Solving for the Maxwell-Boltzmann Distribution Using Scaling

It seems that there are a number of ways to solve for the Maxwell-Boltzmann distribution. In previous notes, we have argued that there are two independent approaches, one global and one local, both based on physical scenarios. The global approach involves maximizing ln of a physical number of arrangements: N! / Product over i n(ei)! ,where n(ei) = Np(ei), subject to the constraint: Sum over i ei p(ei) = Eave. Only if one uses Stirling’s approximation ln(n!) = n ln(n) does one obtain p(ei) = C(T) exp(-ei/T), i.e the Maxwell-Boltzmann (MB) distribution. In (1), it is suggested that this same approach is equivalent to the first law of thermodynamics. We consider this approach briefly here. The second approach which we have discussed in previous notes is a local one restricted to ei+ej = E1 = constant. In such a case, one has a uniform distribution for product p(ei)p(ej) as long as ei+ej=E1. This means any E1 value is possible, even a value greater than Etotal. This leads to p(ei)p(ej) =p(ek)p(el) which yields p(ei) = Cexp(-ei/T) and is based on physical elastic scattering. An approximation, however, is built in here because there is no cutoff on ei, i.e. p(ei)p(ej)=p(ei+ej) allows for ei=.8Etotal, ej=.7Etotal and ei+ej = 1.5 Etotal which is unphysical. Here we discuss a third approach called scaling, which when coupled with an approximation that all ei’s are possible (just as in the subset-uniform distribution approach), yields the Maxwell-Boltzmann distribution. Through scaling, it is possible to show that Eave = T Number so that Eave(T+dT) = (T+dT) Number. Number, however, contains integrals which involve T and so one may calculate the coefficient of the dT contribution to Number and it must be 0. One may then seek a solution for p(ei) which makes it zero and this turns out to be the Maxwell-Boltzmann result. We argue that one may anticipate this based on the local subset-uniform distribution approach.