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!

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!

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.

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.

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.

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.

John Carlos Baez@johncarlosbaez@mathstodon.xyzSo, 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)