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Temperature is a measure of the average kinetic energy of particles in a given place (correct me if I am wrong) so there must be a definite limit to heating like there is an absolute zero to cooling. And what process could achieve this temperature?

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First, let's get a bit of background. "Absolute hot" is the concept that a maximum temperature exists. It describes your question exactly. For a long time, no one was able to figure out whether an absolute hot exists, but in the 20th century major revolutions in theoretical physics gave us answers.

"Absolute hot" isn't the name of the hottest temperature, but it's the concept that one exists. So what is our answer — is there a highest possible temperature that fits the concept of an "absolute hot"? Well, yes. But first we must talk about heat and temperature.

Photons act both like particles and waves. As we add more energy, the photon's wavelength decreases, and its frequency increases. Thus, the wavelength of the light emitted by hot objects generally depends on the temperature of the object. Hotter objects will emit more energetic photons with shorter wavelengths.

According to modern physics, a Planck length is the smallest meaningful length in the Universe (the reason for that is an entirely different question). If we keep raising the temperature of our object, nonstop, the wavelength of the photons it emits will decrease until it reaches a Planck length. Our object would reach a temperature called Planck temperature. That's about $1.417×10^{32} \; \mathrm{K}$.

Planck temperature is theoretically the hottest possible temperature. At that point, the object could not theoretically get any hotter, as the photon's wavelength cannot decrease beyond a Planck length.

But what happens if we were to add more energy to the object? Would it go beyond Planck temperature? Well...God knows. At that point, we wouldn't even call it temperature anymore. Nobody knows what exactly would happen, but it certainly wouldn't match the definition of temperature.

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    $\begingroup$ @Mobal Well, the photon would become hotter than what is physically allowed. I like your question; theoretically, there might not be a limit to how much energy we could add to the photon. But we have no idea what would happen if it went beyond the Planck temperature, as the wavelength shouldn't be able to decrease anymore. It really wouldn't follow the concepts that we've established already, like the Planck length being the smallest possible length. $\endgroup$ Jun 12 '16 at 14:46
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    $\begingroup$ @Mobal I think that's an unsolved problem in physics, even though according to our models, the Planck length must be the shortest possible length (you could read more about it here: goo.gl/4VBO8T). But ultimately, we have no way of testing that. Temperatures that hot would actually form a black hole (called a Kugelblitz). In Einstein's theory of General Relativity, energy is a source of gravity too. $\endgroup$ Jun 12 '16 at 14:56
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    $\begingroup$ uh, perhaps i'm just naive, but i don't think temperature is a well-defined quantity when the energy of a system is fixed (like a single photon, for instance)... $\endgroup$
    – colnegn
    Jun 12 '16 at 22:30
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    $\begingroup$ In reality, wouldn't a photon basically turn into particles well before it reached Planck energy? I understand the answer was theoretical, but when asked "what if you gave it more energy", in reality, long before the photon reached that energy it would give up some energy in the form of particle creation. (or am I mistaken?). I don't think I'm mistaken. $\endgroup$
    – userLTK
    Jun 13 '16 at 1:50
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    $\begingroup$ @userLTK This all took place during the Planck epoch, the earliest period after the Big Bang. There is no currently available physical theory to describe such short times. As you get closer and closer to t(0), our current theories begin breaking down. I'm gonna guess some aspects of GR and QM break down more than others, so physicists could determine the temperature at those times. Still, for the most part, we have no idea what went on back then. $\endgroup$ Jun 13 '16 at 2:40
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As I stated in my comment in Sir Cumference's answer, I might just be naive on the subject, but I believe that:

Assuming that the density of states (the number of states per possible energy) for a given system can be taken to be effectively continuous, one can define the inverse temperature:

$$ \frac{1}{k_BT} = \beta = \frac{dS}{dE}$$

Where E is the system's energy, and S is the system's entropy ($ S= \ln(\Omega(E) $), where $\Omega(E)$ is the density of states (a function of the energy).

This means that for a given density of states of a system: if the derivative of its natural log has any extrema (local maxima or minima), then the system will have infinite temperature at those energies (at the locations of the extrema--remember that $\Omega$ is a function of the energy).

Interestingly, by this definition: at energies where the slope of a system's entropy, S, is negative, the system is defined to have a negative energy (which ironically is more energetic than states with positive temperature!)... although to be honest: I am not totally convinced that systems at negative temperatures are actually at equilibrium, which (I would think) indicates that temperature is ill-defined... but definitely a curiosity worth some consideration.

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