On the Derivation of the Maxwell Boltzmann Distribution

It is known that the statistical factor exp(-e/T), or at least exp(-Ae), appears very naturally from Boltzmann´s equilibrium criterion f(e1)f(e2)-f(e3)f(e4) and holds in both relativistic and nonrelativistic cases. If one lets e1=e and e2 =e+de, one is able to obtain an equation for f(e) alone, namely 1/f df/de=-A where A is constant. This leads to an exp(-eA) factor and suggests that this factor may arise in more general cases than just the collisions of two particles. The purpose of this note is to put forward a physical picture in which there is the ¨appearance¨ of an increasing difficulty of obtaining high energy states. Ultimately, however, it seems that the high ¨decay¨ rate of high energy or momentum particles may be the cause of f(e+de) being smaller than f(e). We argue that high energy particles tend to smash into nearby particles doing work and losing their energy and so have a higher tendency of ¨decaying¨ or ¨shedding energy¨ than lower energy particles. Here it should be noted that ¨high energy¨ is relative to an average energy or temperature. We argue that this is the dynamical reason for a decrease in f(e). Such a dynamical picture does not depend on counting states or the idea of entropy. This is not to say that these are not ¨in the picture¨, but rather suggests that there should be a dynamical driver of equilibrium. We argue that it is important to have a physical picture based on dynamics to describe the Maxwell-Boltzmann distribution, and ultimately the factor exp(-E/T) which is used throughout statistical mechanics.