Moment hierarchy and exact steady-state solutions in a Callaway lattice Boltzmann model for phonon hydrodynamics

Abstract

We analyze the moment hierarchy of a Callaway lattice Boltzmann model for phonon hydrodynamics and derive exact steady-state solutions for a driven planar channel with diffuse walls. The collision operator decomposes the population space into conserved, heat-flux, and fast kinetic sectors, whose coupling through streaming determines the transport response. For the two-dimensional, eight-velocity (D2Q8) lattice Boltzmann model, we derive a fully discrete closure that relates the heat flux to transverse kinetic moments. The higher-order transverse kinetic moment contains both a second-difference term and an explicit contribution from the imposed energy drop. Diffuse-wall population constraints then determine the heat-flux and kinetic-moment profiles without fitted parameters. The resulting solution reproduces the numerical bulk moment amplitudes and wall slip, while an exact wall relation separates the contribution of the higher-order moment from deviations from the leading gradient approximation. Finite-wave-number analysis further resolves the collision-sector mixing associated with longitudinal propagation. These results provide a unified description of collision relaxation, the moment hierarchy, and boundary response within the specified discrete model.