The “checker for high and low energy atoms ” is Maxwell’s Demon https://en.m.wikipedia.org/wiki/Maxwell%27s_demon and is one of the core thought experiments for physics. (Edit I misread checker as a verb….but the point stands)
Within an object all at one temperature, there will be atoms of many energies, but temperature is a function of the collection.
If we think about one object having multiple temperatures, like me having cold hands but normal body temperature, we are implicitly splitting the atoms into two collections.
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