Tsallis Entropy and Degeneracy

 For a Maxwell-Boltzmann situation, one may consider the number of permutations of N particles with n(ei) of them having energy ei i.e. N!/ Product over i n(ei)!. In such a case, n(ei)! removes the degeneracy of the identical n(ei) particles. In order to convert the degeneracies into a sum, one takes ln of the number of permutations. In such a case one has ln(N!)- Sum over i ln(n(ei)!). If N→ infinite, ensuring that n(ei) values are also large, one may apply Stirling’s high n limit to ln(n(ei)!) to yield n(ei)ln(n(ei)). In information theory Sum over i n(ei) ln(n(ei)) is called the average of information where information is given by n(ei). This average, we argue, is really a sum form of the unnecessary degeneracy information for each n(ei) and is called entropy. 

We ask: Is it possible to write a sum of more general degeneracies (other than identical particle permutations) (which should be equivalent to entropy) which, however, in a certain limit become the degeneracy of permutations? 

    We answer yes if one uses the identity:  lim x->0  {p (power x) -1}/ x = ln(p). p ln(p) is the form of the permutation degeneracy in sum form and so it corresponds to the limit of x→0. If x is not 0, one may write a sum of degeneracies depending on p(i) (power x) which in the limit of x→0 become permutation degeneracies. This entropy is called Tsallis entropy and is well-known in the literature (1). We wish to point out that the p(i) (power x) i.e. probabilities to a constant power seem to represent the degeneracies in the system (in sum form), which in a special limit represent the degeneracy of permuting  n(ei) identical particles i.e. ln(n(ei)!) in sum form.