Modification to the Maxwell-Botlzmann Distribution?

In previous notes, we argued the Maxwell-Boltzmann (MB) distribution (as well as the Fermi-Dirac and Bose-Einstein) may be obtained by a time reversal elastic two particle scattering balance. This approach requires no concept of entropy, counting of states or partition functions. For the MB case, one may write f(e1)f(e2)=f(e3)f(e4) ((1a)) to describe e1+e2=e3+e4 ((1b)). Taking ln of both sides of ((1a)) and associating with ((1b)) yields the MB distribution. The scattering approach allows one to visualize a physical mechanism (i.e. elastic scattering) create the equilibrium balance, but it also exposes some possible issues, it seems. First of all, there are cases where f(ei) is very small i.e. the particle energy is very large. For a gas with numerous particles, it is possible that a high energy particle is created in one place (forward reaction of ((1a))), but that the reverse reaction occurs quite a distance away. Overall, there would be a reaction balance and equilibrium, but locally it seems there would be two large fluctuations at two different places. These fluctuations would disturb local equilibrium which would then need to be restored. Furthermore, it seems a reaction balance has to occur locally in a certain amount of time. In such a time period, low energy particles might scatter once, while high energy particles may scatter more than once. This would change ((1a)) it seems, enhancing f(ei) for the higher energy particle. As a consequence of ((1a)), it would enhance lower energy f(ei) as well. Thus, the MB distribution may be changed. E.g. the standard deviation increased. In this note, we try to examine these issues, albeit in a very handwavy manner