Heat equations beyond Fourier: From heat waves to thermal metamaterials

In the past few decades, numerous heat conduction models extending beyond Fourier’s have been developed to account for large gradients, fast phenomena, wave propagation, and heterogeneous material structures typical of biological systems, superlattices, and thermal metamaterials. Navigating through these models has become challenging due to their varying thermodynamic backgrounds and potential compatibility issues. Furthermore, recent discoveries in the field of non-Fourier heat conduction have complicated the interpretation and utilization of specific non-Fourier heat equations, especially when designing materials for the new generation of thermal metamaterials. The situation is further compounded by the existence of numerous modeling strategies in the literature, each offering different interpretations of even the same heat equation. This complexity makes it increasingly difficult to gain a comprehensive understanding of this research field. Therefore, this review aims to facilitate the navigation of advanced heat equations beyond Fourier by discussing their properties and potential practical applications in the context of experiments. We begin with the simplest models and their fundamental principles, progressing toward more complex, coupled phenomena, such as ballistic heat conduction.

We do not delve into the often intricate technical details of each thermodynamic framework or aim to compare each approach from a methodological perspective. Instead, we focus on reviewing models primarily from the Rational Extended Thermodynamics, Extended Irreversible Thermodynamics, and Non-Equilibrium Thermodynamics with Internal Variables frameworks. Additionally, we discuss relevant models from kinetic theory, fractional derivatives, thermomass, and phase lag approaches. We provide background information on these models to highlight their origins, any limitations they may have, and the corresponding stability conditions, if applicable. Furthermore, as the field of non-Fourier heat conduction has become quite segmented, this paper also seeks to establish a common foundation, promoting a comprehensive mutual understanding of the fundamentals of each model and the phenomena to which they can be applied.