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Comment by ChuckMcM | Hacker News Reader

That is a great exposition, it gets a bit bogged down in the money/happiness analogy at times, but overall I felt like I understood what you were saying. To test that, let me echo back what I think you said.

Physicists decided to define the temperature of a system not simply by the amount of energy in the system, rather to define it as the ratio of the energy in the system to the total entropy of that system. In so doing, they created a situation where systems that go to a lower entropy state when energy is added, are described as having a negative temperature.

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Small correction, not the ration of the energy to the entropy but the derivate of the energy by the entropy - that is, the rate at which the energy increases as the entropy increases.

(Of course, it's slightly more confusing since it's actually the inverse of the temperature that's defined as dS/dE, which is the rate at which the entropy increases as the energy increases, a much more understandable definition. And as the physics professor who taught this to a room of students from various departments said of the transition from

  1/T = dS/dE

  T = dE/dS

"The mathematicians in the crowd will be in an uproar now. But we can do stuff like that in physics")

Keep in mind that the relations T = ∂E/∂S and 1/T = ∂S/∂E are normally associated with different levels of abstraction:

T = ∂E/∂S is implied by the fundamental relation of (phenomenological) thermodynamics, whereas 1/T = ∂S/∂E gets derived from a microscopic theory via statistical mechanics.

It's fine mathematics too, it's just like the chain rule (or interpreting the d notation via infinitesimals/limits)

The problem is in the assumption that the inverse derivative exists, which is true for most functions in the real world but not for all (non-smooth) mathematical functions.

That sums it up pretty well.

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