Why are negative temperatures hot?

You've probably heard the statement "negative temperatures are hot!", referring of course to negative absolute temperatures.

But why are they hot? Well, a common explanation is that it's not really the temperature $T$ that is the fundamental quantity, but rather the statistical beta, or "coldness" $\beta=1/T$. So negative temperatures have negative coldness, which is hotter than any positive temperature, since even the hottest positive temperature is only going to give you a small, but positive coldness. So the fact that negative temperatures are hot is a result of the fact that $1/x$ is not really decreasing everywhere, due to its discontinuity.

But why? Why is $\beta$ the fundamental quantity? Why should we arbitrarily consider this to be our metric of hotness and coldness, and not $T$?

This is a really interesting example to teach people to think in a positivist way in physics, and to operationalise things. What does it mean for something to be hot?

Well, you touch it and you say "Ouch!"

Seriously, that's all there is -- if you touch something hot, you say "Ouch!", if you touch something cold, you say "Whee!", or something. That's the fundamental, positivist definition of hotness -- "Does it feel hot?"

Well, why would something feel hot? Because it transfers heat to you. And this is our operational, positivistic definition of hotness -- if one body transfers heat to another body, it is said to be hotter than the other body.

So we need to find out a criterion to decide the direction of heat flow between two bodies. In the past, you've probably taken for granted that heat is transferred from a body with higher temperature to that with lower temperature, but that's just a crappy high school definition. What really causes heat diffusion? Well, when there are a lot of fast-moving particles in one place and slow-moving particles in another, it turns out that a state where the particles are more uniformly spread-out is more likely to happen in future. This is just the requirement that entropy must increase -- it's the second law of thermodynamics.

So if we have body 1 with temperature $T_1$ and body 2 with temperature $T_2$, with heat flow of $Q$ from body 1 to body 2, then the second law of thermodynamics is stated as:

$$\Delta S_1+\Delta S_2>0$$
$$-\Delta Q/T_1+\Delta Q/T_2>0$$
$$\Delta Q\left(\frac1{T_2}-\frac1{T_1}\right)>0$$
In other words -- if $\Delta Q>0$, i.e. if the heat flow is really from body 1 to body 2, then we require $1/T_2>1/T_1$, and if the heat flow is from body 2 to body 1 ($\Delta Q<0$), we require $1/T_1>1/T_2$.

And there you have it! Heat does not flow from the body with higher temperature to the body with lower temperature -- it flows from the body with lower $1/T$ to the body with higher $1/T$. For positive temperatures, these are the same thing -- but negative temperatures have the lowest $1/T$, and are thus hotter.

So those of you want the U.S. to switch to Celsius, or those who report temperatures in Kelvin for no good reason except intellectual signalling... perhaps start reporting statistical betas in 1/Kelvins instead.

"Hey, Alexa, is it chilly outside?"
"The coldness in your area is 0.00375 anti-Kelvin."

"...I think I'll just risk freezing to death."

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