## What is the hottest possible temperature?

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### What is the hottest possible temperature?

http://en.wikipedia.org/wiki/Negative_a ... emperature

This article claims that its infinite since the momentum of an atom can be infinite. I think this is wrong though because its momentum should be limited by the speed of light.
gbagcn2

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### Re: What is the hottest possible temperature?

Actually it is possible to have infinite momentum due to special relativity. I suggest you read the stickied topic about it because it has the answer to your question and many more.
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bjgriffin4

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### Re: What is the hottest possible temperature?

It may help to know that both energy and momentum are infinite for a particle moving at c.
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danpilon54

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### Re: What is the hottest possible temperature?

What? No, if you are moving at c they are just your h\nu and h/\lambda.
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### Re: What is the hottest possible temperature?

This came up in a thread a few months back; basically, there is no theoretical limit on temperature. In reality, you're limited by the apparently finite amount of energy in the universe.
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### Re: What is the hottest possible temperature?

doogly wrote:What? No, if you are moving at c they are just your $h\nu$ and $h/\lambda$

A massive particle moving at c would have infinite momentum if it could exist
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### Re: What is the hottest possible temperature?

what of the plank temperature?
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### Re: What is the hottest possible temperature?

KyleOwens wrote:
doogly wrote:What? No, if you are moving at c they are just your $h\nu$ and $h/\lambda$

A massive particle moving at c would have infinite momentum if it could exist

That not existing is a bit of an obstacle though.
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### Re: What is the hottest possible temperature?

doogly wrote:
KyleOwens wrote:
doogly wrote:What? No, if you are moving at c they are just your $h\nu$ and $h/\lambda$

A massive particle moving at c would have infinite momentum if it could exist

That not existing is a bit of an obstacle though.

You can have a massive particle moving arbitrarily close to the speed of light, and therefore with an arbitrarily high momentum.

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### Re: What is the hottest possible temperature?

gbagcn2 wrote:http://en.wikipedia.org/wiki/Negative_absolute_temperature

This article claims that its infinite since the momentum of an atom can be infinite. I think this is wrong though because its momentum should be limited by the speed of light.

Yeah, because of the lorentz factor of 1/sqrt(1-v^2/c^2) that gets added in special relativity, momentum goes up to infinity as you approach the speed of light.

Also, this is the first time I've heard of "negative" temperature. It's really cool (yuk yuk), but it does seem like an abuse of the boltzmann distributionthermodynamics in general. I like the more basic definitions of temperature.
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Charlie!

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### Re: What is the hottest possible temperature?

Charlie! wrote:Also, this is the first time I've heard of "negative" temperature. It's really cool (yuk yuk), but it does seem like an abuse of the boltzmann distributionthermodynamics in general. I like the more basic definitions of temperature.

Actually it's really hot. Temperature runs from 0 to positive infinity and then from negative infinity to zero. So -5 K is hotter than 5 K. In fact it's hotter than 100,000,000 K or even infinite temperature. So the hottest possible temperature is, in fact, -0.

But that's a bit of a mathematical construct, I admit. In every day situations just goes from 0 upwards and there is no highest possible temperature.
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### Re: What is the hottest possible temperature?

KyleOwens wrote:
doogly wrote:What? No, if you are moving at c they are just your $h\nu$ and $h/\lambda$

A massive particle moving at c would have infinite momentum if it could exist

Is that like when they keep adding energy to particles in the LHC? As I understood it, it can't go any faster, so the excess energy gets turned into extra mass. Or something similar.

ArgonV

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### Re: What is the hottest possible temperature?

Nowadays we generally don't talk about an increase in mass, but rather an increase in momentum. The particles are moving faster; it's just that very fast things take a lot of energy to accelerate just a little bit. So the momentum increase is much greater per increase in velocity for things moving very fast.
Still, I can see the usefulness in a myth of the energy being directed toward increasing the momentum instead of the velocity.
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### Re: What is the hottest possible temperature?

Charlie! wrote:Also, this is the first time I've heard of "negative" temperature. It's really cool (yuk yuk), but it does seem like an abuse of the boltzmann distributionthermodynamics in general. I like the more basic definitions of temperature.

Actually it's really hot. Temperature runs from 0 to positive infinity and then from negative infinity to zero. So -5 K is hotter than 5 K. In fact it's hotter than 100,000,000 K or even infinite temperature. So the hottest possible temperature is, in fact, -0.

But that's a bit of a mathematical construct, I admit. In every day situations just goes from 0 upwards and there is no highest possible temperature.
Erm, there's definitely a disagreement with the equation E = 3/2 kT. It's a pretty fragile mathematical construct. *grumble grumble population inversions grumble grumble.*

Also, I'm not sure that any system with a population inversion will have heat flow to a system with positive temperature. imagine a box of 3 ping pong balls with one ball moving at 1 unit and 2 balls moving at 2 units (a simple negative temperature system). Now inject a ping pong ball moving at 4 units. There is no available interaction that will increase the temperature of the 4-unit ping pong ball. Or maybe I'm not thinking about this in the right way. Oh well, ignore me.
Last edited by Charlie! on Wed May 27, 2009 9:40 pm UTC, edited 1 time in total.
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### Re: What is the hottest possible temperature?

gbagcn2 wrote:This article claims that its infinite since the momentum of an atom can be infinite. I think this is wrong though because its momentum should be limited by the speed of light.

Nope. See, temperature is (simplification time) average kinetic energy. Kinetic energy can go as high as it wants, even with relativity. The reason you're limited by the speed of light is that it would require infinite energy, so velocity is bounded even though energy is not.
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### Re: What is the hottest possible temperature?

Negative temperature is really interesting. Temperature, defined statistically is the inverse of the derivative of the log of the number of accessible states of the system with respect to energy. Therefore, negative temperature implies that if you added more energy to the system, the number of states the system could be in would actually go down. In an infinite system this not possible, and in fact adding energy always increases the number of available states by larger and larger amounts, because there is no "maximum energy state."
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### Re: What is the hottest possible temperature?

When I enter a room.
hotnewrelease

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### Re: What is the hottest possible temperature?

Wiki says:
"At temperatures greater than or equal to TP, current physical theory breaks down"
So the answer is... we don't know?
DrZiro

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### Re: What is the hottest possible temperature?

For practical purposes in open systems, T_P is the hottest temperature that makes any sense. Is it not the temperature at which the average particle energy is one Plank energy?
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### Re: What is the hottest possible temperature?

Not being all mathy, I'd suspect that the hottest temperature yet achieved in our universe was at the point of the big bang. It's all been downhill (downthermometer?) since then.

What can theoretical physics tell us about the temperature of the big bang? Wait, I got Google...
Wikipedia wrote:Extrapolation of the expansion of the universe backwards in time using general relativity yields an infinite density and temperature at a finite time in the past. ... A few minutes into the expansion, when the temperature was about a billion (one thousand million; 109; SI prefix giga) Kelvin and the density was about that of air, neutrons combined with protons to form the universe's deuterium and helium nuclei in a process called Big Bang nucleosynthesis.

Huh. Infinite, even in theory, sounds pretty warm. A billion K doesn't sound much better.

Edit: given that quantum physics might have some things to say about special relativity, we might find that the BB wasn't infinitely anything, just really lots of everything. I seem to recall reading that some have proposed an hourglass shape to the universe's start, rather than a singularity.

Conventional big bang thingy. Infinite temp/density, singularity, start of everything.

One idea of a different big bang -- a big bounce. Not a singularity, just a really really really narrow little crunch where everything bashed into everything else quite hard, then exploded again. Not infinite in any measure, just really big (or really small, depending on the property being measured).

It would be interesting to find out what the temperature might have been at the point of a big bounce. That, then, would be the highest temperature this universe has experienced (as far as we know).
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### Re: What is the hottest possible temperature?

poxic wrote:It would be interesting to find out what the temperature might have been at the point of a big bounce. That, then, would be the highest temperature this universe has experienced (as far as we know).

Well it has to be at least the Planck temperature. Because we know that whatever happened at the Big Bang, our physics must break down there (because normal GR or QM can explain neither a big bang nor a big bounce, nor any alternative), so we must have at least the Planck temperature.

The Planck temperature is 1.416785(71) × 10^32 K

So... hot
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### Re: What is the hottest possible temperature?

Charlie! wrote:But that's a bit of a mathematical construct, I admit. In every day situations just goes from 0 upwards and there is no highest possible temperature.
Erm, there's definitely a disagreement with the equation E = 3/2 kT. It's a pretty fragile mathematical construct. *grumble grumble population inversions grumble grumble.*

Also, I'm not sure that any system with a population inversion will have heat flow to a system with positive temperature. imagine a box of 3 ping pong balls with one ball moving at 1 unit and 2 balls moving at 2 units (a simple negative temperature system). Now inject a ping pong ball moving at 4 units. There is no available interaction that will increase the temperature of the 4-unit ping pong ball. Or maybe I'm not thinking about this in the right way. Oh well, ignore me.[/quote]

E = 3/2 kT is only true for ideal gases. That formula doesn't hold in this context, or, indeed, in many contexts (eg. it doesn't hold for solids, in general). In general, temperature is defined as T = \frac{dE}{dS} with E being the internal energy, and S being the entropy. There is no particular reason that this gradient has to be positive.

For a simple example, take N particles with a ground state and an excited state. If they are all in the ground state, you have an entropy minimum--there is only one possible arrangement that this can take. If you increase one particle into the excited state, both the energy and entropy increase--so the gradient is positive. If the same N particles are now all in the excited state, again with have an entropy minimum--only one arrangement is possible. But here, if we drop one particle into the ground state, we decrease the energy but increase the entropy--the gradient is negative, and hence, so is the temperature. This is true at any point where there are more particles in the excited state than the ground state. For an ideal gas, there are an infinite number of possible states. If the total energy of the system is E, then the particles can arrange themselves in any configuration so long as their total energy remains the same. If we increase the energy to E + dE, then we increase the number of possible energy states for our particles, and thus increase the entropy. Since this is true for any finite energy, the temperature of an ideal gas is always positive. Negative temperature arises only in systems where the number of possible states is fixed (there may even be some pathological examples where the density of states grows sufficiently slowly with energy that a crossover might be possible, but I don't know of any with physical significance).
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### Re: What is the hottest possible temperature?

Surely it is anywhere a singularity occurs, due to the funky infinities involved? Anything with that many infinities must be doing something good with heat.
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### Re: What is the hottest possible temperature?

wst wrote:Surely it is anywhere a singularity occurs, due to the funky infinities involved? Anything with that many infinities must be doing something good with heat.

A quantum mechanical singularity isn't. An object with zero size and precise position is a violation of the Uncertainty Principle. There is broad agreement on this. We don't yet have a full theory of Quantum Gravity, but we believe we understand some of the broad principles.

The Plank Temperature is the temperature of a black hole with a Schwarzchild Radius of the Plank Length. I imagine a argument could be made that trying to exceed this temperature would result in the formation of a black hole, which does have a maximum temperature.

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### Re: What is the hottest possible temperature?

http://news.yahoo.com/s/nm/20100215/sc_ ... emperature

Scientists recorded a temperature of 4 trillion degrees Celsius? That is outrageous.

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### Re: What is the hottest possible temperature?

Diadem wrote:Well it has to be at least the Planck temperature. Because we know that whatever happened at the Big Bang, our physics must break down there (because normal GR or QM can explain neither a big bang nor a big bounce, nor any alternative), so we must have at least the Planck temperature.

The Planck temperature is 1.416785(71) × 10^32 K

So... hot

Seems like there's an xkcd comic in there somewhere. Something involving a physicist complementing a girl about how hot she is....
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### Re: What is the hottest possible temperature?

Tempuratures can be as hot as you want but eventually you get to the point where it's not really "hot gas" or "plasma" but "a few relativistic velocity particles held in a magnetic field, that spread out in flash of radiation if the field goes down"
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### Re: What is the hottest possible temperature?

That's assuming a nondestructive lab setup. Much more interesting would be a configuration of super H bombs arranged on the vertices of an icosahedron or something, that compresses the center to something absurd

With big enough bombs, you could in theory make a micro black hole this way (and there's no theoretical limit to the size of an H bomb), which would evaporate, peaking around the Plank Temperature before it evaporates.

You might be able to make some with a particle accelerator, but that requires funky spacetime geometries (large extra dimensions) or a superpowerful collider, perhaps so much so to be less practical than the bomb approach.

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### Re: What is the hottest possible temperature?

ikrase wrote:Tempuratures can be as hot as you want but eventually you get to the point where it's not really "hot gas" or "plasma" but "a few relativistic velocity particles held in a magnetic field, that spread out in flash of radiation if the field goes down"

Even if you keep it contained, synchrotron radiation ought to leech significant amounts of energy out of the system. I'm not confident such a state of matter is sustainable for particularly long.
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