A system with finitely many states can have negative temperature! Even weirder: as you heat it up, its temperature can become large and positive, then reach infinity, and then 'wrap around' and become large and negative!

The reason: β = 1/kT is more important than T.

(1/n)

Systems with finitely many states act this way because the sum in the Boltzmann distribution converges no matter what the coolness β equals.

When β > 0, states with less energy are more probable.

When β < 0, states with *more* energy are more probable!

(2/n)

For systems with finitely many states, the Boltzmann distribution changes continuously as β passes through zero. But since β = 1/kT, this means a large positive temperature is almost like a large negative temperature!

Temperatures 'wrap around' infinity.

(3/n)

However, I must admit the picture of a circle is misleading. Temperatures wrap around infinity but not zero. A system with a small positive temperature is very different from one with a small negative temperature! That's because β >> 0 is very different from β << 0.

(4/n)

For a system with finitely many states we can take the limit where β→+∞; then the system will only occupy its lowest-energy state or states.

We can also take the limit β→-∞; then the system will only occupy its highest-energy state or states.

(5/n)

So, for a system with finitely many states, the true picture of possible thermal equilibria is not a circle but closed interval: the coolness β can be anything in [-∞, +∞], which topologically is a closed interval.

In terms of temperature, 0⁺ is different from 0⁻.

(6/n)

We often describe physical systems using infinitely many states, with a lowest possible energy but no highest possible energy. In this case the sum in the Boltzmann distribution can't converge for β < 0, so negative temperatures are ruled out.

(7/n)

However, some physical systems are nicely described using a finite set of states - or in quantum mechanics, a finite-dimensional Hilbert space of states. Then the story I'm telling today holds true!

People study these systems in the lab, and they're lots of fun.

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