Thermodynamics of exponential Kolmogorov-Nagumo averages

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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 R\'{e}nyi’s entropy with parameter γ, 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 β that are suddenly quenched to another heat reservoir with inverse temperature β’ = (1-γ)β.

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. 

P. Morales, J. Korbel, F. Rosas, Thermodynamics of exponential Kolmogorov-Nagumo averages, New Journal of Physics (2023) DOI: 10.1088/1367-2630/ace4eb