Thermodynamic conditions ensure the stability of third-order extended heat conduction

TL;DR

This paper demonstrates that thermodynamic principles, specifically positive-definiteness of certain inequalities, guarantee the linear stability of third-order extended heat conduction models, correcting previous overly conservative conclusions.

Contribution

It shows that standard thermodynamic conditions are sufficient for stability in third-order heat conduction, clarifying previous misconceptions and confirming thermodynamics as a stability criterion.

Findings

All coefficients of the dispersion polynomial remain positive for all physical wave numbers.

Standard thermodynamic inequalities suffice for linear stability.

The stability analysis aligns with the rate-equation approach of Giorgi, Morro, and Zullo.

Abstract

In a recent work, Somogyfoki et al. (J. Non-Equilib. Thermodyn. 50, 59-76, 2025) analysed the linear stability of homogeneous equilibrium in third-order non-Fourier heat conduction within the framework of non-equilibrium thermodynamics with internal variables. They identified a stability condition, their equation (49), which could not be derived from the standard thermodynamic inequalities for the 2X2 conductivity blocks, and concluded that the Second Law does not guarantee stability in the most general case. Here we show that this conclusion was due to an overly conservative proof strategy: the standard thermodynamic conditions (concave entropy and non-negative entropy production, as expressed by the $2\times2$ block positive-definiteness inequalities (19)-(20) of the original paper) do suffice for linear stability. The key observation is that all coefficients of the dispersion polynomial remain positive for all physical wave numbers because their structure prevents positive real roots. This result confirms that thermodynamics, understood as a stability theory, ensures fundamental dynamic stability in all thermodynamically consistent third-order extended heat conduction theories. A comparison with the rate-equation approach of Giorgi, Morro and Zullo (Meccanica 59, 1757-1776, 2024) is also presented.