Since the vibration of the particles is what cause temperature and a particle from its own point of view is not moving, is its temperature 0°K from its own point of view?
Is there a thing like relativistic temperature at all? If not, Why not?
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Temperature is not a property of individual particles, it's a statistical property of a collection of particles. For example, nuclear physicists have been known to do thermodynamics with as few as $10^2$ particles, but that's pretty minimal. As you get to lower and lower particle numbers, thermodynamics works more and more poorly, e.g., the second law can be violated with high probability.
Since temperature is not a property of individual particles, it doesn't make sense to talk about a single particle's temperature, in that particle's rest frame.