Consistency of the structure of Legendre transform in thermodynamics with the Kolmogorov-Nagumo average

TL;DR

This paper demonstrates that the fundamental structure of Legendre transforms in thermodynamics remains consistent even when replacing linear averages with nonlinear Kolmogorov-Nagumo averages, preserving core thermodynamic relations.

Contribution

It shows the invariance of thermodynamic Legendre structures and relations under nonlinear averaging schemes like Kolmogorov-Nagumo, extending the robustness of thermodynamic formalism.

Findings

Legendre transform structure remains intact with nonlinear averages

Thermodynamic relations are unaffected by the choice of average

Gibbs equation universality is preserved

Abstract

We show the robustness of the structure of Legendre transform in thermodynamics against the replacement of the standard linear average with the Kolmogorov-Nagumo nonlinear average to evaluate the expectation values of the macroscopic physical observables. The consequence of this statement is twofold: 1) the relationships between the expectation values and the corresponding Lagrange multipliers still hold in the present formalism; 2) the universality of the Gibbs equation as well as other thermodynamic relations are unaffected by the structure of the average used in the theory.