thermodynamics - How is Negative Temperature Hotter than Infinite Temperature?

Recently I have read articles stating that negative Kelvin has been achieved. I was a bit speculative at first as I though how can you get cold than 0 Kelvin, but after doing some research it makes sense on how this is possible.

However the one thing that I still not sure is that it stated that the temperature scale can't be thought as being linear but rather as a loop which goes in the following order, "+0 K, … , +300 K, … , +∞ K, −∞ K, … , −300 K, … , −0 K".

How can negative Kelvin be hotter than infinite temperature?

Answer

The other answer given is correct, but for completeness, I will give the thermodynamic, rather than statistical mechanics, explanation.

Thermodynamics gives us that $$\frac{\partial E}{\partial S}=T$$

This means that when entropy--as a function of energy--increases, the temperature must also increase.

On a macroscopic level, this makes perfect sense because you're never going to think about a system reaching a maximum temperature because energy and entropy are always going to increase together.

Of course, it must be remembered that this definition of temperature was being given long before people were thinking about quantum mechanics. Even more, they were only concerned with temperatures because they realized that they could use a temperature difference to get work out of a system. A good example of this is the Carnot heat engine. The reason I reference the Carnot heat engine is that it demonstrates that when temperature was being defined mathematically, which was around this time, they were concerned with heat flow. We know that heat flows from a hot body to a cold body, so this means, almost exclusively, that heat flows from high energy to low energy.

For instance, in a system with only two energy states containing $N$ particles, the state of maximum entropy is when $\frac N2$ particles are in both states 1 and 2. The state of maximum energy, however, is when all $N$ particles are in energy level 2.

That means, if we push beyond the state of maximum entropy, energy will increase as entropy decreases, hence, $\frac{\partial E}{\partial S} < 0$.

So, negative kelvin can be "hotter" than infinite temperature because our definition of temperature has nothing to do with "hotness" as we feel on the macroscopic level, it is merely an exchange between energy and entropy.

Furthermore, negative kelvin is hotter than infinite temperature, not because it would feel hotter, but because heat flows from the negative temperature object to the positive temperature object because we can demonstrate mathematically that the negative temperature object is in a higher energy state. It makes sense to think about it that way because 90% of chemistry is just looking for the low-energy state.

In regards to the issue of our temperature scale going in a loop, there is a solution to this using the thermodynamic beta, $$\beta=\frac{1}{kT}$$Consider this graph which shows that temperature as a function of energy is asymptotic and thus discontinuous, while $\beta$ is continuous as a function of energy.

Also worth pointing out is that $\beta$ has units of energy, and is thus more physically relevant than temperature because energy is fundamental to all matter, while temperature is basically nothing more than a mathematical construct to describe "hot" and "cold".