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Thermal transport in crystals: from the quantum Dyson equation to mesoscopic phonon hydrodynamics
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DOI：10.1080/00018732.2026.2685376.png)
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Abstract
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Thermal transport in dielectric, non-magnetic crystals is mediated by quantized lattice vibrations, which drift and interact when driven out of equilibrium by a temperature gradient. This phenomenon can be described at multiple theoretical levels, ranging from fully quantum descriptions to semiclassical and mesoscopic, continuum approaches. This review rigorously discusses the theoretical steps and approximations connecting these different levels of theory, bridging the gap between quantum phonon Dyson and Kadanoff-Baym equations and semiclassical Boltzmann transport formalism, and then discussing the coarse-graining procedures that yield the mesoscopic viscous heat equations for non-diffusive, hydrodynamic heat transport in devices. We show how the established Guyer-Krumhansl and dual-phase-lag equations emerge as special linear-isotropic-band and inviscid limit of the viscous heat equations, respectively; most importantly, we demonstrate that the viscous heat equations predict not only the canonical Poiseuille flow and second sound, but also more exotic effects such as negative thermal resistance, steady-state thermal backflow and vortices. We highlight how combining these frameworks with first-principles simulations enables us to transparently connect microscopic phonon physics to observable non-diffusive heat-transport phenomena, and to guide experimentalists in their detection, amplification, and control. We discuss how to recast the viscous heat equations in terms of Helmholtz and biharmonic equations that we solve analytically. We rely on this to discuss the similarities and differences between the macroscopic behavior of the ‘phonon fluid’ and that of other hydrodynamic systems (such as classical fluids and electron fluids), discussing their hallmark on compressibility and vorticity, and their influence on strength and nature of phonon hydrodynamics. We conclude by providing a roadmap on how to generalize the tools used to describe phonon hydrodynamics to other quasiparticles, motivating future advances in the study of collective quantum transport phenomena in solids.
Keywords:
Quantum heat conduction
phonon Boltzmann transport
viscous heat equations
thermal conductivity
thermal viscosity
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13.8
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6.3K
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