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I read a few days ago that in the LHC temperatures of billions of degrees were achieved. I'm curious to know what does it really mean such a temperature? The concept of temperature is easy to grasp when the numbers are familiar, such as 100 C, but when it's in the range of millions it's difficult to understand.

Also, if you can provide some explanation of how the temperature is estimated it would be very helpful.

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It's very, very, very hot. :) Usually, temperature is estimated based on the energy of the radiation emitted by the hot body, just like what they do with stars. –  user172 Nov 8 '10 at 14:24
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:) it must be...but joking aside, maybe the better question would be "what's the physical meaning of temperature" –  Albert Nov 8 '10 at 14:26
    
I just hope it is not maximum achievable energy divided by Boltzmann's constant (-; –  mbq Nov 8 '10 at 14:29
    
Billions of degrees seems to be a vast understatement! The Boltzmann relation tells us that it corresponds to only a few hundered KeV, while the the LHC can reach into the TeV range. (I did this in a rush however, someone please verify this.) –  Noldorin Nov 8 '10 at 14:50
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The temperature of the quark-gluon plasma created in a nucleus-nucleus collision at the LHC is of the order of several hundred MeV, which corresponds to trillions of degrees. –  Igor Ivanov Nov 8 '10 at 15:12

3 Answers 3

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You posed two distinct questions:

  1. how is temperature defined as a physical quantity?
  2. how is temperature measured in these circumstances?

For the first question, temperature is defined as a thermodynamic quantity relating the change of entropy and the change of internal energy of a system. This is not very intuitive, I agree, but it is THE definition, and it make physicists sure they are talking about the same well-defined quantity.

A somewhat less correct but a much more intuitive definition of temperature is the amount of energy of the chaotic motion per particle. If your particles move chaotically very fast, near the speed of light, so that energy per particle is very large, you temperature is large as well.

As for the second question, physicists measure temperature of heavy-ion collisions indirectly, on the basis of several characteristics they observe in their detectors. The simplest way is by detecting of energetic photons and fitting them to thermal spectrum; another way is by studying the geometry of flow of hundreds of particles produced in the collision and fitting them to some models.

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As you can check in any good textbook, in thermodynamics temperature is defined via the following relation: 1/T = dS/dU. Thermodynamics operates with macroscopic parameters and does not know anything about the actual microscopic constituents of the system. This is in fact the power of thermodynamic formalism, as it can be applied not only to a system of particles that physically move in space, but also to spin systems, magnetic materials, black holes etc., that is systems in which the relevant degrees of freedom are internal. –  Igor Ivanov Nov 8 '10 at 17:41
    
Thank you for the answer, but how's the definition of temperature as "relating the change of entropy and the change of internal energy of a system" with "fitting energetic photons to thermal spectrum"? –  Albert Nov 8 '10 at 18:37
    
@Igor: sorry, I posted my original comment in error... I think I typed it out and then stopped to think about whether it was true, and somehow I clicked "submit" anyway despite thinking to myself that I shouldn't submit it. I must have been really tired or something ;-) Anyway, I deleted the comment to avoid misleading people. –  David Z Nov 8 '10 at 18:42
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Although unless I'm mistaken, for systems which admit the concepts of kinetic energy and particle number, the definition in terms of entropy is equivalent to one in terms of average kinetic energy. –  David Z Nov 8 '10 at 18:44
    
@David: no problem. And yes, there exists the equipartition theorem which states that we have kT/2 per degree of freedom including usual motion. Though, the exact coefficient (1/2 here) relies on the quadratic dependence of the hamiltonian on this degree of freedom; for ultrarelativistic particles it is just kT. –  Igor Ivanov Nov 8 '10 at 20:01
  1. We can't really understand weight beyond a few tons(or whatever you can lift ;) and less than a few grams.
  2. Our eyes can perceive only the visible spectrum.
  3. Our ears can't hear frequencies only within certain range.
  4. Similarly we can't feel temperature beyond some limit.

These limitations are imposed on our organs by evolution.

Its just too hot to be explained intuitively. We need some math ;-)

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The limitations that immediately came to mind for me were "Why doesn't the LHC melt?" But I already knew the answer to that - it's that the matter that has been heated so much has next to no mass, and as such isn't going to produce enough thermal radiation to warm a whole pizza. :) –  Ernie Nov 18 '10 at 0:30

Physics is about ideas and about experience, so when a certain number is impossible to perceive, the concept of understanding itself must be taken into account, i.e. we enter the realm of philosophy.

The problem does not only occur with temperature. What is a distance of 10^16 m? What does it mean to go back in time 10^15 years and from there 10^-3 sec forward?

Such numbers - as temperature in the range of millions degrees - emerge from calculations where experimental values have been inserted in a formula. The result (e.g. 10^6 degree Celsius) is not a reality in the sense of a perception by a human.

The understanding lies in the ability to tackle the mathematical side, the philosophical problem of reality is not taken into account.

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Why would this answer not be useful, care to comment? –  Gerard Nov 8 '10 at 21:29
    
Temperature is well defined without reference to human perception. That definition agrees with the human meaning in the range where they both apply, but is not restricted to that range. –  dmckee Jan 26 '12 at 2:47
    
@dmckee I fully agree. The OP explicitly asked about the meaning of temperatures in unfamiliar ranges, so I emphasized the ontological problem involved in that. –  Gerard Mar 20 '12 at 9:06

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