Misusing the entropy maximization in the jungle of generalized entropies

Abstract

It is well-known that the partition function can consistently be factorized from the canonical equilibrium distribution obtained through the maximization of the Shannon entropy. We show that such a normalized and factorized equilibrium distribution is warranted if and only if the entropy measure I{(p)} has an additive slope i.e. ∂I{(p)}/∂pi when the ordinary linear averaging scheme is used. Therefore, we conclude that the maximum entropy principle of Jaynes should not be used for the justification of the partition functions and the concomitant thermodynamic observables for generalized entropies with non-additive slope subject to linear constraints. Finally, Tsallis and Rényi entropies are shown not to yield such factorized canonical-like distributions.

Rescuing the MaxEnt treatment for q-generalized entropies

“…A very interesting result was recently reported in [24]. Let a and β stand for the two Lagrange multipliers appearing in a canonical ensemble MaxEnt treatment.…”

“…We show here by construction, appealing to conventional techniques of Functional Analysis, not used in [24], that the functional forms (1.1), (1.2) are indeed recoveredà la Tsallis (or Renyi), thus surmounting the formidable obstacle posed by [24] to the use of MaxEnt for these two entropies. We are not claiming that the arguments of [24] are wrong, but that they are Calculus-based, while the proper MaxEnt scenario requires functional Analysis.…”

“…These properties, it is shown in [24], are NOT possessed by the PDs that maximize either Tsallis' or Renyi's entropies. This constitutes an intriguing result that deserves further consideration.…”

An extensive study of Bose–Einstein condensation in liquid helium using Tsallis statistics

“…Extremizing s q subject to suitable constraints yields more general canonical ensembles(see Appendix B), where the probability to observe a microstate with energy ǫ i is given by: [1,44,45] …”

Strong correlations between the exponentαand the particle number for a Renyi monoatomic gas in Gibbs' statistical mechanics

“…Why? Because MaxEnt workings with q-generalized entropies have recently received serious questioning [2,3], and we wish to disentangle our findings from MaxEnt.…”