Abstract

The group entropies introduced by Piergiulio Tempesta [1] offer a systematic axiomatic approach to entropies, considered as functionals on probability spaces. Beside of satisfying the axiomatic group structure functionals are also required to be extensive for the relevant asymptotic behaviour of W(N), the number of allowed microstates of the system consisting of N constituents. In this way entropies fall into different classes determined by W(N). For exponential W(N) ~ exp(N), the group entropies reduce to either the Boltzmann or the Rényi entropy. Sub-exponential W(N) leads to the Tsallis q-entropy and super-exponential to new entropies. The latter case has, e.g., been suggested to be relevant to the thermodynamics of black holes [2]. It is interesting to note that the maximum entropy principle leads to q- exponential probability distributions for all cases of W(N), even when the entropy is different from the Tsallis entropy³. The group entropies are directly relevant to information theory for instance when applying the approach of permutation entropies to time series where the number of patterns easily grows faster than exponential as a function of the length of the time series [4]. Turning to thermodynamics, we will want to relate the group entropies to Clausius entropy (defined in terms of heat exchange) and to derive the first law of thermodynamics. We will further discuss thermodynamics equilibrium conditions for systems described by group entropies. References [1] P. Tempesta, Group entropies, correlation laws, and zeta functions. Phys. Rev. E 84 , 021121 (2011). [2] H.J. Jensen and P. Tempesta, Group Entropies as a Foundation for Entropies, Entropy 26, 266 (2024). [3] Constantino Tsallis, Henrik Jeldtoft Jensen, Extensive composable entropy for the analysis of cosmological data. Phys. Lett. B, 861, 139238 (2025). [4] J M Amigó, R Dale and Piergiulio Tempesta, Permutation group entropy: A new route to complexity for real-valued processes, Chaos 32, 112101 (2022).