Thermodynamics of an ideal generalized gas:II Means of order $\alpha$

TL;DR

This paper explores the thermodynamics of an ideal generalized gas using power means of order α, linking entropy, inequalities, and efficiency in nonextensive, multifractal systems.

Contribution

It introduces a novel thermodynamic framework based on power means of order α, connecting entropy, inequalities, and efficiency in nonextensive systems.

Findings

Power means are monotonic functions of their order, underpinning second laws.

Final states in L-potential equilibration maximize entropy; in entropy equilibrium, they minimize L.

Entropy change relates to Shannon and Rényi entropies in nonextensive, multifractal systems.

Abstract

The property that power means are monotonically increasing functions of their order is shown to be the basis of the second laws not only for processes involving heat conduction but also for processes involving deformations. In an $L$-potentail equilibration the final state will be one of maximum entropy, while in an entropy equilibrium the final state will be one of minimum $L$. A metric space is connected with the power means, and the distance between means of different order is related to the Carnot efficiency. In the ideal classical gas limit, the average change in the entropy is shown to be proportional to the difference between the Shannon and R\'enyi entropies for nonextensive systems that are multifractal in nature. The $L$-potential, like the internal energy, is a Schur convex function of the empirical temperature, which satisfies Jensen's inequality, and serves as a measure of the tendency to uniformity in processes involving pure thermal conduction.