# When Less is More, part 2: Flipping the Sign

In part 1, we looked at absolute temperature scales.  Before an aside about not setting your kitchen on fire, we defined absolute zero as the point where atoms have no more energy from their motion (or kinetic energy).  We’ve now also nicely set ourselves up for the “big deal”.  If absolute zero is where there’s no kinetic energy, how do you get below that?  Can atoms somehow have negative kinetic energy?  The simple (and relevant to this discussion) answer is no.

Instead, we need to add something we’ve been neglecting from our discussion of temperature.  So far we’ve only talked about heat.  But another concept is used in the “relevant” definition (I’ll have more to say on this below) of temperature for this experiment:  entropy, which is basically the amount of disorder in a system.  If you took an advanced physics or chemistry class in high school (or maybe a first-year class in either of those fields in college), you might have learned the second law of thermodynamics, which states that entropy tends to increase, and must increase in closed systems.  It also defines how entropy changes in physical processes:  at a given (absolute) temperature T, if an amount of heat dQ flows into a system (we can also use a negative sign for heat flowing out), then the entropy S changes by an amount dS by

$\Large dS=\frac{\ dQ}{\ T}$

Of course, definitions can sometimes work both ways, and that’s what physicists decided to do.  Based on this, they solve for T, and we “officially” define temperature as

$\Large T=\frac{\ dQ}{\ dS}$

And so with this definition, we can see where a negative temperature comes from: anytime the heat flow and entropy change have different signs, T should be negative.

We can also be more general in that last equation, and look at temperature as a function of how entropy changes with respect to energy (if you’ve taken calculus, I mean the derivative of entropy with respect to energy), and temperature is

$\Large \frac{\ 1}{\ T}=\frac{\ d}{\ dE} S(E)$

Of course, “anytime” would still be rare in our everyday experience.  At the low energies and macroscopic scales of normal human life, heat (or energy) and entropy increase together.  But scientists can set up quantum systems (read: basically any system on an atomic scale) where we break that trend.  This is where a more basic understanding of entropy comes in.  If you have a system where atoms can have two different energies, the greatest entropy is when half of the atoms are in each energy state.  This is because that allows for the greatest number of combinations of atoms.  Look at this plot of the number of total possible combinations for making a combination of k atoms from a set of n total atoms based on this formula  , in this case where n is 50.

Note the giant peak at 25, or n/2

As you can see, the number of combination explodes close to the halfway point.  Also note the symmetry of the combination curve in this case.  So let’s say with our 50 atoms, k equals the number of atoms in the second, higher energy state.  (For those who know about electron orbitals, realize here that we’re not talking about the energy levels of the electron; we’re actually talking about the energy state the whole atom.  This can come up in systems where atoms are placed in magnetic fields.)  In most everyday systems, you’d have a lot of atoms in the lower state, and so we have a small k.  Adding energy kicks atoms up to the higher state, so k increases, and you see the number of possible configurations (and therefore the entropy), also increases.  But we have the opposite happen if we were to start with a high energy system, where k is already close to 50.  In that case, adding energy would decrease the entropy, and so by the third equation, we have a negative temperature.  Ta da!

The idea of a “simple” energy state system having a negative temperature is actually kind of old hat; i.e., we did it already.  (In fact, if you know how lasers work, this basically describes the population inversion of electrons; we just don’t typically ascribe a “temperature” to electrons confined in solids.)  So what’s the big deal about this new research?  While the atoms were in a gas state, the team was able to get them in a lattice arrangement in space and actually prevented motion of atoms.  By rapidly adjusting the magnetic field and lasers to trap most of the atoms in a high energy state, they made what should have been an energetically unstable arrangement (like a pencil balanced on its tip) stable.

Now that we have a spatially spread out system in a negative temperature, we can also test lots of interesting physics.  For example, the combination of stability on millisecond time scales (which is LONG for particle physics) and high energies can allow for unique probing of the Standard Model, such as the formation of structures defined by the strong force.  The researchers also point out that in this negative temperature regime, they also experience negative pressure, which is similar to the force believed to drive the expansion of the universe.

So that paragraph ends the NEW science, but I promised I would go more into the definition of temperature.  Or more accurately, why many people (look at the comments  on the review) seem to complain about how this really isn’t a negative absolute temperature.  Especially as those slightly more in the know freak out when they learn that systems with negative absolute temperature behave as if they were hotter than a temperature of positive infinity.  I’ll explain that confusing fact briefly.  If you put a negative temperature object next to a positive temperature object and there’s no energy source, heat will flow from the negative temperature object.  As I mentioned, the 2nd Law of Thermodynamics says entropy increases in closed systems.  Adding energy to the negative temperature system would decrease the system entropy (positive temperature systems lose entropy when they lose energy, and the negative system loses entropy when it gains energy).  To maximize the entropy, heat would need to flow out of the negative temperature object to the positive temperature object, bringing both objects closer to that giant entropy peak in the middle.

Those who complain that the temperature is only negative because of a weird definition resort to arguments about kinetic energy.  Unless you take a course on statistical mechanics, you almost never see the definition of temperature we just derived here.  Instead, you typically talk about how temperature is a measure of the average kinetic energy of the atoms in a system, and absolute zero is then described as the point where all atomic motion stops.  This then leads people to an obvious question, “How can atoms move less than being stopped?”

It’s a good question.  The problem is that the basic definition people are using isn’t the right one.  As one of the wonderful SciBloggers explains, the kinetic energy is important, but not in the way that simplification leads most people to believe.  It’s not the mere average of atomic energies that matters, it’s the nature of the probability distribution of energy around the average.  In statistical physics, this is described by the (Maxwell-)Boltzmann distribution.

Plot of number of atoms at each velocity for a variety of temperatures based on the Boltzmann distribution (given in Celsius, not Kelvin). A plot for energy would be similar. (From Wikipedia)

In the figure, you’ll see that the average velocity does increase with temperature.  Another trend is that the distribution widens with increasing temperature.  And if you analyzed the graphs, you’d see that most atoms have less energy than the average energy.  For a negative temperature system, these features are slightly tweaked.  The average energy increases with temperature (so -1 K has a higher average energy than -100 K).  The distribution widens at lower temperatures.  And the big deal is that most particles in a negative temperature system have a higher kinetic energy than the average.  This would look like the graph of the Boltzmann distribution if we say the temperature is negative, and so that’s why we go with that.  So it’s not that physicists have lied, it’s just kind of bizarre compared to our normal experience.

There’s also one slightly more intuitive explanation that I’ve been using to explain the concept of negative temperature.  First, I need to dispel another aspect of the definition that people are confused on.  Atoms at absolute zero don’t have zeroes of everything else.  Quantum mechanics says there is a zero-point, or minimum, energy that all objects have in a system.  And I’m pretty sure the uncertainty principle requires anything with a non-zero energy to have a momentum, which means it must move.  Instead, physicists define absolute zero as a minimum entropy point.  In order to minimize the entropy of a negative temperature system, we have to add energy to it.  In other words, we have to ADD heat to a negative temperature system to bring it to absolute zero.