Thermoelectric hydrodynamics of electron-phonon fluids: miscibility, drag, and vorticity

Layered semimetals such as graphite display non-diffusive, hydrodynamic heat transport [1] in the same temperature range where charge-doping-dependent thermoelectric anomalies have been observed. Here we introduce a first-principles theory based on the coupled electron–phonon Boltzmann transport equation to quantitatively describe the coupling between these phenomena, and its control from atomistic to device scales—particularly how charge doping drives the transition from phonon-only hydrodynamics to thermoelectric hydrodynamics of electron–phonon mixtures [2]. We show that composite "relaxon" electron–phonon excitations control the miscibility and the viscosity tensors of the fluid components, as well as the influence of electron–phonon drag on thermoelectric transport. We then demonstrate that the coupled Boltzmann equation can be coarse-grained into a set of mesoscopic Viscous Thermoelectric Equations, formally unifying Gurzhi's electron hydrodynamic equation [3] and the Viscous Heat Equations [1,4], while extending them to cover the coupled electron–phonon bifluid regime. We leverage this framework to elucidate when electron and phonon fluids remain unmixed or mix, rationalize thermoelectric experiments in graphite, and demonstrate that dramatic deviations from diffusive behavior such as simultaneous vortical heat and charge backflow can be induced, controlled, and amplified.

[1] Dragašević, Rajkov, Simoncelli, arXiv:2303.12777 (2025)

[2] Coulter, Rajkov, Simoncelli, arXiv:2503.07560 (2025)

[3] Gurzhi, Sov. Phys. Usp. 11 (1968)

[4] Simoncelli, Cepellotti, Marzari, Phys. Rev. X 10, 011019 (2020)