Boltzmann Statistics and Negative Temperature

Maxwell-Boltzmann Statistics
Maxwell-Boltzmann statistics is a thermodynamics model that describes the distribution of non-quantized particles over possible energy states given the bulk temperature of the particles. It's pretty complicated stuff.

A Boltzmann distribution is always of the form exp(-E/kT), where E is an energy level, k is Boltzmann's constant, and T is absolute temperature. Boltzmann's constant, k, has units of J/K, hence, it relates energy (at the particle level) to temperature (in the bulk level).

The form in which I have seen it most is n2/n1 = exp(-(E2-E1)/kT), where n2 is the number of particles in energy state 2, n1 is the number of particles in energy state 1, and E2/E1 are the corresponding energy levels.

Population Inversion
Consider the following Boltzmann distribution:

n2/n1 = exp(-(E2-E1)/kT)

where E2 is greater than E1. In this situation, E2-E1 will always be positive, and so the ratio n2/n1 will always be less than unity. This means that the number of particles occupying a higher energy state will always be less than the number of particles occupying a lower energy state. That makes sense.

However, what if we could create a situation in which n2/n1 > 1? That would mean that the higher energy states were more populated than the lower energy states. Such a situation is called population inversion.

Negative Temperature
Looking at the equation above, the only way for population inversion to occur is if the temperature goes negative. This concept seem bogus at first, but it actually makes a lot of sense.

Negative temperature is "hotter" than infinite positive temperature. On the "hotness" scale, it goes like this:

+0K < +100K < +∞K = -∞K < -100K < -0K To understand the difference between positive and negative temperature, I will give a few ascii diagrams showing different temperature levels. The numbers on the left indicate energy states (higher number = higher energy state) and the boxes on the right indicate filled states.

+0K:

4
3
2
1
0 ███████

+100K:

4
3 █
2 ███
1 █████
0 ███████

10^10K:

4 █████
3 ███████
2 ███████
1 ███████
0 ███████

+∞K=-∞K:

4 ███████
3 ███████
2 ███████
1 ███████
0 ███████

-10^10K:

4 ███████
3 ███████
2 ███████
1 ███████
0 █████

-100K:

4 ███████
3 █████
2 ███
1 █
0

-0K:

4 ███████
3
2
1
0


Notice that adding a negative temperature to a nearly infinite positive temperature results in a negative temperature that could not be achieved by the addition of more positive temperature. Combining a negative temperature with any positive temperature results in heat flowing from the negative temperature to the positive temperature. That is why negative temperature is "hotter" than any positive temperature.

I may be wrong in parts of my explanation, but this is the basic idea.

Population Inversion and Lasers
Population inversion is a requirement for lasing. When light strikes the active layer, the probability of stimulated emission occurring is directly proportional to the probability of absorption occurring, with the proportionality constant being the ratio of the electrons in the conduction band to the electrons in the valence band. Basically, the ratio of emission to absorption is:

n2/n1 = exp(-(E2-E1)/kT)

And therefore, for lasing to occur (amplification of light; more emission than absorption), population inversion is a requirement.