Geometrization of continuum mechanics and nonequilibrium thermodynamics: Heat conduction

Abstract

We propose a geometric formulation of non-equilibrium heat conduction. It combines the variational (Hamiltonian) structure of symmetric hyperbolic thermodynamically compatible (SHTC) equations with the exterior calculus of differential forms. The central object is the thermal impulse one-form on the four-dimensional matter-time manifold. Its time component is the temperature, and for reversible (ballistic) heat transport it is the exact differential of a scalar potential (i.e. it is holonomic). Non-equilibrium effects are described by a second one-form, the small-scale thermal impulse, which is not required to be exact. Its time component is a fluctuating temperature, and its exterior derivative, the four-thermal vorticity, satisfies a Bianchi-type kinematic identity. The variational principle together with this identity yields a non-equilibrium two-temperature model. Its field equations have the same structure as Maxwell's equations in a medium, with the thermal impulse field, the thermal vorticity field and a thermal Poynting vector contributing to the heat flux. Dissipation is introduced through relaxation terms that conserve the total energy and produce entropy. A formal asymptotic analysis shows that, in the relaxation limit, the model reduces to a Guyer-Krumhansl-type heat conduction with independent transverse and longitudinal length scales. The transverse nonlocal term comes from the spatial non-holonomy of the small-scale thermal impulse, while the longitudinal term comes from its non-holonomy in time, i.e. from the relaxation of the fluctuating temperature. We also discuss the gauge-theoretic interpretation of the model. Its kinematics is that of an abelian gauge theory, while the local gauge symmetry is explicitly broken by the existence of two temperatures. Finally, we discuss the consequences of the geometric structure for structure-preserving numerical methods.