Renyi Entropy, Additivity and Maxwell-Boltzmann Free Energy

In (1), Shannon’s entropy, S = - Sum over i  fi ln(fi) is described as an average of the information function ln(fi) and it is suggested that one may create different averages of information and hence different entropy functions. (1) suggests that one consider Renyi entropy in this light and further notes that it is additive like Shanonn’s entropy for Pnew = Sum i,j p1(i) p2(j). (1) also argues that if one uses the Maxwell-Boltzman (MB) probabilities in the Renyi entropy expression:  ln{Sum over i fi power q} / (1-q), then one obtains an expression in terms of MB free energies F(T) and F(T/q). 

   Here, we try to consider Shannon’s entropy in terms of two body elastic scattering with time reversal balance. In such a case, additivity first arises due to conservation of energy, e1 and e2, as the ln function converts f(e1)f(e2) into a sum. Even in the case of Tsallis entropy, one may introduce a function ln(G) which has additive properties. {The entropy then is Sum over i P(fi) ln(G(fi)) which is not additive.} Given the Shannon MB case, the presence of ln means that the term:  ln{ Sum over i fi} = ln(Z(T)) appears naturally as the ln of the normalization constant. Furthermore, given: ln(f(ei)) = -ei/T, multiplying both sides by q yields:   ln { fi power q } = -ei/*T./q) Thus, the Shannon MB case is already linked with a power law probability found in the Renyi entropy expression. In other words, the additive form of the defined Renyi entropy is in a sense the ln of a normalization constant of an MB distribution with fi power q, fi=exp(-ei/T)   , with fi power q as a new set of unnormalized probabilities. This, we argue, shows the origin of the expression, given in (1), of Renyi entropy using MB probabilities leading to a funciton of F(T) and F(T.q) where F is the Shannon-MB free energy, i.e. -kT ln(Z).