Tsallis, Kaniadakis Logarithms and Information

According to (1), C. Tsallis introduced a mathematical generalization to Shannon’s entropy given by: Sq=1/(1-q) { Sum over i p power q-1} . In the limit of q->1, Shannon’s entropy is obtained. This is equivalent to considering entropy as the average of information: Fq(p) = 1/(1-q) ( 1- p power 1-q) so that lim q->1 Fq(p)=ln(p). Kaniadakis also has an analogous Fq(p) or Fk(p)= 1/2k (p power k - p power -k) such that lim k->0 Fk(p) = ln(p). In addition, for lim q->1 pq(ei) -> unnormalized Maxwell-Boltzmann as does lim k->0 pk(ei). Here we note that one may create a generalized F(p) expression which tends to ln(p) in a certain limit, namely: Lim a->0 1/{(1+b)a} { p power a - p power -a*b } = ln(p) ((1)) ((1)) involves more information than either the Tsallis or Kaniadakis cases. If one applies: F(p) = x where x=-ei/T which holds for the Tsallis, Kaniadakis cases and should hold for F(p) of ((1)), then the two simplest solutions which remove the information of b are: b=0 and b=1 ((2)) These are in fact the Tsallis and Kaniadakis cases. We suggest that both of these cases derive from the more information based expression ((1)) and that two distributions which appear in nature try to minimize information (based on the imposed lim parameter->0 F(parameter, p) = ln(p). This is similar to the manner in which the Maxwell-Boltzmann distribution tries to minimize information, but with no limits tending to ln(p) being introduced. This begs the question: Why should one be interested in particular power law limits which in a certain limit yield ln(p)? We suggest that it is in fact the power law form p power b which is of key interest, but that lim parameter->0 F(parameter, p) = ln(p) must hold as must F(parameter, p) = -ei/T. These ideas need to be coupled with the notion that there are two solutions to b in ((1)) which remove information. We ask: What math model (based on some kind of physics) yields two solutions, one for b=0 and one for b=1 and involves p power b? We note that the Tsallis case only involves one power, i.e. b=0. Then p power a is an eigenfunction of the operator p d/dp. Thus, p d/dp F(q,p) = a1*F(q,p) + a2 ((2)). In other words, F(q,p) and p d/dp F(q,p) are very similar. What about the b=1 solution? ((2)) does not hold in such a case, but as we have noted in another preprint,, if a1*F(q,p)= -ei/T *a1, then there is another way to obtain -ei/T*a1, namely F(q, p(a1*ei)) = - a1*ei/T. The form F(q,p(ei)) = -ei/T does not guarantee two solutions, but considering the operator whose eigenfunction is p(ei) power b does. In other words, one has ((2)), but also F(p) + p dF/dp = a1 F(p(a2*p) as the b=1 solution. This math combination of F(p) and p dF/dp is, however, linked to a single operator D (change operator) acting on: Sum over i p(ei) F(q, p(ei)) ((3)) where ((3)) is the average of the information function F(q,p(ei)). The fact that this returns a1*ei or more generally a1*ei + a2 is equivalent to considering p(ei)->p(ei)+dp(ei) inserted into ((3)) yielding the same results as when it is inserted into Sum over i p(ei)=1 and sum over i ei p(ei) = eave. Values of b not= 0,1 tend to lead to equaotins which are not linear or quadratic and may not have analytical solutions. Given that F(q,p)= -ei/T, a power law form of F(q,p(ei)) leads to a power law form of p(ei) and so physically one must anticipate or feel that nature favours such a distribution. Mathematically, one can, however, say that it is linked to minimum information in ei for ei/T>>1, i.e. a linear ei/T and is based on as little information as possible, power laws with only a single parameter q, or k being allowed (i.e. we consider k and -k as the same information). It does not allow for a and a*b as two different powers as this apparently contains too much information.