Tsallis Entropy from Maximization of Entropy and the Condition of Stability Part 2

 In Part I, we argued that F(p(ei/T)) = -ei/T, where p(ei/T) is an unnormalized probability. We suggested that one may find Tsallis entropy from p(ei) power q dF/dp(ei) =1  ((1)), where F= lnq(p(ei)) and Tsallis entropy is:  Sum over i   p(ei) power q lnq(p(ei)). We argued that this was due to stability considerations.

   Here we consider the importance of information especially that contained in an a priori constraint. We start with the Maxwell-Boltzmann case and introduce a function F such that F(p(ei)) = -ei/T, where ei/T is key energy information. We note that such information is also contained in an a priori constraint:   Sum over i p1(ei) ei = Etotal. Here p1(ei) is normalized and p(ei) is not. We write:Sum over i p(ei) F(p(ei)) = Sum over i  (-ei/T) p(ei) and suggest taking d/dp(ei). The point we make is that d/dp(ei) acting on p(ei) leaves F(p(ei)) free to display the relevant information linked with the average energy constraint and so the second part of the derivative: p(ei) dF(p(ei)/dp cannot reveal any more ei/T information and must be a constant, i.e.  p(ei) dF/dp(ei) = 1, for example. This restriction of information immediately yields F(p) = ln(p). 

   Very similar arguments apply to the Tsallis case, but here the information linked with an a priori constraint is now:  Sum over ei p1(ei) power q and so considers  Sum over i p(ei) power q F(p(ei)).

   We note that even though there are similarities with the math procedure of extremization, it is really the informational content of the a priori constraint that is considered here. We do not explicitly set out to maximize any function (with some physical meaning) subject to constraints.