Speculation on Information, Scaling and the Maxwell-Boltzmann and Tsallis Distributions

We suggest that if one wishes to use a probabilistic description of a gas of independent particles, then one may write: Eave= {Integral c1 sqrt(e)de e p(e/T) } / {{Integral c1 sqrt(e)de e p(e/T) } where p(e/T) is not normalized. e/T is used, with T, an unknown parameter, because probability has no units. If one makes the assumption that e ranges from 0 to infinite, then E=T*Number. Thus, T is a parameter proportional to Eave and also Etotal. One may then ask whether there exists a math function S(T, ei, p1(ei/T)) (p1 is normalized) which describes the system in the following sense. If T->T+dT and p1(ei/T) - p1(ei/T) + dp1(ei/T), can dS/dT dT describe the changes in the physical constraints Eave= sum over i ei p1(ei/T) C(T) where C(T) is the normalization of p(ei/T) and Sum over i p1(ei/T) = 1? We suggest that if scaling is considered, one may only have S(ei/T, p(ei/T)) as a change in the scale used to measure ei and T should not change the statistical function. If this is the case, g(p(ei/T) = -ei/T, where g is -1 * the inverse of p1. As a result, one may seek F(g(p(ei/T)), p(ei/T)) which is strictly a function of p(ei/T) and may be able to describe the two constraints upon a change in T. One way to have this occur is for F=Sum over i p1(ei/T) g(p1) such that: g(p1) = -ei/T and p1 dg/dp1 = 1, where p1 is the normalized probability .This leads to the Maxwell-Boltzmann distribution. If one assumes a physical constraint of Sum over i ei p(ei) power q and F = Sum over i p(ei/T) power q g(p), then this leads to the Tsallis distribution. We have discussed some of these ideas in previous notes, but the main point we present here is that scaling, i.e. the form ei/T in F means that F is strictly a function of p(ei/T). One does not have to start with the assumption of an a priori function F(p(ei/T)) as we did in previous notes. Furthermore, p(ei/T) contains both ei and global Eave information and so it is quite plausible that a statistical function of p(ei/T) could describe the constraints in the problem. We also consider the question of energy conservation in individual scattering collisions.