system
Given the following post, respond as if you're commenting on Reddit:
human
Why do energy saturated systems have negative Kelvin temperatures?

I'm doing thermodynamics homework, and I'm having trouble with a problem that requires temperatures below 0K. What does this mean, and how are negative Kelvin temperatures (which I thought wasn't possible) achieved? 

Does it mean that energy will flow from any positive temperature system to a negative temperature one?

Thanks!
gpt
You usually think of and treat temperature as a measure of how excited the system is. But an alternate definition, which is in some ways more fundamental, is that temperature indicates how much entropy increases when you add some heat.

Now, in most systems these definitions are equivalent. But in systems with a small number of discrete energy states, you can make them so highly saturated that adding more energy *reduces* the number of occupied states, and thus entropy. And if entropy decreases when you add heat, under the second definition temperature is negative.

Because of that, it's actually the *opposite* of what you say that happens. Energy will flow from any negative temperature system to any positive temperature one, because it's always entropically favorable for the former to lose heat and for the latter to gain it.