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The Einstein analog:
So the Rate of Excitations is given by:
Suppose we have some distribution of relative velocities given by , where is the fraction of collisions occurring with relative velocities . Then:
where is the “collisional rate coefficient” . Then the Rate of de-excitation is given by:
We recognize now that is the rate of excitations of A using B moving at relative velocity . If we have detailed balance, then this has to be the same as the rate of de-excitation .
Where is the reduced mass . However many are created by collisional excitation, the same number are used for the reverse de-excitation. This is detailed balance.