Saturday 5 January 2013

Negative Temperatures

A friend of mine saw this article from Nature

Ultracold atoms pave way for negative-Kelvin materials

and asked me how is that possible, I mean, how is it possible to have Kelvin temperatures below zero? I gave him a very quick answer, but I think he deserves a more enlightening explanation, so I decided to write this article to explain what's the meaning of a negative Kelvin temperature.

First of all, I have to say that his concern is justified. Although there is nothing very unusual with negative Celsius or Fahrenheit temperatures, the Kelvin scale is constructed in a different way. Without going to irrelevant details, in general the Kelvin temperature is associated with the average kinetic temperature of the molecules (or atoms) of a material. The zero in the Kelvin scale would mean zero kinetic energy, or no molecular movement (either vibrational, rotational or translational). Therefore, in principle, there should be no negative Kelvin temperature as we cannot have less than no energy at all by what energy is supposed to be. (Some of you will say that you can talk about negative energy, but that is not what I mean here. Think of this as differences in energy relative to the ground state, which have to be positive. If someone wants, I can write another post about this later.)

Well, to find out what is wrong here, I need to introduce some concepts, but I'll do that in the most basic way possible. The first concept is that of a derivative. This is a very simple mathematical concept and the only reason why it's not taught in school much earlier is because people are silly. A derivative is a rate of change of one quantity relative to other. For instance, speed is the derivative of space relative to time as it is the amount of space you travel in certain amount of time. You divide one by the other, basically. Acceleration is the derivative of speed with respect to time. There are other examples, and some subtleties, but the idea is that the speed, for instance, allows you to calculate how farI have traveled if you know for how long I've been traveling.

Let's keep using speed as our best example for a derivative. To calculate how much I traveled, I need sticks marking the distance from a zero point in the road. Then I define that the space I traveled is given by the number in the marking I stopped minus the one in which I began. If it's positive, I traveled forwards, if it's negative, I traveled backwards. For instance, if I started on kilometer 4 and finished on kilometer 9, I traveled 9-4=5 kilometers forwards. If I start on kilometer 2 and end on kilometer -3, I traveled -3-2=-5, or 5 kilometers backwards. The convention is that when I calculate the derivative, the sign must be there, be it negative or positive. So if I traveled -6 kilometers in 3 hours, I would say that my average speed is -2 km/h, the minus sign meaning that I traveled backwards with respect to the marking sticks.

By what I explained above, a negative derivative means that one quantity is decreasing while the other increases and a positive derivative means that the it's increasing. But what does it have to do with temperature? To make the connection, I need to introduce some concepts of thermodynamics... and a small model.

Let me start by defining the model and we can use it to understand the rest. The first piece of the puzzle is to understand that what the guys from the paper are looking at is NOT kinetic energy. That's because energy can also be used to define what is called a cost function. You probably learn that all systems try to find the state of least possible energy. This is called a ground state. Whenever the system has more energy than that, it is in an excited state. So, energy is a cost to move the system from its ground state.

When we study thermodynamics, sometimes we look at systems where kinetic energy has no usefulness at all, but there are other measures of cost that will give us an idea about how far the system is from the ground state. One of these systems is a one dimensional line of spins in a homogeneous magnetic field. Calm down, I will explain each term.

The word spin is associated with a property of elementary particles which is very subtle, but what is important for us here is that a charged particle with spin works as a tiny magneto. In order to make it clear on paper we draw a particle with spin as an arrow, where the tip of the arrow is supposed to indicate the north pole of our magneto. A homogeneous magnetic field is a magnetic field which has the same value everywhere in space. As a magnetic field has also a direction, if it's homogeneous the direction is also the same everywhere. Now, if you put a charged particle with spin in a magnetic field and let it go, it will try to align itself with the field. The energy associated with that spin will then be a measure of how misaligned the spin is. The worst case, of course, is when the spin is pointing in the totally opposite direction. I'll draw a picture to make it clearer:


It goes like this. The black lines with arrows represent a homogeneous magnetic field. The green arrows are spins (of electrons, for instance) pointing in some direction. The letters below represent the energy of the spin state in the field. When the spin is pointing in the same direction as the field, the energy is the minimum possible. In the picture I gave the value zero, but the actual number is irrelevant. The important thing is that as you move to the right, the misalignment increases and so does the energy. Remember, physically the spins want to align, so we attribute a number to the state that indicates the amount of misalignment and call it energy. We're almost there.

Now you need to imagine a line of spins in a magnetic field. Each one pointing in a different direction. It helps to introduce right now the concept of entropy. I will not give it in details, but here the association of entropy with disorder is a useful picture. So, if all the spins in a line are pointing to the same direction like this


We can say that the state is highly organized and that the entropy is low. If one of the spins is flipped like this


this state is said to have a higher entropy as it is a bit more disorganized than the previous one. The correct definition of the entropy is achieved by the following procedure. I will use some conventions. I will assume that our magnetic field is ALWAYS pointing upwards and that the spins can have only two positions, upwards and downwards. Then, for each spin, there are only two possible values of the energy to which we will give the value E=0, when it points upwards as the field, and E=1, when it points downwards opposite to the field. The total energy of the line is the sum of the energies for each spin. Therefore, our first line above has energy zero and the second line has energy 1.

The entropy S is related to the number of possible states with the same energy. In fact, it's the logarithm of it. It's easy to see that there is only one state with energy zero (everyone up!) and the logarithm of 1 is zero, so the entropy of our first line is S = 0. There is also only one state with energy 9, as we have nine arrows, it's everyone pointing down. So, again, this state has zero entropy. There are exactly 9 states with one spin down and the rest up, meaning, 9 states with energy 1. These states have, therefore, entropy S = log 9, which is higher than zero, of course.

The final blow is the fact that, a long time ago, we understood that we could define temperature as the derivative of the energy with respect to entropy. This could be done for kinetic energy, but we physicists like to generalize things, and we decided to generalize this definition of temperature even for our line of spins above! Now, let's see the consequences of this generalization. One important thing to bear in mind is that this generalization is not talking about average amounts of kinetic energy anymore. It coincides with that in the case of moving molecules, but surely has nothing to do with it for our line of spins as they are not moving!

Remember that derivatives are rates of change. If we define temperature as the rate of change of energy with entropy, then what happens with the sign of the temperature? If everyone is pointing up and we flip one spin down, we increased the energy and, at the same time, we increased the entropy. So, the temperature is positive and so far so good. 

We finally arrived at the trick! If we start with everyone pointing down, the energy is the highest possible (E = 9) but the entropy is zero (S = 0) because there is only one state with energy 9. If we flip one of the spins up, then the energy will decrease to 8, so the variation in energy will be -1, negative! But the entropy will increase because there are 9 possible states with one spin up and the rest down. So the entropy will increase from zero to log 9. In the same way as our speed, the energy now decreased with and increasing entropy, therefore the derivative is roughly -1/log 9, which is negative! Ta-dah! Negative temperature. Nothing to do with average kinetic energy. Just a trick that plays with the definition.

What the guys do is basically the same thing we did with our line of spins, technically called a spin chain, but in a larger scale. :)

UPDATE: I finally found the footnote I was looking for with a paper of 1956 which deals with negative absolute temperatures. It's a technical paper, but what I would like to leave clear is that, contrary to what I read in many places, this is known to exist for more than half a century:

Thermodynamics and Statistical Mechanics at Negative Absolute Temperatures
Norman F. Ramsey - Phys Rev 103, 20 (1956) - Free PDF

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