Hydrodynamic states of phonons in insulators

TL;DR

This paper extends the Chapman-Enskog method to study phonon hydrodynamics in insulators, revealing local equilibrium violations and deriving equations for dissipative processes, with results matching known low-temperature heat conductivity.

Contribution

It introduces a generalized kinetic approach for phonon hydrodynamics, accounting for non-conservation of phonon momentum and dissipation effects, providing new integral equations for transport coefficients.

Findings

Local equilibrium is violated even at linear velocity approximation.

Derived steady state heat conductivity matches Akhiezer's classical result.

Integral equations for viscosity and heat conductivity are solvable at low temperatures.

Abstract

The Chapman-Enskog method is generalized for accounting the effect of kinetic modes on hydrodynamic evolution. Hydrodynamic states of phonon system of insulators have been studied in a small drift velocity approximation. For simplicity, the investigation was carried out for crystals of the cubic class symmetry. It has been found that in phonon hydrodynamics, local equilibrium is violated even in the approximation linear in velocity. This is due to the absence of phonon momentum conservation law that leads to a drift velocity relaxation. Phonon hydrodynamic equations which take dissipative processes into account have been obtained. The results were compared with the standard theory based on the local equilibrium validity. Integral equations have been obtained for calculating the objects of the theory (including viscosity and heat conductivity). It has been shown that in low temperature limit, these equations are solvable by iterations. Steady states of the system have been considered and an expression for steady state heat conductivity has been obtained. It coincides with the famous result by Akhiezer in the leading low temperature approximation. It has been established that temperature distribution in the steady state of insulator satisfies a condition of heat source absence.