Entropy and Entropy Production in Multiscale Dynamics

TL;DR

This paper explores heat conduction across different levels of modeling, analyzing how entropy and thermodynamics evolve during the approach to equilibrium, emphasizing the roles of classical and non-classical entropy production.

Contribution

It introduces a multiscale analysis of entropy and thermodynamics in heat conduction, connecting Cattaneo and Fourier models and highlighting entropy production's role in equilibrium.

Findings

Entropy production characterizes the approach to equilibrium.

Classical entropy arises in the Fourier model.

Vanishing entropy production alone does not define equilibrium.

Abstract

Heat conduction is investigated on three levels: equilibrium, Fourier, and Cattaneo. The Fourier level is either the point of departure for investigating the approach to equilibrium or the final stage in the investigation of the approach from the Cattaneo level. Both investigations bring to the Fourier level an entropy and a thermodynamics. In the absence of external and internal influences preventing the approach to equilibrium the entropy that arises in the latter investigation is the production of the classical entropy that arises in the former investigation. If the approach to equilibrium is prevented, then the entropy that arises in the investigation of the approach from the Cattaneo level to the Fourier level still brings to the Fourier level the entropy and the thermodynamics even if the classical entropy and the classical thermodynamics is absent. We also note that vanishing total entropy production as a characterization of equilibrium state is insufficient.