Lattice Boltzmann method for semiclassical fluids

TL;DR

This paper develops a lattice Boltzmann method tailored for semiclassical fluids, incorporating Bose-Einstein and Fermi-Dirac distributions, introducing new polynomials, and validating the model with electron flow experiments.

Contribution

It introduces new D-dimensional polynomials for the lattice Boltzmann method based on semiclassical distributions, enhancing numerical efficiency and applicability to electron hydrodynamics.

Findings

New polynomials generalizing Hermite polynomials are introduced.

The method accurately reproduces electron flow behaviors such as Ohm's law.

Validation through Riemann problem and Poiseuille flow confirms model effectiveness.

Abstract

We determine properties of the lattice Boltzmann method for semiclassical fluids, which is based on the Boltzmann equation and the equilibrium distribution function is given either by the Bose-Einstein or the Fermi-Dirac ones. New D-dimensional polynomials, that generalize the Hermite ones, are introduced and we find that the weight that renders the polynomials orthonormal has to be approximately equal, or equal, to the equilibrium distribution function itself for an efficient numerical implementation of the lattice Boltzmann method. In light of the new polynomials we discuss the convergence of the series expansion of the equilibrium distribution function and the obtainment of the hydrodynamic equations. A discrete quadrature is proposed and some discrete lattices in one, two and three dimensions associated to weight functions other than the Hermite weight are obtained. We derive the forcing term for the LBM, given by the Lorentz force, which dependents on the microscopic velocity, since the bosonic and fermionic particles can be charged. Motivated by the recent experimental observations of the hydrodynamic regime of electrons in graphene, we build an isothermal lattice Boltzmann method for electrons in metals in two and three dimensions. This model is validated by means of the Riemann problem and of the Poiseuille flow. As expected for electron in metals, the Ohm's law is recovered for a system analogous to a porous medium.