Energy Entropy, Thermodynamics and Function-Inverse Function Pairs

Recently in the literature, various derivations of entropies such as Tsallis entropy and Kaniadakis entropy have appeared using extremization of a Shannon like entropy expression or the thermodynamic relationship dE=TdS ((1)). In a series of notes, we have argued that Shannon’s, Tsallis’ and Kaniadakis’ entropies and distributions as a function of energy are related to function-inverse pairs. In this note, we wish to consider ((1)) in a general way to show this relationship between function and inverse functions pairs. In particular, we wish to show that for the Maxwell-Boltzmann, Juttner, Kaniadakis and Tsallis distributions, E internal = TS with function-inverse function pairs used to create S, the entropy. From this, dE=TdS follows with T being a placeholder because f(e/T).