10 ms·
Negative temperature
- mjfl 5y agothe hottest temperature is negative 0 :)
- endymi0n 5y agoBeautiful. If computer science and physics had a child, this would be their Euler's Identiy.
- imoverclocked 5y agounsigned int t = 0; t—-; t > 0 … makes sense to me! :-P OTOH, I did learn a new way of thinking about temperature from this article. It’s still counter-intuitive and pretty fascinating.
- heisenzombie 5y agoHence why the one true temperature scale is thermodynamic beta. “Boy howdy, it’s a balmy 38 reciprocal electron-Volts outside! who wants to go for a swim?”
- xen0 5y agoThanks, but I'll be sticking to Kelvin* when setting my oven. * Or Rankine if you're American.
- bregma 5y agoI always assumed going below absolute zero was undefined behaviour. I guess someone just needs to rewrite all of physics in Rust.
- snthpy 5y ago"Temperature is loosely interpreted as the average kinetic energy of the system's particles. The existence of negative temperature, let alone negative temperature representing "hotter" systems than positive temperature, would seem paradoxical in this interpretation. The paradox is resolved by considering the more rigorous definition of thermodynamic temperature as the tradeoff between internal energy and entropy contained in the system, with "coldness", the reciprocal of temperature, being the more fundamental quantity. Systems with a positive temperature will increase in entropy as one adds energy to the system, while systems with a negative temperature will decrease in entropy as one adds energy to the system."
- nabla9 5y agoIt's very common to have apparent paradoxes when you have multiple definitions, one more intuitive and one more rigorous or specific. Popular science journals use group velocity, wave velocity vs. signal velocity confusion to write a "faster than light" article every year.
- dspillett 5y agoNot unlike the silly algebra tricks that “prove 0==1” and other such, which some of us played with at school. We had a good maths teacher who joined in, reasoned & unreasoned a few of them, and pointed us to off-curriculum things we might want to read. A nice lesson on the joy of thinking and applying critical thinking.
- toxik 5y agoFTL is easy. Send one object due north at 3/4 c. Send one object due south at 3/4 c. They are now separating at ~1.5c from your point of view.
- somebodynew 5y agoIsn't the speed at which one of the objects is moving away from the other 24c/25 per the relativistic velocity addition formula?
- walnutclosefarm 5y agoIt's quite possible to have the apparent rate of separation of two particles, from the standpoint of third frame of reference, be superluminal. It happens in astronomical observations. See http://spiff.rit.edu/classes/phys200/lectures/superlum/superlum.html http://spiff.rit.edu/classes/phys200/lectures/superlum/super... for a nice example and explanation. But you are quite right that from the frame of reference of either of the two particles sent traveling at .75c in opposite directions, the other particle is receding at subluminal velocity, not FTL.
- AlexandrB 5y ago
- yread 5y agoThere is also https://en.wikipedia.org/wiki/Negative_pressure https://en.wikipedia.org/wiki/Negative_pressure
- jeffwass 5y agoReminds me of a thermodynamics/statistical mechanics class homework problem from way back. Two systems with negative temperatures A and B, where |A| > |B|, are put in contact. Which way does the heat flow? As the article alludes, it can be easier to think about inverse temperature (which they called coldness), so heat will flow from less cold system B to more cold system A. In this context, negative epsilon would be the ‘hottest’ temperature, and a negative infinite temperature the same as a positive infinite temperature.
- carlob 5y agoOne of my favorite stat mech facts is that you can get to inverse temperature as a Lagrange multiplier using only probability and information theory: Given a system that has many states each with energy E_i, what is the highest Shannon entropy distribution that has average energy E? If you define a Lagrange multiplier \beta to enforce the constraint \sum_i p_i E_i = E and maximize the entropy you get the Boltzmann distribution!
- contravariant 5y agoOne minor nitpick is that in this derivation your choice of coordinates matters. Luckily (classical) physics provides a canonical set of coordinates which even comes with a measure that is preserved under the equations of motion. This magical property of the phase space can't be emphasized enough if you ask me.
- happytron 5y agoCan you expand on this? Is there a nicer coordinate-free way to define entropy?
- contravariant 5y agoThe definition of entropy is not coordinate independent. Very briefly the reason for this is that the uniform distribution isn't coordinate independent. There is a related coordinate independent way to measure distances from one distribution to another, which is the Kullback leibler divergence, and looks like: D_KL(P || Q) = \int dP/dQ log(dP/dQ) dQ = E_P[log(dP/dQ)] (This is by far the most general definition and uses the Radon-Nikodym derivative to be applicable to discrete, continuous and even weirder distributions. For reasons that I'll try to explain I also view this as a generalization of the notion of entropy) If you let Q be a uniform distribution this is equivalent to the entropy (up to some sign changes and a constant). However uniform distributions aren't coordinate independent and therefore the notion of entropy isn't. My personal conclusion is that there's no way of doing statistics without picking a(n improper) prior, since you inevitably need to pick some coordinates which ends up doing the same thing. You can then equivalently talk about the KL-divergence of a distribution from this prior or the entropy of a distribution, both end up being more or less the same thing.
- y3sh 5y agoSo these negative Kelvin temperatures are just values in excess of +infinite Kelvin? Reminds me of complexity classes in theoretical CS. It is odd though to join two together on +- scale, because it jams a non-enumerable set into enumeration by making infinity a single tick on the scale.
- gspr 5y ago> It is odd though to join two together on +- scale, because it jams a non-enumerable set into enumeration by making infinity a single tick on the scale. Can you elaborate? I fail to see what you mean.
- y3sh 5y agoThink of it like an unbounded array of integer temperatures values -- if you keep adding memory you can keep adding more positive values to the end of the array infinitely, each with an indexable location (i.e. enumerating the set). But this concept breaks if we just say we'll throw all the negative values on after infinity; if we're infinitely adding positive values to the array, we'll never have the chance to stop and start with those negative Kelvin values on after. When this happens it's called non-enumerable (or more formally fails the test of diagonalization). It seems that the authors of this system chose to make +infinity an arbitrary enumarable point to show the negative Kelvin values in excess of that point. Having said that, the set of all integers (negative, 0, positive) should be enumerable (because you can just *=-1 each index), but not when infinity is a member of that set. I think there's a numberphile and veratisium on why not all infinity's are equal if curious to explore.
- gus_massa 5y agoAdding infinity to a set does not change it from numerable to non-numerable. There are plenty of ways to add infinity to a set, for example, you can take the natural numbers N and just add infinity: 0, 1, 2, 3, ..., ∞ It's also possible to add another set of natural numbers after infinity, but in this case infinity is usually call ω: https://en.wikipedia.org/wiki/Ordinal_number https://en.wikipedia.org/wiki/Ordinal_number 0, 1, 2, 3, ..., ω, ω+1, ω+2, ω+3, ... Your example with an array is just: 0, 1, 2, 3, ..., ∞, ..., -3, -2, -1 It's possible to define a "<" relationship there and a topology and most of the other usual stuff, but it's a numerable set. There are similar constructions for non-numerable sets, like the real numbers. You can add one infinite on both sides and you get something that is topologically equivalent to the border of a circle, or add two infinites (one on each side) and you get something that is topologically equivalent to a closed interval, or add even more infinites https://en.wikipedia.org/wiki/Compactification_(mathematics) https://en.wikipedia.org/wiki/Compactification_(mathematics)
- state_less 5y agoIs the early universe, around the Big Bang, an example of high energy and low entropy state? Seems like we’ve been cooling off since, so we were hotter earlier and negative temperatures would be hotter than non-negative temperatures.
- toxik 5y agoIn the limit of the universe being a point, yes. Entropy must then have been zero since phase space is a point. But that’s just conjecture.
- roywiggins 5y agoThe very early universe more or less must have had much lower entropy than we do today, you don't need to wind back all the way. A uniformly mixed goop of a universe is going to be very low entropy due to all the different ways it could collapse under the influence of gravity.
- AnimalMuppet 5y agoCan you (or anyone) explain how entropy interacts with gravity? A uniformly mixed goop should be the highest entropy state, with stuff partitioned into clumps being a lower entropy state... apart from gravity. But if you add gravity to the mix, then... what?
- roywiggins 5y agoMy understanding is fully based on pop-sci, but: Eventually the influence of gravity starts generating black holes, which are really high entropy objects (in fact, for a given region of space, you can't actually fit any more entropy into it than a black hole does). So it seems to me that stars and planets and so on are intermediate steps between a hot, low-entropy universe at one end and a cold, high-entropy universe consisting of black holes. Eventually the black holes maybe evaporate leaving an extremely fine mist of radiation, but by that point space will have stretched out the the point that the undiffentiated radiation cannot collapse again. I found this from an actual physicist: "our gravitationally bound ball of gas has a negative specific heat! In other words, the less energy it has, the hotter it gets." https://math.ucr.edu/home/baez/entropy.html https://math.ucr.edu/home/baez/entropy.html
- rotskoff 5y agoStatistical physicist here. Negative temperatures occur when a system has a finite number of high energy states. On average, when temperature increases, both the energy and the "randomness" or entropy of a configuration increase as well. Of course, if there are only a few high energy states available, then the randomness will not increase, it will decrease. That's negative temperature! Because we define temperature as the rate of change of the energy with respect to the energy, in systems like the one I described, this rate of change becomes negative.
- Davidzheng 5y agoRate of change of entropy i presume is what you meant
- IgorPartola 5y agoSo the basic idea is that I’d you have a two particle system (a and b) and each particle can be in a low (a and b) or high energy (A and B) state then you can have the following combinations: ab, Ab, aB, and AB. Since particles are indistinguishable Ab and Ba are the high entropy states but ab and AB are low entropy states?
- Rayhem 5y agoBasically. Instead of a and b, let's just look at L(ow) and H(igh) energy states. If I have the ground state, L L L L there's only one way to arrange the system: everything in the low state, and the entropy is log(1) = 0. If I add one quantum of energy, I can have L L L H, L L H L, L H L L, H L L L and the change in energy is dE = 1, while the change in entropy is dS = log(4) - log(1). "Temperature" is really just a scaling factor between dE and dS, so in this case T > 0. Adding another quantum, L L H H, L H L H, H L L H, L H H L, H L H L, H H L L Again, dE = 1 (by construction), and dS = log(6) - log(4) which is less than log(4). dS > 0 so T is also greater than 0. Adding one more quantum, however, L H H H, H L H H, H H L H, H H H L and we have dE = 1 and now dS = log(4) - log(6)! We've added energy (conventionally made it "hotter"), but the system has become more ordered. Adding one last quantum, H H H H dE = 1 and dS = log(1) - log(4). This is as hot as the system can get, it cannot accept any more energy and can only give it away which is why "negative temperatures are hotter than all positive temperatures". If this system is brought into contact with any conventional positive temperature system, statistical fluctuations mean at least one of the energy quanta we added will flow towards it, cooling this system and heating the other.
- secondcoming 5y agoObligatory sixtysymbols video: https://www.youtube.com/watch?v=yTeBUpR17Rw https://www.youtube.com/watch?v=yTeBUpR17Rw
- petercooper 5y agoI'm so bad at (physical) science I had to see if "simple" Wikipedia had an entry for this because I couldn't follow it at all and.. it does! :-) https://simple.wikipedia.org/wiki/Negative_temperature https://simple.wikipedia.org/wiki/Negative_temperature
- Cthulhu_ 5y agoThanks for the link; I glazed over when the article said "Confined point vortices are a system with bounded phase space as their canonical momenta are not independent degrees of freedom from their canonical position coordinates." It's like I understand the individual words like phase, canonical and coordinates, but the combination is... an entirely different field of study.
- lekevicius 5y agoHuh, I stopped trying to understand at that sentence. Maybe Wikipedia could start linking to "simple" articles for these topics? Or at least include some more links to more basic concepts. That sentence is a fortress even for a curious person.
- petercooper 5y agoI do think Wikipedia should make the link to the "simple" branch more obvious on standard English Wikipedia. Currently they treat it merely as an obscure alternative language, but it could really help a lot of people if it were more prominently linked on each article that has a simple variant.
- crdrost 5y agoHaha, no yeah, that's a very jargony sentence indeed. Phase space is jargon for the set of all allowed configurations of a system. In the simplest case, an ideal gas of n spherical atoms, each atom is specified by 3 independent numbers for position and 3 for momentum, so you need 6 n numbers total to specify the whole configuration... That is the phase space. This phase space is only partly bounded, it is bounded in position (V^n where V is the volume each particle can occupy) but not in momentum (the temperature can always go higher). Because each of these numbers can vary independently from each other, they are called “independent degrees of freedom.” Now, point vortices are an abstract mathematical limit of physical vortices as they get really small. Generally we take these mathematical limits because it's much easier to work with and we are lazy. The article is saying that Onsager chose to explain his theory with point vortices because you can't give them arbitrarily high momentums, in fact their positions have to be tied up with where they are going, so that they are not independent anymore like they were with the gas. So Onsager knew that if he chose this system then the boundary on volume (well, area, the vortices live in 2D) would become a boundary on momentum space and then you couldn't dump infinite energy into the system. And this is important because it means that when you add more energy, the number of allowed configurations with that energy eventually shrinks. This is the negativity of negative temperature, increase something and something else decreases. The thing that is decreasing, a volume in phase space, is also called the entropy. Increase energy, and entropy decreases... most normal systems work the opposite way, entropy increases. As systems share things to maximize total entropy, this overcharged vortex system is going to share energy with any normal system at any temperature, because that normal system will increase its entropy with energy coming into it, and the overcharged vortex system will gain entropy when it loses energy. So it is hotter than any other thing in the world. (I am erasing the jargon word “canonical” from this explanation... It just refers to a particular procedure we use to define momentum coordinates based on your position coordinates. When you are not dealing with a point particle with defined mass in a flat space, you have to make these sorts of choices.)
- nayuki 5y agoA practical demonstration of negative temperature from The Action Lab: https://www.youtube.com/watch?v=jdjTYlReE-I https://www.youtube.com/watch?v=jdjTYlReE-I
- pkaye 5y agoLooks like our universe doesn't have overflow handling for temperatures. It just wraps around.
- aftbit 5y agoI was nerd-sniped by the units for thermodynamic beta (inverse temperature): gigabytes per nanojoule. I knew entropy and information science are related but I didn't realize I'd ever find myself looking at GB/nJ.