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 entropy[3].
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).
