Thermodynamics of exponential Kolmogorov-Nagumo averages

TL;DR

This paper explores the thermodynamics of systems characterized by exponential Kolmogorov-Nagumo averages, revealing their entropy, equilibrium behavior, and non-equilibrium dynamics, with implications for multifractal thermodynamics and stochastic thermodynamics.

Contribution

It introduces a thermodynamic framework for systems using exponential Kolmogorov-Nagumo averages, linking Rényi entropy, equilibrium distributions, and non-equilibrium laws.

Findings

Rényi entropy describes the thermodynamic entropy of these systems.

Equilibrium states follow a Boltzmann distribution despite the generalized averaging.

Non-equilibrium dynamics obey the second law and fluctuation theorems.

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 R\'{e}nyi'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 $\beta' = (1-\gamma)\beta$. 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.