Nonequilibrium Statistical Mechanics and Hydrodynamics for a Granular Fluid

TL;DR

This paper develops a theoretical framework for granular fluids using nonequilibrium statistical mechanics, deriving hydrodynamic equations and transport coefficients for inelastic hard sphere models, highlighting differences from normal fluids.

Contribution

It introduces a formal derivation of hydrodynamic equations for granular fluids from microscopic principles, emphasizing the role of inelastic collisions and the concept of a 'normal' state.

Findings

Derived exact microscopic balance equations for granular fluids.

Obtained Navier-Stokes equations with Green-Kubo transport coefficients.

Connected kinetic theory with macroscopic hydrodynamics for low-density granular gases.

Abstract

Granular fluids consist of collections of activated mesoscopic or macroscopic particles (e.g., powders or grains) whose flows often appear similar to those of normal fluids. To explore the qualitative and quantitative description of these flows an idealized model for such fluids, a system of smooth inelastic hard spheres, is considered. The single feature distinguishing granular and normal fluids being explored in this way is the inelasticity of collisions. The dominant differences observed in real granular fluids are indeed captured by this feature. Following a brief introductory description of real granular fluids and motivation for the idealized model, the elements of nonequilibrium statistical mechanics are recalled (observables, states, and their dynamics). Peculiarities of the hard sphere interactions are developed in detail. The exact microscopic balance equations for the number, energy, and momentum densities are derived and their averages described as the origin for a possible macroscopic continuum mechanics description. This formally exact analysis leads to closed, macroscopic hydrodynamic equations through the notion of a "normal" state. This concept is introduced and the Navier-Stokes constitutive equations are derived, with associated Green-Kubo expressions for the transport coefficients. A parallel description of granular gases is described in the context of kinetic theory, and the Boltzmann limit is identified critically. The construction of the "normal" solution to the kinetic equation is outlined, and Navier-Stokes order hydrodynamic equations are re-derived for a low density granular gas.