I have a seemingly simple question that is still puzzling to me. The temperature of an atom is proportional to its kinetic energy. If an atom now moves upwards in a gravity field, its kinetic energy changes: $E_h = E_0 - hgm$. So $T_h < T_0$, where $T_h$ is the temperature of the atom at height h and $T_0$ is at height 0. The reason this is puzzling to me is that it seems the particle could induce a flow of energy between other particles at height h and at height 0, even if both initially also had the same temperature $T_0$: energy would flow down, the temperature at the top would decrease towards $T_h$. This seems paradoxical to me. How would the system really behave?
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6$\begingroup$ "The temperature of an atom is proportional to its kinetic energy." No. The average energy divided by the number of particles is not the same thing as the energy of an individual particle. $\endgroup$– hftCommented Feb 12, 2023 at 20:19
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1$\begingroup$ To understand how the system behaves, you can compute the partition function for N atoms in a gravitational field. $\endgroup$– hftCommented Feb 12, 2023 at 20:21
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1$\begingroup$ See also the adiabatic lapse rate. $\endgroup$– ChemomechanicsCommented Feb 12, 2023 at 21:54
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$\begingroup$ @Chemomechanics Thanks! I am surprised though, I thought that the temperature difference between lower and higher atmosphere was due only to the air near the ground being heated by the ground (-> indirectly by the sun). It still sounds strange to me that a temperature difference should be created "spontaneously" since I thought different temperatures almost always equalize spontaneously. $\endgroup$– TuringPatternCommented Feb 13, 2023 at 19:39
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$\begingroup$ @hft thanks, I will try that out, see if I can get it done with my limited knowledge of statistical mechanics $\endgroup$– TuringPatternCommented Feb 13, 2023 at 19:40
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Temperature is a measure of the average kinetic energy of an ensemble of particles. So, defining one atom's temperature makes no sense.
