). Entropy reaches a peak with half of the atoms in the high energy state, since this gives the largest number of possible atom state combinations.

Temperature is defined as the slope of the energy/entropy curve. The curve goes vertical at max entropy. If I understand right, at this point the temperature overflows like an integer variable, going from +inf to -inf and approaching zero from the negative end. (It's not really a continuous curve, but I don't know enough to guess at what difference that makes.)

So it sounds like the recent news about a negative-temperature gas was more about creating a new material with these sorts of quantum states. The negative temperature part caught the attention of the reporters (and the rest of us), but isn't the real scientific discovery. That's my reading of it, anyway.

## Re:Layman (Score:4, Informative)

Here's my take on a layman explanation:

It's a water model in the classical world. It doesn't model everything from the quantum world, but makes it easier to understand the concepts.

Imagine a long vertical tube, closed off at the bottom.

When it's empty, it has minimal entropy (a measure for the amount of disorder).

When you add an amount of water (which models energy here), the water level rises and so does the entropy.

Now the definition of temperature is amount of heat energy per amount of entropy (T=dQ/dS). In the above situation, both amounts are positive, so the temperature is also positive.

Now imagine we close off the tube at the top too. This will leave an amount of air trapped there.

When we add an amount of water (using a valve to make sure the air doesn't escape), at first the system will behave exactly the same.

But when the water level gets near the top, the air gets pressurized and starts pushing back. And this increasingly so until it's almost full.

If we would make a hole in the middle of the tube, the water would squirt out until a pressure equilibrium was reached. We could extract work from this, to power a little water wheel. This means the "full" state had a lower entropy than the "middle" state.

So in this system, entropy went from a low value to a certain (maximum) higher value, and then back to a low value. This for an increasing amount of water (low, medium, max).

So what does this mean for temperature as defined above?

We kept adding the same amount of water (dQ in our model).

The change in entropy (dS) this caused is the slope of a hill (low, max, low), so at first it is a positive amount, which gets smaller and smaller, to become zero at the equilibrium point. After that, adding more water (energy) will cause the entropy to go down again, so dS will become a small negative amount at first and a larger negative amount near "full".

When we plug this in in the equation for temperature (T=dQ/dS) we get:

Going from "empty" to "middle": dQ is positive and the same, dS is positive and gets smaller, approaching zero. So T starts at some positive value, then gets higher and higher approaching positive infinity.

Going from "middle" to "full": dQ is still positive and the same, the change in entropy dS is zero at first and then becomes smaller and smaller (negative). So T starts out at negative infinity and then gets higher and higher approaching some negative value.

This illustrates how the temperature scale goes for increasing heat energy: ... +inf -> -inf ... -0

+0

So a system with negative temperature has more energy than the same system with any positive energy.