Years ago I addressed "hot" in terms of molecular speed, applying relativistic limits, calculating an upper limit on temperature and finding a large, but not inconceivable, maximum. Alas I've lost the paper (which was probably wrong anyway, but it was a fun exercise). The question has long bugged me: is there a real upper limit on temperature?
> applying relativistic limits, calculating an upper limit on temperature and finding a large, but not inconceivable, maximum.
There is no relativistic limit to temperature. You probably calculated temperature by using the speed of the particle (and assumed it to max out a c), but at relativistic speeds this is wrong, you should use the energy of the particle instead.
The question is: Does particle A impart energy to particle B, or does it take away energy? Whichever is faster imparts energy to the slower one. It makes no difference that the speed is limited - one particle can always have more energy than the other, and can therefor impart energy to it. It's not possible to actually reach that speed, so no matter how fast a particle is moving, you can always increase its speed.
Time dilation should cause some very interesting artifacts though, I have no idea how to go about calculating what would happen. Length dilation is even more interesting - you may find your particles are so small that they pass each other and don't collide.
However it's not actually the particle that collides, it's the electric fields that collide. But electric fields travel at the speed of light - so the fields won't really have time to notice each other.
And finally at these energies a collision is unlikely to bounce, instead it would create some new particles, using up the energy and cooling rapidly.
Is there any simple way of explaining why our model would break down at that point?
The first (utterly naive) guess that springs to my mind would be that at that point the vibration would be sufficiently fast that when sampled at a Planck time interval, there'd be no motion at all. Like overflow of an unsigned int. This is almost surely wrong. :)
The abstract definition of temperature is that you have some set of modes (mode = possibly-occupied quantum states with fixed energy) and that there is enough interaction between these modes such that their probability of being occupied has a characteristic dependence on their respective energy. This characteristic dependence is a thermal distribution (which is either Bose-Einstein or Fermi-Dirac statistics).
According to wikipedia (heh), quark-gluon plasma is not really a plasma so much as a liquid. Presumably, that means the modes are still roughly momentum modes, and you can literally think of a hotter QGP as faster quarks and gluons sliding around and slamming into one another. It's a Fermi liquid, though, which means the exclusion principle has significant effect. (I'm a physics grad student, but not a high-energy theorist, so take that for what it's worth.)
There's certainly a relativistic limit - i.e. if the mean velocity of randomly oscillating particles in the gas / liquid approaches c. (I leave aside the question of what is used to confine such highly energetic matter)
That in itself wouldn't limit the temperature. The average speed of the particles could approach c and their average energy would continue increasing.
Molecules will break apart above a few thousand degrees.
If you take a gas (say Helium) above a certain temperature (probably ~1e4 K) the collisions between the atoms will knock the electrons off and you'll be left with a plasma.
Confining a plasma could either be done with magnets (e.g. a fusion reactor), or with more plasma! (in the case of stars, where the gravity keeps itself from exploding).
Above 1e7 K you start to get fusion of Hydrogen, and I think above 1e11 K pretty much any nucleus will fuse/break down.
So it's difficult to say there's a maximum temperature, as it's quite hard to define.
What's your maximum temperature? About 41'C... above that and you'll die...
I've been in a dry sauna as the temperature gradually reached its max peak of 105 C. At that temperature, the trick is to not move so that there's a relatively static layer of air around you. You also have to breathe slowly so as not to burn your nostrils/throat. And you perspire like crazy and the evaporation of your perspiration cools your body. It's still crazy hot though and you can't last very long before having to exit.
Agree that it doesn't limit the temperature, although it makes the thermodynamics / statistical physics more complicated. You can certainly keep pumping more and more energy into the system even if the particles' mean velocity approaches c.
And yes, these are well above biologically-friendly temperatures. :)
I believe the limit is energy density. As the mass of the particle gets high enough (e=mc^2), its gravity gets stronger until it becomes a black hole and then explodes in a puff of Hawking radiation.
Years ago I addressed "hot" in terms of molecular speed, applying relativistic limits, calculating an upper limit on temperature and finding a large, but not inconceivable, maximum. Alas I've lost the paper (which was probably wrong anyway, but it was a fun exercise). The question has long bugged me: is there a real upper limit on temperature?