Fluctuating Navier-Stokes equations for inelastic hard spheres or disks

TL;DR

This paper derives fluctuating hydrodynamic equations for inelastic hard spheres or disks starting from the Boltzmann equation, revealing unique properties of the fluctuating forces in such granular systems.

Contribution

It provides a derivation of fluctuating Navier-Stokes equations for inelastic particles, including novel constitutive relations and force correlations with finite relaxation times.

Findings

Fluctuating fluxes have the same form as average fluxes with the same transport coefficients.

Random forces are not white and have finite relaxation times.

Amplitude of random forces involves new coefficients not determined by macroscopic transport coefficients.

Abstract

Starting from the fluctuating Boltzmann equation for smooth inelastic hard spheres or disks, closed equations for the fluctuating hydrodynamic fields to Navier-Stokes order are derived. This requires to derive constitutive relations for both the fluctuating fluxes and the correlations of the random forces. The former are identified as having the same form as the macroscopic average fluxes and involving the same transport coefficients. On the other hand, the random force terms exhibit two peculiarities as compared with their elastic limit for molecular systems. Firstly, they are not white, but have some finite relaxation time. Secondly, their amplitude is not determined by the macroscopic transport coefficients, but involves new coefficients.