Does hot air rise -- from a statistical-mechanical viewpoint
Question #6329 asks whether and why hot air rises. The consensus answer is straightforward: - hot air is less dense than cold air - therefore, by the principal of buoyancy, it rises. However, it is a thermodynamical explanation rather than a statistical-mechanical one. I would like to ask the same question from a statistical-mechanical point of view.
Furthermore, while the buoyancy argument makes sense to me for an enclosed volume of hot air, it does not make sense to me for an unenclosed volume.
For an enclosed volume H of hot air, the statistical-mechanical explanation for it rising seems straightforward. As the air in an enclosed volume gets hotter, $PV=nRT$ says that it expands but its pressure lowers. The air below H is thus at a lower temperature but higher pressure than H. Thus, the average kinetic energy per unit volume is higher below H than in H. Thus, if there were a barrier below H, more impulse would be transferred to the barrier from below than from above. The same would be true for the barrier above H; but since atmospheric pressure is lower at higher altitude, the energy transferred would be less. Thus the container would rise.
However, for our unenclosed-gas example, there is no such barrier -- and a non-existent barrier cannot absorb any impulse! So now let's reconsider the problem, but this time without a barrier surrounding the hot region H.
Here's my explanation. Please poke holes into it :-)
Before we heated it, the air in the region of space H had the same temperature and pressure as the surrounding air. Then, some agent added energy to H to heat up those molecules. Now the molecules in H are hotter than their surroundings. However, the number density of molecules in H has not changed -- i.e., we are not yet at equilibrium.
With the molecules in H suddenly moving faster (and, as usual with gases, moving in random directions), all of the boundaries of H will experience a diffusion gradient. At every boundary, more molecules will leave H than enter. To be clear, these boundaries of H have no actual physical barrier of any sort; they are merely theoretical boundaries.
The energetic molecules in H thus start to leave H in all directions. Up, down and sideways. They are replaced by a smaller number of less energetic molecules from outside. As that happens, the number density of molecules in H diminishes. At some point, we reach thermal equilibrium -- but with most of the heat energy having left the region H.
So where has it gone? The air below H was originally warmer and at a higher pressure than the air above H. Hence, the diffusion gradient at the boundary above H will be steeper than the gradient at the boundary below H. Hence, more heat energy will flow upwards than downwards. However, there will definitely be heat energy that moves both downwards and sideways from H.
Conclusion: while hot air does tend to generally move up, it also moves downwards and sideways too.
Is this a correct conclusion and reasoning?