Isn't all this just punning on definitions? If the particle velocities in a gas are Maxwell-Boltzmann distributed for some parameter T, we can say that the gas has "Maxwell-Boltzmann temperature T". Then there is a separate Jaynes-style definition about "temperature" in terms of the knowledge someone has about the gas. If all you know is that the velocities follow a certain distribution, then the two definitions coincide. But if you happen to know more about it, it is still the case that almost all interesting properties follow from the coarse-grained velocity distribution (the gas will still melt icecubes and so on), so rather than saying that it has zero temperature, should we not just note that the information-based definition no longer captures the ordinary notion of temperate?
You are essentially right. The point is that 'average kinetic energy of particles' is just a special case that happens to correspond to the Jaynes-style definition, for some types of systems. But the Jaynes-style definition is the 'true' definition that is valid for all systems.
But if you happen to know more about it, it is still the case that almost all interesting properties follow from the coarse-grained velocity distribution (the gas will still melt icecubes and so on)
Again, as I mentioned in my previous replies, the gas will melt ice cubes, but is only in thermal equilibrium with 0 K ice cubes.
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.
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