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Comment by aatish | Hacker News Reader

Hi. I'm the author of the post. I realize that it's a somewhat challenging read, at least compared to my usual style, but my goal was to stay as true to the physics as possible. I was a little disappointed with some of the popular science coverage, because some sources made it seem like this is the first time that physicists reached negative temperatures (not true), or that negative temperatures aren't physical (they are, but they're weird!). I hope you find it interesting.

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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.

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")

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.

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.

That sums it up pretty well.

I thought you did a great job of using metaphor to explain a dense topic. In fact, my main complaint is that your allusions to human behavior actually confused, or at least got in the way of the explanation...I think the metaphorical part should be less elaborate, because your primary audience is going to be (I'm being solipsistic here) are going to be people like me with a casual interest in physics who still remember the implications of absolute zero at the atomic level...no need to extend the metaphor in a way to appeal to people who think absolute zero is 0 celsius.

Thank you. What I liked about the metaphor is that it fairly accurate. But you may be right about the overuse - it's hard for me to work out the right balance when explaining physics, and I perhaps err on over-explaining. Then again, no one really complains if you make something too easy to understand! :)

I would improve it by defining the relationships between money/happiness and energy/entropy before giving the example. I would have found it a lot easier to read.

Thanks, that's a nice, concrete suggestion and I see your point.

Nice explanation for the layman, and it's certainly not the first time - messed around with negative temperatures in BECs as an undergrad!

Thing is, when you understand why it happens, it seems perfectly rational and expected. It's all about information.

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