Where do the hot molecules go in the bath?

Watching the hot water swirl into a bath of cooler water, I wondered about the process of mixing of the two bodies at a molecular level: where do the hot molecules go?

Of course temperature is believed to reflect the average kinetic energy of molecules in a substance.

The energy of the hotter molecules is distributed into the cooler molecules via mechanical transfer, and perhaps some coupling via photon absorption.

That made me wonder: if the temperature of a substance represents the average energy of molecules, what is the variance in energy?

Are there substances with a larger variance? That would be interesting: it would have a larger population of much higher and much lower energy molecules, which could (without invoking Maxwell’s Daemon) partake in reactions requiring more energy.

normal curves

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