What gives a better intuition is thinking in inverse temperature.
Regular temperature is, ‘how weakly is this thing trying to grab more energy so as to increase its entropy’.
Inverse temperature is ‘how strongly...’ and when that gets down to 0, it’s natural to see it continue on into negatives, where it’s trying to shed energy to increase its entropy.
The energy/entropy plot makes total sense, the energy/temperature doesn’t really because I don’t have a good feel for what temperature actually is, even after reading the “Temperature” section of your argument (it previously made sense because Mathematica was only showing me the linear-like part of the graph). Can you recommend a good text to improve my intuition? Bonus points if this recommendation arrives in the next 9.5 hours, because then I can get the book from my university library.
Depends on your background in physics. Landau & Lifshitz Statistical Mechanics is probably the best, but you won’t get much out of it if you haven’t taken some physics courses.
I made a plot of the entropy and the (correct) energy. Every feature of these plots should make sense.
Note that the exponential turn-on in E(T) is a common feature to any gapped material. Semiconductors do this too :)
Why did you only show the E(T) function for positive temperatures?
This is a good point. The negative side gives good intuition for the “negative temperatures are hotter than any positive temperature” argument.
What gives a better intuition is thinking in inverse temperature.
Regular temperature is, ‘how weakly is this thing trying to grab more energy so as to increase its entropy’.
Inverse temperature is ‘how strongly...’ and when that gets down to 0, it’s natural to see it continue on into negatives, where it’s trying to shed energy to increase its entropy.
No reason. Fixed.
The energy/entropy plot makes total sense, the energy/temperature doesn’t really because I don’t have a good feel for what temperature actually is, even after reading the “Temperature” section of your argument (it previously made sense because Mathematica was only showing me the linear-like part of the graph). Can you recommend a good text to improve my intuition? Bonus points if this recommendation arrives in the next 9.5 hours, because then I can get the book from my university library.
Depends on your background in physics. Landau & Lifshitz Statistical Mechanics is probably the best, but you won’t get much out of it if you haven’t taken some physics courses.