Assess the validity of the viscous heat equations in the ballistic–hydrodynamic crossover

Assess the validity of the viscous heat equations in the crossover regime where ballistic propagation, boundary scattering, and momentum-conserving phonon collisions interact, using systematic benchmarks against full space-dependent solutions of the phonon Boltzmann transport equation.

Background

The viscous heat equations are obtained by coarse-graining the linearized phonon Boltzmann transport equation and are intended for near-equilibrium, moderately nonlocal regimes. Their derivation neglects spatial gradients of the nonequilibrium phonon population and incorporates finite-size effects only approximately through effective boundary conditions and Bosanquet-type rescaling of transport coefficients.

The review notes that the crossover near the boundary of this regime may involve ballistic propagation, boundary scattering, and momentum-conserving collisions simultaneously. The phonon Knudsen minimum is identified as an example of such behavior, but the reliability of the viscous heat equations in this regime has not been established. Direct comparison with full space-dependent Boltzmann transport calculations is proposed as the appropriate test.

References

One example is the phonon “Knudsen minimum” [49]. The validity of the VHE in this crossover regime should therefore be assessed in future work, for example by systematic benchmarks against full space-dependent solutions of the LBTE.

— Thermal transport in crystals: from the quantum Dyson equation to mesoscopic phonon hydrodynamics  (2608.13339 - Lucente et al., 13 Aug 2026) in Section 4.12, “Limitations of the viscous heat equations”

Note that the negative dimensionless temperature represents transient cooling below the ambient temperature rather than an unphysical absolute temperature, although the actual physical realizability of such a ballistic sub-ambient state remains an open question.

— Exact analytical solution for the non-selfadjoint problem of Maxwell--Cattaneo--Vernotte heat conduction with heat-transfer boundary condition  (2608.13164 - Szücs et al., 13 Aug 2026) in Section 3.1, subsection “Parameter dependence of the solution,” discussion accompanying Fig. 7