The Statisical Stability Method for the MB/Tsallis/Kaniadakis Cases and Normalization

In (1), we introduced a statistical stability method which yields the Maxwell-Boltzmann, Tsallis and Kaniadakis distributions as solutions of a single differential equation: F(p(ei)) + p(ei)dF/dp(ei)= a1ei + a2 ((1)). Here p(ei) is unnormalized. The basic idea is to introduce an information function F(p(ei)) such that F(p(ei)) = -ei/T and its average: Sum over i p(ei) F(p(ei). It is argued that p(ei)-> p(ei)+dp(ei) yields the same changes in this average as it does in the constraints Sum over i p1(ei)=1 and Sum over i ei p1(ei)=Eave. Here p1(ei) is normalized. In this note, we try to clarify some issues linked with normalization. First, we note that p(ei) is unnormalized while p1(ei) is normalized. We consider an equilibrium solution, for example, C(T)exp(-ei/T). Here T is a fixed number based on Eave. We consider changing only the form exp(-ei/T) to exp(-ei/T) + dp(ei). Then Sum over i dp(ei) = 0 and Sum over i ei dp(ei) =0 as well because there is a fixed amount of average energy. Thus, we do not really need to worry about the normalization factor it seems and one can consider the constraints Sum over i p(ei) and sum over i ei p(ei) as changing in the same way as Sum over i p(ei) F(p(ei)) with all p(ei) values unnormalized. One then only considers terms proportional to ei and a constant in the change to Sum over i p(ei) F(p(ei)) and it does not matter what values a1,a2 have in a1*ei + a2. As a result, the stability method seems to be a little bit different than a math extremization problem. The second normalziation consideration is one linked with the Kaniadakis distribution. If one considers the differential equation ((1)) and F(p(ei)) = -ei/T=x for the Kaniadakis case, then one might be tempted to use the unnormalzied Kaniadakis form p(ei) = { sqrt(1+kkxx) + kx} power 1/k ((2)). This is the inverse of 1/2k ( p power k - p power -k) = F(p), but if one uses this solution directly into the de ((1)), one does not obtain a solution proportional to x= -ei/T. The catch, it seems, is that the Kanidakis solution from ((1)) involves using both terms F(p) and p dF/dp to create a scaled b1 F(p(a*ei)) solution and this solution leads to a distribution which is ((2)) multiplied by a factor d power 1/k.. This extra factor changes the result of F(p)+pdF/dp = a1*ei. Thus, even though p(ei) Kaniadkais is unnormalized, one may have a solution of p(ei)= d power 1/k {sqrt(1+kkxx) + kx} power 1/k just as well as p(ei) = {sqrt(1+kkxx) + kx} power 1/k. This seems to be a subtle point which we wish to highlight.