[the__alchemist]

This is so cool! I just learned about this last week. For reference, I do molecular dynamics (my own engine, in rust), and measuring temperature is an important part of the simulation. (So you can nudge it to a target temperature, for example). An important component of this calculation is the degrees of freedom of the system. Calculating this depends on your model. For example, are you representing atoms that can each move on their own? Rigid molecules of multiple atoms that can rotate? Are you removing center-of-mass velocity from the system.

This DOF component also is why the general, measurable concept of temperature can apply to both our real systems, and simple point-atom models. (Or coarser ones). It is, not surprisingly, at the heart of why negative temperature exists!

[pama]

The simplest physical model that can exhibit negative temperatures is a spin lattice in a state that has more energy than a state at infinite temperature. Adding more energy to such a system reduces the entropy.

[dnautics]

negative temperature in this case is a sampling thing. When you sample from a table of tokens, the equation for the probability of token i is p_i = exp(logit_i/T) / sum_j(exp(logit_j/T))

Not really related to molecular dynamics temperature except superficially in terms of phenomenology (higher temperature crosses activation barriers in the joint probability landscape). Negative temperature makes no sense in MD

[zozbot234]

In a way, negative temperature is higher than the highest positive temperature. High positive temperatures just gives you a uniform distribution on all possible tokens, highly negative temperatures is the same behavior. As you reach the low-negatives, you place more and more weight on unlikely tokens.

This makes more intuitive sense if inverse temperature is the physically relevant quantity, since you then have a smooth change as you cross from positive inverse temperature into negative, with zero standing for a uniform distribution and high positive (resp. negative) inverse temperatures just placing more and more weight on likely (resp. unlikely) tokens.

[dnautics]

This is such a good way to put it (and it cleanly falls out of the exponential equation)

> inverse temperature is the physically relevant

right there in the equation!

[ggggffggggg]

This was super clear and interesting, thanks!

[the__alchemist]

Yea.... after a reread, I think this article may be getting at something else. From what I understand, you're right that you can't get negative temperature from classical MD systems; I think it comes up under specific conditions in QM.

[amluto]

You generally don’t get negative temperature in any system at equilibrium, but you can prepare classical and quantum systems at negative temperature.

Classical: put 100 balls in a box and shake the box continuously. The balls will be distributed through the box with more balls toward the bottom than the top, and the distribution will have some temperature. Now magically freeze all the balls (keep their velocities but pause time for a bit) and turn the box upside down. When you resume the system, the temperature will be (briefly) negative.

Quantum: take a bunch of atoms with two electronic states each. Put 75% in the higher energy state and 25% in the lower energy state. Now the temperature is negative. Most lasers actually work this way, and the classic way to make them is to have more than two states and to carefully excite atoms via the third state. The math is surprisingly straightforward.

There’s a nuclear analogue. If you could manage to prepare a sample of something like Technetium-99 plus Technetium-99m state with more (higher energy) 99m than (lower energy), then the effective temperature of the nuclear state would be negative. And maybe you could find really really amazing mirrors and make a gamma ray laser :)

[adrian_b]

Right. This is how you make a laser.

When passing through a material that is at thermal equilibrium, light is attenuated, because a part of it is absorbed. On the other hand, if the material is not at thermal equilibrium and you prepare it to have a negative temperature (by "pumping", so that more molecules/atoms/ions are in states with higher energy than they are in states with lower energy), then when light passes through the material it can be amplified (by stimulated emission), instead of being attenuated, like through a material at positive temperature.

Of course, in any laser/maser material, only a very small fraction of its constituents have an energy distribution corresponding to a negative temperature, there is no known method that could force a whole piece of a material to have a negative temperature (because the energy would redistribute spontaneously inside the material towards what corresponds to a positive temperature, faster than energy could be provided from the outside; lasers use some special energy states that have a low probability of decaying spontaneously, so they persist enough time after pumping).

[dgoldstein0]

Negative temperature happens in physical systems when there's a constrained state space and energy in the system comes near the maximum - as then adding energy reduces the number of possible states the molecules are in. Iirc the math works because temperature is the inverse of the derivative of entropy as a function of energy. So you need a system where entropy (number of possible states) decreases with more energy.

It's pretty rare to have such a system though.

[layer8]

> This is so cool!

Negative temperature tends to be that. ;)

[turzmo]

Negative temperatures are hotter than positive ones, in fact