Can you have a high temperature with very little heat?

[MaxKang]

From what I know, temperature is defined to be the average kinetic energy of molecules within a system while heat is said to be the total kinetic energy of molecules.

I know this might be something we can never achieve in real life but here's how my thought process went.

Imagine you have a box with radiation shielding such that there is no radiation process involved. If you can somehow suck out all the molecules leaving just one molecule behind, depending on how slow or fast this molecule travels you can have very little "total kinetic energy" with quite high temperature(average kinetic energy). I guess when the molecule bounces off the wall there could be a sudden change in temperature.

I am aware that temperature is only defined when you have a reasonable amount of molecules but if we can somehow come up with the setting I mentioned, then what happens in real life? do we feel hot? cold?

 

[andrew s 1905]

Indeed you can. In medium within and between galaxies and clusters of galaxies there are very hot gases that emit x-rays but have very low density. See here https://en.wikipedia.org/wiki/Intracluster_medium.
Regards Andrew

 

[akhmeteli]

From what I know, temperature is defined to be the average kinetic energy of molecules within a system while heat is said to be the total kinetic energy of molecules.

Temperature only equals average kinetic energy (with some caveats) in classical physics. This is not so in quantum physics. Example - ideal Fermi gas.

 

[MaxKang]

Temperature only equals average kinetic energy (with some caveats) in classical physics. This is not so in quantum physics. Example - ideal Fermi gas.

But we still should be able to sense how "cold" or "hot" it feels to be in that environment should we not? Does the quantum theory predict that we will indeed feel cold in an environment where there is only a very little amount of molecules?