Thermodynamics of exponential Kolmogorov-Nagumo averages

Abstract

This paper investigates generalized thermodynamic relationships in physical systems where relevant macroscopic variables are determined by the exponential Kolmogorov-Nagumo average. We show that while the thermodynamic entropy of such systems is naturally described by Rényi's entropy with parameter $\gamma$, an ordinary Boltzmann distribution still describes their statistics under equilibrium thermodynamics. Our results show that systems described by exponential Kolmogorov-Nagumo averages can be interpreted as systems originally in thermal equilibrium with a heat reservoir with inverse temperature $\beta$ that are suddenly quenched to another heat reservoir with inverse temperature ${\beta }^{\prime }=(1-\gamma )\beta$. Furthermore, we show the connection with multifractal thermodynamics. For the non-equilibrium case, we show that the dynamics of systems described by exponential Kolmogorov-Nagumo averages still observe a second law of thermodynamics and the H-theorem. We further discuss the applications of stochastic thermodynamics in those systems -- namely, the validity of fluctuation theorems -- and the connection with thermodynamic length. namic length.

14 pages, 1 table. arXiv admin note: text overlap with arXiv:2203.13673. Authors note: the original manuscript arXiv:2203.13673 was splitted into two different manuscripts

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• Perspective on Physical Interpretations of Rényi Entropy in Statistical Mechanics
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• On nonlinear compression costs: when Shannon meets Rényi
• Geometric Structures Induced by Deformations of the Legendre Transform