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

Abstract

In a recent work, Somogyfoki et al. ( J. Non-Equilib. Thermodyn. vol. 50, pp. 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 2 × 2 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 × 2 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 (Ván, Phil. Trans. R. Soc. A , vol. 381, 2023, Art. no. 20220276), 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 , vol. 359, pp. 1757–1776, 2024) is also presented.