Linearized Boltzmann Equation and Hydrodynamics for Granular Gases

TL;DR

This paper derives hydrodynamic equations and transport coefficients for granular gases from the linearized Boltzmann equation, confirming previous results and discussing implications for linear response theory.

Contribution

It provides a hydrodynamic framework based on the spectrum of the linearized Boltzmann operator for granular gases, linking microscopic dynamics to macroscopic equations.

Findings

Hydrodynamic eigenfunctions and eigenvalues are calculated in the long wavelength limit.

The derived Navier-Stokes equations and transport coefficients agree with Chapman-Enskog results.

Implications for linear response methods and Green-Kubo formulas in granular fluids are discussed.

Abstract

The linearized Boltzmann equation is considered to describe small spatial perturbations of the homogeneous cooling state. The corresponding macroscopic balance equations for the density, temperature, and flow velocity are derived from it as the basis for a hydrodynamic description. Hydrodynamics is defined in terms of the spectrum of the generator for the dynamics of the linearized Boltzmann equation. The hydrodynamic eigenfunctions and eigenvalues are calculated in the long wavelength limit. The results allow identification of the hydrodynamic part of the solution to the linearized Boltzmann equation. This contribution is used to calculate the fluxes in the macroscopic balance equations, leading to the Navier-Stokes equations and associated transport coefficients. The results agree with those obtained earlier by the Chapman-Enskog method. The implications of this analysis for application of methods of linear response to granular fluids and derivation of Green-Kubo expressions for transport coefficients are discussed.