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Comment by drexlspivey | Hacker News Reader

Does temperature have a theoretical upper limit ?

Not in classical thermodynamics, but temperatures above the Planck temperature aren't understood with current quantum models. It would probably require a theory of quantum gravity to shed further light on this.

No, it does not. In a closed system if you add enough energy the temperature will eventually become negative (after ‘passing through’ +inf).

As you reach a state where almost all particles are in their maximum energy states (this is assuming there is one) you will slowly approach negative zero (which again you can never quite attain).

Statistical mechanics can be confusing at first.

I am not sure about your first statement...

In particular even with a quantum non-interacting gas with particle-in-a-box modes, the Hamiltonian is not bounded from above and there is no reason to expect a negative temperature, no?

There exist systems, like spin systems, where energy is bounded from above and so entropy decreases as you add energy, which is the definition of negative temperature... But I find it dubious that every system is such, unless I am missing something nonintuitive about say relativistic effects or so

No you're quite right, that's why I said "...maximum energy states (this is assuming there is one)". Of course with no upper bound on energy this isn't possible, that's the definition straight up, just like you say.

So what happens if I stick my hand in a negative temperature state of matter?

You'd be fine, because any kind of matter that you could stick your head into is incapable of reaching negative temperatures.

Sure you can, gas lasers exist. Stick your hand into the gain medium and have fun getting fried!

Wikipedia article on the concept of negative temperature: https://en.m.wikipedia.org/wiki/Negative_temperature

The critical bit for me:

> This is only possible if the number of high energy states is limited. For a system of ordinary (quantum or classical) particles such as atoms or dust, the number of high energy states is unlimited (particle momenta can in principle be increased indefinitely). Some systems, however [...], have a maximum amount of energy that they can hold, and as they approach that maximum energy their entropy actually begins to decrease.

In my (limited) understanding, it's somewhat like the phenomenon that a communication channel bit error rate over 0.5 actually results in less information loss (imagine a BER of 1: that's just a NOT gate).

If your energy states are limited, adding energy actual brings you closer to an ordered state (that of everything being in the highest state).

But, this is not a situation you get by simply heating something up with a blowtorch, no matter how hot it is.

This is simply reaching the integer overflow of our simulated world

interestingly, no. Temperature can become infinitely high, and you can still add heat, at which point it becomes infinitely low. You can carry on adding heat, and the temperature will get back to absolute 0. https://chemistry.stackexchange.com/questions/36885/how-is-n...

You can't heat something to negative temperature, although negative temperature things will transfer energy to positive temperature things. Negative temperature can be achieved through lining up many small magnets against a larger magnetic field. Disordering the magnets will reduce the potential energy, running opposite to the usual trend where increasing disorder involves the occupation of higher-energy states.

How does it become infinitely high? Temperature is a measure of average particle speed, and that’s limited to the speed of light

(I’m not a physicist so I’m willing to be corrected, but this doesn’t jibe with my low level compulsory physics courses from uni :) )

Temperature has to do with how the entropy changes with the addition of energy. It helps to use the “coldness” or “thermodynamic beta” scale, β = ΔS/ΔΕ is the coldness of a system, the thermodynamic temperature is defined as 1/(k β) where k is a conversion factor, the Boltzmann constant, to convert between units of energy and kelvins.

For most normal systems, entropy increases with an addition of energy, and they have a positive coldness. Confusingly, the lower the entropy change, the less cold or hotter we would regard it: if you bring two systems into contact, they share energy to maximize their total entropy, so something which has low coldness = low entropy change will donate a lot of energy to something with a higher coldness = higher entropy change, the smaller negative will be balanced out by a larger positive.

You can extrapolate this to an infinite temperature, this would be an object with β = 0 or zero coldness, it can take or lose energy without changing its entropy at all. An example is an assembly of electron spins in a magnetic field, when 50% of them are aligned with and 50% are aligned against the magnetic field: this is the most entropic that the spin system could possibly be, so there is no way to increase it and to first order changes in energy do not decrease it. It has zero coldness or infinite temperature.

Add a little bit of energy and it is in the state where it actively wants to lose energy, putting more energy into the system requires aligning more of the spins along the magnetic field. This is a negative coldness, which is also regarded as a negative temperature by this T =1/(k β) formula.

> Temperature can become infinitely high, and you can still add heat, at which point it becomes infinitely low. You can carry on adding heat, and the temperature will get back to absolute 0.

Perhaps the correct measure is not temperature, but inverse temperature (i.e. 1/T)?

Not a physicist, if temperature of particles is movement (I think chemical bonds will break long before the following limit), then the upper limit is just below the speed of light. To make it even hotter would require infinite amount of energy. Now what that temperature is in kelvin, I don't know.

Temperature is more about the kinetic (and other) energy of the particles than the velocity — you can keep adding energy indefinitely, or at least until you hit some kind of planck-scale weirdness point, even though the velocity is only asymptotically approaching c.

Enough energy in one place will create a black hole.

For each specific fusion reaction there is an optimal temperature (for maximum reactivity). Usually around a billion kelvin or so, plus or minus a few orders of magnitude.

Yes, 1.42 x 10^32 K but it's not as straightforward as naming a single limit [1].

[1] https://www.popsci.com/article/science/ask-anything-whats-ho...

Children's TV level astrophysics (it's not astrophysics, but that's the context where stuff like that is presented?) suggests that if Brownian motion approached 1c, mass would grow towards the infinite. So you could always add even more energy?

(sorry, can't provide anything beyond that level)

More children's TV level of astrophysics: so you keep adding energy to your particles, they approach 1c, they gain and gain mass... until they collapse into nanoscopic black holes and immediately evaporate into hawkins radiation, right?

I mean that's the general upper limit on stuff in the universe: it eventually collapses into a black hole.

The mean velocity in a gas is https://en.wikipedia.org/wiki/Thermal_velocity

v ~= Sqrt( k_B * T / m) ~= constant * Sqrt(T)

(There is another constant in the formula that depends on what definition of mean you use, but it's safe to ignore it for this discussion.)

So if T is big enough, the result of this formula is faster than light.

But this formula is useful only for a not relativistic gas. Once the temperature is so big that relativistic effects are important, you must use another formula. (The other formula is more difficult to calculate, but when the temperature is low the result is almost identical to the formula I wrote.)

Temperature has no theoretical upper limit, but if it's high enough weird things can happen as described in a sibling comment. More details in https://en.wikipedia.org/wiki/Planck_units#Planck_temperatur...

That question has an answer in another question.. does energy have a theoretical upper limit? Lowest temperature is absolute zero, no movement.

Yes it does - the plank temperature, where the wavelength of light emitted is the plank length. Current theories predict that this would be the maximum possible temperature. Of course, this is so hot as to be totally irrelevant to anything practical.

I mean the obvious upper limit is the particles moving at C, but I suspect quantum physics means that the limit is actually below that (my uneducated low level undergrad physics courses make me assume some relationship to the plank constant)

You'd eventually have so much energy that the mass energy relation would give you a black hole.

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