# Infinite temperature

#### Schrodinator

Was reading Kroemer on high temperature mechanics, infinite temperatures and their dependance on negative kelvin temperatures.

One question does that mean that its actually physically possible to measure something as having infinite heat or temperature? That seems paradoxical to me? Or is it just a theoretical limit of the calculus within which temperatures may exhibit finite temperatures.

A quick google turned up absolute hot, which is AKA the planck temperature or the temperature at the big bang, so this would seem to contradict an actual measurable infinity.

Am I getting unnecessarilly confused as I always assumed infinity was a purely mathematical concept, not a physical reality? For example in kind of the same way 0k is not achievable infinite k is not achievable in experiment.

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#### Drakkith

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Infinite temperature would require more energy than is contained in the universe since you would need infinite energy. Or at least it would require ALL of the energy if our universe is infinite. But I think I just gave myself a headache with that lol.

#### SpectraCat

Science Advisor
Was reading Chromer on high temperature mechanics, infinite temperatures and their dependance on negative kelvin temperatures.

One question does that mean that its actually physically possible to measure something as having infinite heat or temperature? That seems paradoxical to me? Or is it just a theoretical limit of the calculus within which temperatures may exhibit finite temperatures.

A quick google turned up absolute hot, which is AKA the planck temperature or the temperature at the big bang, so this would seem to contradict an actual measurable infinity.

Am I getting unnecessarilly confused as I always assumed infinity was a purely mathematical concept, not a physical reality? For example in kind of the same way 0k is not achievable infinite k is not achievable in experiment.
Infinite temperature (or negative temperature) is something that can only be achieved for a system with a finite number of discrete energy levels, and a finite number of particles populating those levels. In that case, infinite temperature is characterized by case where all of the levels, from lowest to highest, have populations that are equal within small statistical fluctuations. If you create a population inversion in such a system (as in a laser), then the system can be said to have negative temperature by the same definition.

However it is important to realize that these are highly-specific cases which can only be achieved with the input of more energy than comes out, so everything is consistent with the 2nd law (and 3rd law) of thermodynamics. In the broader sense of "how hot can I get this arbitrary sample of bulk matter", infinite temperature is fundamentally unobtainable, as is negative absolute temperature.

#### Schrodinator

Infinite temperature (or negative temperature) is something that can only be achieved for a system with a finite number of discrete energy levels, and a finite number of particles populating those levels. In that case, infinite temperature is characterized by case where all of the levels, from lowest to highest, have populations that are equal within small statistical fluctuations. If you create a population inversion in such a system (as in a laser), then the system can be said to have negative temperature by the same definition.

However it is important to realize that these are highly-specific cases which can only be achieved with the input of more energy than comes out, so everything is consistent with the 2nd law (and 3rd law) of thermodynamics. In the broader sense of "how hot can I get this arbitrary sample of bulk matter", infinite temperature is fundamentally unobtainable, as is negative absolute temperature.
Thanks that was helpful. So essentially the integral in question is limited to finite real temperatures ie non infinite kelvin temperature. Does that mean that in an experiment as the wiki opined the absolute hot is the temperature of the universe at the big bang?

Absolute hot is a concept of temperature that postulates the existence of a highest attainable temperature of matter. The idea has been popularized by the television series Nova. In this presentation, absolute hot is assumed to be the high end of a temperature scale starting at absolute zero, which is the temperature at which entropy is minimized and classical thermal energy is zero.

Current cosmological models posit that the highest possible temperature is the Planck temperature, which has the value 1.416785(71)×1032
kelvin. The Planck temperature is assumed to be the highest temperature in conventional physics because conventional physics breaks down at that temperature. Above ~1032K, particle energies become so large that the gravitational forces between them become as strong as any other force and are identical in the Grand Unified Theory.[citation needed]

Some forms of string theory allow a temperature of 1030K, known as Hagedorn temperature.[citation needed]

Quantum physics formally assumes infinitely positive or negative temperatures in descriptions of spin system undergoing population inversion from the ground state to a higher energy state by excitation with electromagnetic radiation. The temperature function in these systems exhibits a singularity, meaning the temperature tends to positive infinity, before discontinuously switching to negative infinity. However, this applies only to specific degrees of freedom in the system, while others would have normal temperature dependency. If equipartitioning were possible, such formalisms ignore the fact that the spin system would be destroyed by the decomposition of ordinary matter before infinite temperature could be reached uniformly in the sample.
 See also
http://en.wikipedia.org/wiki/Absolute_hot

I ask because this article is uncited, and wiki is not always that reliable. However the bolded part does agree with what you said.

The bolded bit is from the work I am reading. This is where I got confused. C. Kittel, H. Kroemer (1980). Thermal Physics (2 ed.). W. H. Freeman Company. ISBN 0-7167-1088-9 btw.

I also seem to be having a bit of a disagreement with a friend on this who insists, that infinite temperature is a natural consequence of negative kelvin states and that such a material would be measured to be infinitely hot ie the thermometer would actually read ∞ .

I presume I can now tell him he's mistaking a maths model for an experimental model (although usually they are pretty much the same thing).

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#### Schrodinator

Infinite temperature would require more energy than is contained in the universe since you would need infinite energy. Or at least it would require ALL of the energy if our universe is infinite. But I think I just gave myself a headache with that lol.
Infinite temperature and infinite energy aren't the same thing. The infinity arises from the spin states of the medium, however as the wiki above opines these spin states would decohere before infinite temperature could exist. . Since the highest achievable temperature should be the termperature at t>0 after the big bang, that should be absolute hot, at this energy level as it says all the forces are the same and all is energy. Obviously though a spin like a circle can be infinite and still finite If you see what I mean, damn now I have a headache. lol.

http://www.cartage.org.lb/en/themes/Sciences/Physics/Cryogenics/Temperature/negativeKelvin/Infinitetemperature/Infinitetemperature.htm [Broken]

This link is quite informative and you can see where the infinity arises in the maths.

System at infitine temperature
At equally populated levels, temperature, but not the energy, of a system is infinite

I would like to elaborate another nice conclusion from the formula. If both levels are equally populated, N(hi) = N(lo) and the logarithm vanishes, so that

The value of the temperature becomes infinite (note that DeltaE does not change). Since number of particles are always finite, only an finite amount of energy is needed to get infinite temperature in this model. Infinite temperature cannot be reached anyway by heating, although not because of the energy needed, but because heat flows from high temperature to low temperature. To heat a body, a hotter body is required.
As Einstein famously said: Last edited by a moderator:

#### Naty1

Sofar as is known, there is no observable that is infinite in our universe.

But sometimes the mathematical models we use imply that there is something like that at the big bang and at black hole singularities.

On the other hand there are LOTS of different types of infinity...so who knows....

check here for a quick survey:
http://en.wikipedia.org/wiki/Infinity

#### Schrodinator

Yes most people understand that an infinity is a problem to reality. Sadly I cannot convince some people that maths is not equal to reality in all cases, I end up being accused of being a crackpot because I don't think physical infinities can be measured and it's really annoying.

I know what infinities are mathematically, and that some are countable and finite like the universe and some are beyond all limits, sadly trying to explain to people that calculus or set theory limits reveals nothing but ignorance in people if they think it applies to experimental values is a waste of time, they just insist maths = reality; for people studying science it must really piss them off,it does me! Gah maths was invented it is not the ultimate reality what the hell is your major malfunction?

"It is good that we have met with a paradox, now we may begin to make progress."

Niels Bohr.

Einstein was right human stupidity is infinite, uncountable and beyond all bounds. The singularity is where physics breaks down. Most think it is not infinite gravity as that would break conservation of energy laws. Most people then go on to say the maths is not apt, some however go on to say reality is not apt. It's these fools I have to deal with. I'm by no means an expert but I get so fed up of dealing with supposed experts who will argue till they are blue in the tooth, that if the maths says one thing it must be reality! Are you a moron? Is the only question I have for them. I do understand what Einstein meant about infinite stupidity at this point, that was the one thing he could be sure of. Last edited:

#### Drakkith

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The singularity is where physics breaks down. Most think it is not infinite gravity as that would break conservation of energy laws.
Why would that break the conservation of energy laws?

#### Schrodinator

Why would that break the conservation of energy laws?
Infinite gravity requires infinity mass/energy. Singularities have infinite gravity in an infinitely small space, well at least the equations do, again another example of maths likely being just flat out wrong. Any infinity physically contains more energy than is in the universe by definition. Still I've even had discussions with professionals who thinks because the maths has a singularity or infinity it actually is infinite in reality. Which is worrying.

#### Drakkith

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2018 Award
Infinite gravity requires infinity mass/energy. Singularities have infinite gravity in an infinitely small space, well at least the equations do, again another example of maths likely being just flat out wrong. Any infinity physically contains more energy than is in the universe by definition. Still I've even had discussions with professionals who thinks because the maths has a singularity or infinity it actually is infinite in reality. Which is worrying.
Why would that require infinite energy? The gravitational force from a mass will warp spacetime around it. The denser something is, the more spacetime is warped. (Meaning that if you compress the earth to, say twice its density, the gravitational force is the same at a distance equal to or greater than the original surface, but from there to the new surface the gravity is increased.) Once something is black hole density, it is compressed effectively into an infinite density, correct? I don't see how that requires infinite energy. The strength of the gravitational field overall hasn't changed.

#### Schrodinator

Why would that require infinite energy? The gravitational force from a mass will warp spacetime around it. The denser something is, the more spacetime is warped. (Meaning that if you compress the earth to, say twice its density, the gravitational force is the same at a distance equal to or greater than the original surface, but from there to the new surface the gravity is increased.) Once something is black hole density, it is compressed effectively into an infinite density, correct? I don't see how that requires infinite energy. The strength of the gravitational field overall hasn't changed.
I'm really not in the mood to quibble over semantics so I'll just end it here. Believe I meant whatever you like. Knock yourself out I'm bored already.

The integrals in the equations throw up infinity x infinity, this implies infinite quantities and energy in an infinitesimal space end of story. Not going to say any more because frankly I wasted my time even typing this.

I have an answer to my question, you are more than welcome to start a new thread and make out I said something contentious. I will not post though.

#### Drakkith

Staff Emeritus
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2018 Award
I'm really not in the mood to quibble over semantics so I'll just end it here. Believe I meant whatever you like. Knock yourself out I'm bored already.

The integrals in the equations throw up infinity x infinity, this implies infinite quantities and energy in an infinitesimal space end of story. Not going to say any more because frankly I wasted my time even typing this.

I have an answer to my question, you are more than welcome to start a new thread and make out I said something contentious. I will not post though.
I'm not sure why you are so disgruntled, but ok.

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