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On the identification of two internal tensorial variables and a heat transport equation with inertial, thermal viscosity and vorticity terms

  • Liliana Restuccia EMAIL logo , David Jou and Michal Pavelka ORCID logo
Published/Copyright: December 8, 2025
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Journal of Non-Equilibrium Thermodynamics
From the journal Journal of Non-Equilibrium Thermodynamics

Abstract

In a rigid crystal, phonons are quasiparticles that carry heat, and they can be seen at various levels of description like phonon kinetic theory, phonon hydrodynamics, Guyer–Krumhansl equation, or Fourier heat conduction. In a previous paper, we extended the Guyer–Krumhansl equation by adding two internal state variables, a symmetric tensor and an antisymmetric tensor. These internal variables express the viscous and vortical motion of phonons. In the present paper, we analyze the model by means of asymptotic expansion, and we find a geometric analogue of the model. While the zeroth-order expansion reduces the model to the Guyer–Krumhansl equation, the first-order expansion gives a more complicated heat flux evolution equation, of which the stability conditions of equilibrium solutions are studied. The geometric formulation of the model gives the heat flux as a functional of the entropy flux, in contrast to the usual setting of Extended Irreversible Thermodynamics, and also the two fluxes cease to be proportional via the temperature (in contrast to the Clausius formula for entropy flux).

Keywords: internal variables theory; heat transfer; vorticity; Poisson bracket; phonon hydrodynamics; non-equilibrium thermodynamics
Corresponding author: Liliana Restuccia, Department of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences, University of Messina, Viale F. Stagno d’Alcontres, Salita Sperone 31, 98166, Messina, Italy, E-mail: 

Acknowledgments

MP was supported by Czech Science Foundation, project 23-05736S. MP is a member of the Nečas Centre of Mathematical Modeling.

  1. Research ethics: Not applicable.

  2. Informed consent: Not applicable.

  3. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  4. Use of Large Language Models, AI and Machine Learning Tools: None declared.

  5. Conflict of interest: The author states no conflict of interest.

  6. Research funding: MP was supported by Czech Science Foundation, project 23-05736S.

  7. Data availability: Not applicable.

Appendix A

In this Appendix we include for the reader’s convenience some mathematical formulas useful in the calculation throughout the manuscript.

Each antisymmetric tensor Q a can be represented by an axial vector Q (a). Then, it holds that

(49) Q i j a J j ( q ) = ( Q ( a ) × J ( q ) ) i ,

where the matrix formulation of the quantities reads Q i j a = 0 Q 12 a Q 13 a − Q 12 a 0 Q 23 a − Q 13 a − Q 23 a 0 ,   J j ( q ) = ( J 1 ( q ) , J 2 ( q ) , J 3 ( q ) ) T ,   ( Q ( a ) × J ( q ) ) = Q 2 ( a ) J 3 ( q ) − Q 3 ( a ) J 2 ( q ) Q 3 ( a ) J 1 ( q ) − Q 1 ( a ) J 3 ( q ) Q 1 ( a ) J 2 ( q ) − Q 2 ( a ) J 1 ( q ) .

Therefore, from equation (49), we obtain

( Q 12 a + Q 3 ( a ) ) J 2 ( q ) + ( Q 13 a − Q 2 ( a ) ) J 3 ( q ) = 0 ,

− ( Q 12 a + Q 3 ( a ) ) J 1 ( q ) + ( Q 23 a + Q 1 ( a ) ) J 3 ( q ) = 0 ,

(50) − ( Q 13 a − Q 2 ( a ) ) J 1 ( q ) − ( Q 23 a + Q 1 ( a ) ) J 3 ( q ) = 0 ,

so that equation (49) is equivalent to requiring that each vector J (q) is a solution of the homogeneous system of equations (50), i.e., we have   Q 1 ( a ) = − Q 23 a , Q 2 ( a ) = Q 13 a , Q 3 ( a ) = − Q 12 a , or

(51) Q i ( a ) = − ϵ ijk Q j k a ,

where ϵ ijk is the Levi-Civita symbol.

In the case when Q a = ( ∇ J ( q ) ) a , the associated axial vector Q (a) is

(52) Q ( a ) = 1 2 ∇ × J ( q ) , or in components   Q i ( a ) = 1 2 ϵ ijk ∇ j J k ( q ) = 1 2 ϵ ijk J k , j ( q ) ,

i.e., taking into account (51) 1 and (51) 2, we have   ∇ × J ( q ) 1 = ( J 3,2 ( q ) − J 2,3 ( q ) ) = 2 Q 1 ( a ) = − 2 Q 23 a ,   ∇ × J ( q ) 2 = ( J 1,3 ( q ) − J 3,1 ( q ) ) = 2 Q 2 ( a ) = 2 Q 13 a ,   ∇ × J ( q ) 3 = ( J 2,1 ( q ) − J 1,2 ( q ) ) = 2 Q 3 ( a ) = − 2 Q 12 a .

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Received: 2025-01-24
Accepted: 2025-09-26
Published Online: 2025-12-08

© 2025 Walter de Gruyter GmbH, Berlin/Boston

https://doi.org/10.1515/jnet-2024-0100
Keywords for this article
internal variables theory; heat transfer; vorticity; Poisson bracket; phonon hydrodynamics; non-equilibrium thermodynamics
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