# A physical explanation for negative kelvin temperatures

Just to get the thoughts rolling...

Consider a two state system with discrete energy levels $E_1$ and $E_2$ where $E_2 > E_1$ which contains $N$ particles.

We can easily deduce that the state of maximum entropy is when energy level $E_1$ and $E_2$ each contain $\frac{N}2$ particles.

Based on the (flipped-upside-down) thermodynamic definition of temperature, $$(\frac{\partial E}{\partial S})=T$$ we see that when energy as a function of entropy increases, temperature also increases.

This means, for our two state system, every time a particle populates level $E_2$, there is a corresponding increase in temperature because that population increases entropy.

What about when we go beyond the point of maximum entropy where there are $\frac N2$ particles in $E_1$ and $E_2$?

Well, then we are adding energy to the system, but the entropy is decreasing.

Based on the Boltzmann factor, this can't be possible unless the temperature of the system is negative. That is, $$\frac{P_1}{P_2}=e^{-\frac{(E_1-E_2)}{kT}}$$ which approaches infinity unless one says the system has a negative absolute temperature.

Now we're getting to the point: this increase in energy which is connected to a decrease in entropy is what has been called a negative kelvin temperature. This is most often connected to population inversion in lasers.

Forgive me for saying things you already know.

All of this raises a lot of questions though, the most striking to me is that there does not seem to me to be a good physical explanation for how this can be possible. For instance, when people say that the negative temperature system is "hotter" I get what that means as far as saying that heat will flow from a negative temp. system to a pos. temp system.

I'm fine with that except that it invokes the idea of a fairly macroscopic quantity--hotness--for a purely quantum phenomenon. So, is there any macroscopic physical meaning to this statement that the negative temperature is "hotter"?

Concerning population inversion in lasers: doesn't this mean that the negative temperature required for that laser to exist ought to be so "hot" no mere mortal can handle such a device.

After all, negative absolute temperatures are often explained as being "hotter than infinite temperature." (I've heard this other places, but also see the neg. temp. Wikipedia article for quote.) I understand that again from a mathematical perspective, but what is the physical connection? A statement like that must surely manifest itself in some way that is noticeable beyond the quantum level.

I guess I really want to understand how it could be this "hotter than infinite temperature" system isn't perceptible in the macroscopic realm? I mean, lasers are cool, but they aren't just melting through everything as if they have infinite temperature.

My bad for the obvious misuse of the word "physical" when I probably meant something more like "macroscopic". I hope the point is clear anyways.

Thoughts?

So, is there any macroscopic physical meaning to this statement that the negative temperature is "hotter"? Doesn't this mean that the negative temperature required for that laser to exist ought to be so "hot" no mere mortal can handle such a device?

If you go far up enough in the atmosphere, you'll reach regions where the temperature is over 3000 K. Yet rockets aren't exploding in flames when they get there, because the atmosphere is very sparse. It will transfer energy to the rocket, because its temperature is very high, but only incredibly slowly.

In general, human beings cannot detect temperature; we can only detect heat transfer. This is why metal surfaces often feel cold; they're the same temperature as everything else, but they conduct better, so you lose heat faster when touching them. Similarly, if you go up really high into the atmosphere, you won't feel like it's hot. Instead, you'll feel nothing at all.

So our intuitive sensory notions of "hot" or "cold" have very little to do with temperature. So it's indeterminate how "hot" a negative temperature system will feel, because that depends on how quickly it transfers heat to your body. All the temperature means is that it will transfer heat.

• So those "negative kelvin" particles are infinitely hot with little kinetic energy, or kinetic enough to blow a hole through a person without him even noticing because they are so small? Aug 23, 2016 at 13:53
• @CeesTimmerman It's purely a statistical property. No particle is special at all, it's just the configuration of all of them that makes it negative temperature. Aug 23, 2016 at 16:13
• One possible confusion is that you can't get negative temperature for the system you're probably thinking about, like a gas, since that would require infinite energy. I'm talking about a system where the energy is bounded. Aug 23, 2016 at 16:14
• This page says cosmic photons have an energy of 2.78 kelvin, so how much energy would the particles that show up in a bubble chamber have? Aug 23, 2016 at 21:29
• @CeesTimmerman This seems to be unrelated to the original question. If you want an answer to that, you should post it as a separate question! Aug 23, 2016 at 21:30

The temperature is really only negative in the sense of the classical definition of temperature. What is actually happening in a population inversion is the particles aren't following Boltzmann distribution of energies anymore. Comparing the temperature of a Boltzmann distributed system to a non-Boltzmann system might not be meaningful at all. People say that negative temperatures are hotter than any positive temperatures because, when brought into contact, heat will flow from the negative temperature object to the positive one. This is simply because the negative temperature object has more particles in a higher energy state (which is unstable). The particles will tend to redistribute themselves into thermal equilibrium (a Boltzmann distribution).

Lasers don't exactly have negative temperature. They are often brought up because they are one example of where a population inversion occurs. The population inversion occurs in the energy level of electrons, not in the thermal energies. So if you considered the "temperature" of electron energy levels, lasers would have a negative temperature. This is different from the physical temperature though.

You've touched on a very important point - "negative temperature" is a misnomer. Start with a simple thought experiment.

1. Physics tells us that temperature changes happen continuously. In order to change from 10K to 20K, we must first pass through 11K, 12K, and all other intermediate temperatures.
2. In order to achieve a "negative temperature," you would need to travel through 0K, which is not possible.
3. If it was possible to "skip over" 0k and produce something that was -1K, then we could transfer its energy to an equivalent mass of 1K and produce a net temperature of 0k, which is again impossible.

The term "negative temperature" is really a trick of the mathematics involved, to use that entropy-temperature ratio you mentioned to describe some spooky phenomena where the entropy and temperature don't seem to line up exactly. This could be interpreted in many different ways. Maybe our definition of entropy or temperature is skewed, maybe the entropy-temperature relationship is fundamentally flawed in some way (or missing some extra component). But either way, I don't think there is any "physical explanation" for it, except that wave functions are complicated and the universe is basically made of magic.

To go from positive to negative temperature, you must go thru infinity. It may be more fruitful to think of beta = 1/T, which does go from positive thru zero to negative. Beta is the derivative of the number of available quantum states with respect to energy: add a unit of energy and how many states can it occupy? Think of a coin funnel half full of marbles: the marbles form a flat surface; add one marble, and how many places can it go? And how does that answer change as the funnel fills? It gets wider as it fills, therefore positive beta and positive temperature. If the sides were infinitely steep (so, a cylinder), then no change as it fills, therefore zero beta and infinite temperature. If it sloped the other way, like the top of Willy Wonka's tower or the upper part of a spherical tank, then negative beta and negative temperature.

I learned the above at Caltech. Minus the coin funnel example.

And why wouldn't there be thermal coupling so the material's temperature matches what's going on inside it?

"Negative temperature" can only occur in systems with a maximum energy state.

Thermodynamics dictates a precise relation between energy and state occupancy at a given temperature. Some detailed derivations are here. At positive temperature the higher states are less populated. Setting aside the rules for fermions and bosons, which are more relevant at low temperatures, the occupancy drops exponentially with energy.

Negative temperatures can occur if there is a finite number of states. Suppose we have 4 equally spaced energy levels and 8 (non-identical) particles. We have E units of energy to distribute the particles; the maximum possible E is 24. The postulate of thermodynamics is that all distinguishable states (choices of which particles get how much energy) are equally likely. For E=2, 8 states give both units to a single particle and 28 give one unit to each of two particles. The average occupancy averaged over all states, for the first three energy levels is 6.22:1.55:0.22. Lower energy states are more occupied. If we take a limit of a large number of particles and large E (keeping the average energy per particle fixed) we will get a precise exponential decay of occupancy with energy level and a positive temperature.

If we repeat the 8 particle situation with E=22 we have flipped our problem upside down: we start with particles in the highest state and then allocate two units of "negative" energy. This means we have a negative temperature. For our 4-state system we get positive T below 1.5 E/particle and negative T above 1.5 (T is infinite at exactly 1.5 units; similar to the singularity in the curve y=-1/x). Thus negative temperatures are hotter than infinitely hot.

However, if we have an infinite number of energy levels we always get a positive temperature. If the energy levels are equally spaced (as is the case for the quantum harmonic oscillator) the temperature will be proportional to the energy; a constant heat capacity.

Real systems don't have a maximum energy state. Every system has translational degrees of freedom at high energies. There is no upper bound to kinetic energy (the speed of light gives a limits the speed but there is no limit to the momentum or the energy). Theoretically, an electron could even have more kinetic energy than the Planck energy. Add a huge amount of energy to matter and you will fill these degrees of freedom, tearing the matter apart in the process.

Negative temperatures can transiently exist if you ignore some degrees of freedom. Suppose you have n electronic states which rapidly exchange spin energy with each-other but they only sluggishly interact with a heat bath of molecular translation, rotation, and vibration. You inject energy with a flash lamp or an electronic discharge and then wait long enough to thermalize the electronic states, but not long enough to themalize them against the heat bath. You may end up with a negative temperature among the electronic states. Of course, such a situation is unstable and the energy will eventually dissipate to the heat bath, as incoherent radiation, or a laser beam.