Breakdown of hydrodynamics in the inelastic Maxwell model of granular gases

TL;DR

This paper analyzes the eigenfunctions and eigenvalues of the linearized Boltzmann equation for inelastic Maxwell molecules, revealing conditions where hydrodynamic descriptions break down due to slow-decaying kinetic modes.

Contribution

It provides a detailed spectral analysis of the linearized Boltzmann operator, identifying non-hydrodynamic modes and conditions for the failure of hydrodynamic descriptions in granular gases.

Findings

Identification of non-hydrodynamic modes

Existence of a critical inelasticity parameter

Breakdown of hydrodynamics below critical inelasticity

Abstract

Both the right and left eigenfunctions and eigenvalues of the linearized homogeneous Boltzmann equation for inelastic Maxwell molecules corresponding to the hydrodynamic modes are calculated. Also, some non-hydrodynamic modes are identified. It is shown that below a critical value of the parameter characterizing the inelasticity, one of the kinetic modes decays slower than one of the hydrodynamic ones. As a consequence, a closed hydrodynamic description does not exist in that regime. Some implications of this behavior on the formally computed Navier-Stokes transport coefficients are discussed.