Tsallis Entropy from Maximization of Entropy and the Condition of Stability

In  a previous note (1), we argued that in the Maxwell-Boltzmann (Shannon) case, one may either maximize Shannon’s entropy:  - Sum over i p(ei) ln(p(ei)) subject to the two a priori constraints: Sum over i p(ei)=1 and Sum over i ei p(ei) = Etotal or use reaction balance: e1+e2=e3+e4 and p(e1)p(e2)=p(e3)p(e4). The first approach may be described as maximizing the probability of N trial runs and so is linked with minimal information which is often thought of as a universal statistical principle. We argued in (1), however, that if one only uses reaction balance, there is no notion of maximizing an overall probability of N trial runs. In such a case, there seems to be a different principle at work. One may write:  p(ei) d/dp(ei) F(p(ei))=1  ((1)) such that F(p(ei))= -ei/T, the information. This leads to F(p(ei)) = ln(p(ei)). We argue that ((1))is a statement of stability. Namely, if one argues that for a given set p(ei), if one has F(p(ei)), then Sum over i {p(ei)+delta(p(ei)} F(p(ei) + delta(p(ei)) = Sum over i p(ei) F(p(ei). This then seems to be a way to justify ((1)) not just for the Maxwell-Boltzmann case, but also for the Tsallis one. 

    In particular, in (1) we argue that one may obtain the Tsallis lnq(p(ei)) from the conditions:   lnq(p(ei)) = -ei/T,  lnq(p(ei) for q=1 = ln(p(ei) and p(ei) power q d/dp(ei) lnq(p(ei) = 1 ((2)). We suggest that the latter condition is one of stability and hence seems justified even if one cannot associate Tsallis entropy with the maximization of the probability of N runs. We also note that these conditions govern the nonadditivity of Tsallis entropy:   S(A,B) = S(A)+S(B) + (1-q)S(A)S(B). 

    Next, we consider (2) in which the Tsallis non-additivity condition (and that of other forms of entropy) is said to follow from the notion of a conserved quantity (e.g. ei) and the zeroth law of thermodynamics. Secondly, (2) argues that of the many entropies which one may create, Tsallis entropy has the property of being stable, namely given an observable C(p(ei)), a small change in p(ei) should also mean a small change in the observable. We argue that both the conditions of (2) actually follow from the conditions ((2)).