Why is there no absolute maximum temperature?

Question

If temperature makes particles vibrate faster, and movement is limited by the speed of light, then I would assume that temperature must be limited as well.

Why is there no limit?

Answer 1

I think the problem here is that you're being vague about the limits Special Relativity impose. Let's get this clarified by being a bit more precise.

The velocity of any particle is of course limited by the speed of light c. However, the theory of Special Relativity does not imply any limit on energy. In fact, as energy of a massive particle tends towards infinity, its velocity tends toward the speed of light. Specifically,

$$E = \text{rest mass energy} + \text{kinetic energy} = \gamma mc^2$$

where $\gamma = 1/\sqrt{1-(u/c)^2}$. Clearly, for any energy and thus any gamma, $u$ is still bounded from above by $c$.

We know that microscopic (internal) energy relates to macroscopic temperature by a constant factor (on the order of the Boltzmann constant), hence temperature of particles, like energy, has no real limit.

Answer 2

There is an absolute maximum temperature, and it is $0^{-}$. :)

Okay, that sounds silly, but look it up in L&L: Statistical Physics I.

Think about an Ising paramagnet in an external field: At "zero" temperature (or actually $0^{+}$) the free energy of a system will be minimized by a unique minimum energy configuration. As we raise the temperature, the number of microstates with slightly higher energy grows rapidly, so we have a lower free energy in these entropically favorable configurations. Now we continue all the way to infinite temperature, at which point the system becomes completely disordered.

But wait, what if we drive the system to even higher energy? In that case there are fewer microstates and so the derivative that defines temperature goes negative, and the temperature that corresponds to these configurations is $-\infty$. This actually corresponds to the principle of "population inversion" in lasers. Anyway, higher and higher energy configurations (with their continually decreasing entropy) correspond to decreasing negative temperatures, until all of the spins point against the external field at $T=0^-$.

Answer 3

The speed of light is an upper limit for the speed of a massive object, but there is no upper bound on the kinetic energy of an object. In fact, that's why the speed of light is an upper limit (one of many reasons, anyway)-- an object moving at the speed of light would have infinite kinetic energy.

The temperature is a measure of the average kinetic energy of particles in a sample. Since kinetic energy does not have an upper limit, temperature does not have an absolute maximum.

(In equations, the kinetic energy is: $K=(\gamma - 1)mc^2 = (\frac{1}{\sqrt{1-v^2/c^2}}-1)mc^2$ which becomes infinitely large as v gets very close to the speed of light c.)

Answer 4

**Here's another perspective. **

The temperature o an object (particle) is a function of its energy. Theoretically there is no limit to the energy we can keep adding into a system.

However objects emit radiation that is dependent on their temperature. Object with higher temperature emit radiation with shorter wavelength.

According to quantum mechanics the shorter length in the universe is the Planck Distance (Planck length =1.616×10^(−27) nm) Therefore the upper limit of temperature will be the corresponding temperature of the body that emits electromagnetic waves with wavelength equal to the plank distance. Hence the higher temperature that can be achieved is `1.417×10^32 K which is also known as the Planck Temperature. As i mention in the beginning theoretically we can still keep adding energy to the object. However if we do then the laws of physics break down. This amount of energy well instantly cause a kugelblitz ( a black hole formed by energy).

Answer 5

If there is a maximum possible physical temperature it is well above anything we can reach experimentally and would require a complete theory of quantum gravity to understand it fully.

Neutron stars are some of the hottest objects in the universe today with temperatures up to around 10 trillion degrees Kelvin ($10^{12} K$). Similar temperatures have been reached in heavy ion collisions recently at the Large Hadron Collider for very small volumes and times. At these temperatures even the protons and neutrons in nuclear matter are torn apart leaving just a plasma of quarks and glouns.

But these temperatures are cool compared to the earliest moments of the big bang. According to our incomplete theories something really odd happens when you get to the Planck temperature which is around $10^{32} K$, so a good 20 orders of magnitude higher than anything we can produce.

When talking about these very high temperatures it is wrong to think in terms of the kinetic theory of gases or similar classical theories. You can't just apply relativistic mechanics and expect it to have any validity. Temperature is a feature of equilibrium thermodynamics and you can't reach equilibrium without interactions, so a discussion of fast moving non-interacting particles can not provide an answer to the question. You need relativistic quantum field theory and ultimately you have to think beyond even that.

At Planck scale temperature spacetime itself must be highly energised by gravitational interactions with hot matter. Some people think that spacetime passes through some kind of phase transition at this point, but if it does we have very little understanding of what kind of phase state lies beyond or whether temperatures can be raised further. Such understanding is in the realm of quantum gravity which is not yet fully developed. Such physics may describe the very earliest moments of the big bang and perhaps nowhere else in the universe.

Answer 6

While special relativity does not, a priori, place any constraints on the maximum temperature a system can attain, the situation changes when we consider the quark-gluon plasma - a stage you will eventually reach if you heat up any hadronic matter sufficiently. Rolf Hagedorn realized that for hadronic matter there exists a maximum temperature above which the partition function of the system is not well-defined. In other words you can only heat up hadronic matter to a maximum given by the Hagedorn temperature $T_H$.

Since hadronic-matter constitutes the vast majority of the matter we interact with (excluding dark matter and dark energy), in some sense $T_H$ is the maximum temperature that ordinary matter can attain, though this is by no means the end of the story ...

Of course, even with special relativity alone, one can see that when the temperature of a gas of particles becomes comparable to the rest energy of the particles in question, any attempt to increase the temperature beyond that point will only lead to pair creation. This was, vaguely speaking, the reasoning behind Hagedorn's work.

You might also find this Nova column on the Hagedorn phase enlightening.

Answer 7

We have two reasons for there not being a limit. As everyother commenter here has said, SR does not limit the energy per particle. Actually energy per degree of freedom would be a more precise statement. In any case temperature does not equate directly to the energy per particle DOF, but rather to the staistical probabilities, namely that the relative probabilities of a particular state being occupied is proportional to e(- deltaE/kT ). (Even that only applies to the low density limit, fermions are limited to one particle per allowable state, so in some high density low temperature limits (solid state, and degenerate state (some stellar interiors, white dwarfs etc.)) the lowest energy states are almost fully occupied. But, in any case, temperature applies to the probability distribution of the occupation of states with different energies, average energy per particle is just the normalized integral of this density time energy.

Answer 8

If we make some reasonable assumptions on the behavior of small black holes (since we don't understand quantum gravity, this is speculative), there is an absolute maximum temperature. If we go to the Planck temperature, $1.4 \times 10^{32}$K, thermal radiation will produce Planck-mass black holes, which radiate at the Planck temperature. If we try to increase the temperature higher, the thermal radiation will produce larger black holes, which will cool things down because they will absorb the Planck-scale radiation and re-radiate at a lower temperature.