Equivalence of Partition Functions Leads to Classification of Entropies and Means

We derive a two-parameter family of generalized entropies, Spq, and means mpq. To this end, assume that we want to calculate an entropy and a mean for n non-negative real numbers {x1,...,xn}. For comparison, we consider {m1,...,mk} where mi = m for all i = 1,...,k and where m and k are chosen such that the l and l norms of {x1,...,xn} and {m1,...,mk} coincide. We formally allow k to be real. Then, we define k, log k, and m to be a generalized cardinality kpq, a generalized entropy Spq, and a generalized mean mpq respectively. We show that this family of entropies includes the Shannon and Rényi entropies and that the family of generalized means includes the power means (such as arithmetic, harmonic, geometric, root-mean-square, maximum, and minimum) as well as novel means of Shannon-like and Rényi-like forms. A thermodynamic interpretation arises from the fact that the l norm is closely related to the partition function at inverse temperature β = p. Namely, two systems possess the same generalized entropy and generalized mean energy if and only if their partition functions agree at two temperatures, which is also equivalent to the condition that their Helmholtz free energies agree at these two temperatures.