Scaling Approach to Finding the Maxwell-Boltzmann Distribution and Maximization

   In a previous note, we suggested that one may obtain the Maxwell-Boltzmann distribution through a scaling approach which does not use reaction balance or maximization of entropy. In other words, it does not appear at first glance to use any extremization idea. We will show here that this approach works in any dimension for the nonrelativistic case. The question then becomes: Why should a scaling approach select a maximum entropy or reaction balance result (which is based on a uniform distribution of product p(ei)p(ej) probabilities).  How is it that an extremum result appears if no a priori extremization is employed.

   We argue that in fact a maximization strategy is applied in the scaling approach because one sets the coefficient of dT to 0 (instead of considering all powers of dT) and only considers p -> p+dp/dT dT. In general it is the sum of the full series which arises when T-> T+dT which is 0, not the first coefficient. We argue that setting the first coefficient to 0 is equivalent to choosing an extremum, i.e. one which does not change a function to first order when one changes the variable in question, here p(ei/T) probability to first order.

   We note that the scaling approach is based on the following idea.

E = T  {Integral de e power b  c1  e p(ei/T) } / { Integral de e power b c1 p(ei/T) } ((1))

Here b = .5 for three dimensions and p(ei/T) is unnormalized. In general, the factor of T is a fixed number. If one changes T→ T+dT then the sum of all powers of dT in an expansion should add to zero. It is a special case if the first order term is 0. We show that this special case is the MB distribution in the nonrelativistic case for any power b. By forcing the Number in E= T Number to remain unchanged by changing p(ei/T) → p+dp/dT dT, one is choosing an extremum solution. (It is not clear, however, that this approach works in the relativistic case because one does not have e power b, while reaction balance or maximization of entropy subject to constraints still does.)