[Paper Review] A case study of non-Fourier heat conduction using Internal Variables and GENERIC

Mátyás Szücs, Michal Pavelka+4 more|arXiv (Cornell University)|Mar 4, 2021

Thermoelastic and Magnetoelastic Phenomena78 references18 citations

TL;DR

This paper unifies non-Fourier heat conduction models using Non-Equilibrium Thermodynamics with Internal Variables (NET-IV) and the GENERIC framework, demonstrating that entropy current multipliers—previously lacking physical interpretation—can be understood as relaxed higher-order state variables. The key contribution is the derivation of extended heat conduction laws (e.g., Maxwell–Cattaneo, Guyer–Krumhansl, ballistic–diffusive) via both NET-IV and GENERIC, revealing that the conjugate of a vectorial internal variable, not the variable itself, should be identified with heat current density.

ABSTRACT

Applying simultaneously the methodology of Non-Equilibrium Thermodynamics with Internal Variables (NET-IV) and the framework of General Equation for the Non-Equilibrium Reversible-Irreversible Coupling (GENERIC), we demonstrate that, in heat conduction theories, entropy current multipliers can be interpreted as relaxed state variables. Fourier's law and its various extensions -- the Maxwell-Cattaneo-Vernotte, Guyer-Krumhansl, Jeffreys type, Ginzburg-Landau (Allen-Cahn) type and ballistic-diffusive -- heat conduction equations are derived in both formulations. Along these lines, a comparison of NET-IV and GENERIC is also performed. Our results may pave the way for microscopic/multiscale understanding of beyond-Fourier heat conduction, and open new ways for numerical simulations of heat-conduction problems.

Motivation & Objective

• To resolve the long-standing ambiguity in the physical interpretation of Nyíri’s entropy current multipliers in non-Fourier heat conduction.
• To unify the NET-IV and GENERIC frameworks for deriving generalized heat conduction equations.
• To demonstrate that entropy current multipliers correspond to relaxed higher-order state variables in the thermodynamic state space.
• To provide a consistent GENERIC formulation of ballistic–diffusive heat conduction, previously missing in the literature.
• To enable structure-preserving numerical simulations by leveraging GENERIC’s geometric structure for beyond-Fourier models.

Proposed method

• Applying NET-IV with vectorial and tensorial internal variables to generalize the entropy current density, leading to extended heat conduction laws.
• Using the GENERIC framework to derive time evolution equations from a Hamiltonian (reversible) and a dissipation potential (irreversible) structure.
• Identifying the conjugate of the internal variable—not the variable itself—as the physical quantity corresponding to heat current density.
• Extending the state space with higher-order tensorial variables to generate nonlocal and higher-order multipliers in the entropy current density.
• Performing reductions of higher-order models to recover known equations (e.g., MCV, GK, BD) via fast relaxation of the highest-order variable.
• Deriving the same evolution equations and constitutive relations through both NET-IV generalization and GENERIC reduction, confirming consistency.

Experimental results

Research questions

• RQ1 What is the physical meaning of Nyíri’s entropy current multipliers in non-Fourier heat conduction?
• RQ2 How can the GENERIC framework be extended to include ballistic–diffusive heat conduction models?
• RQ3 Why is the identification of heat current density with the conjugate of an internal variable more theoretically consistent than direct identification with the variable itself?
• RQ4 Can the same generalized heat conduction equations be derived via both NET-IV (generalization) and GENERIC (reduction)?
• RQ5 What is the role of higher-order tensorial state variables in generating nonlocal and higher-order constitutive relations?

Key findings

• Entropy current multipliers in non-Fourier heat conduction are physically interpreted as relaxed higher-order state variables that have attained a quasi-stationary state.
• The conjugate of a vectorial internal variable—not the variable itself—must be identified with the heat current density, correcting a common theoretical misidentification.
• The ballistic–diffusive heat conduction model is successfully formulated within the GENERIC framework, completing the set of known beyond-Fourier models.
• The same generalized heat conduction equations (e.g., MCV, GK, BD) are derived both via NET-IV generalization and via GENERIC reduction, demonstrating equivalence.
• The framework preserves thermodynamic consistency through approximations and reductions, ensuring that entropy production remains non-negative.
• The unified approach enables structure-preserving numerical methods and opens pathways for microscopic/multiscale modeling of heat conduction.