Microscopic statistical description of incompressible Navier-Stokes granular fluids

TL;DR

This paper develops a microscopic statistical framework for incompressible Navier-Stokes fluids using a modified kinetic equation derived from the Master kinetic equation, demonstrating entropy conservation and decay to equilibrium in specific flow scenarios.

Contribution

It introduces the Modified Master Kinetic Equation (MMKE) that ensures incompressibility and Navier-Stokes validity, extending the statistical description of granular fluids.

Findings

MMKE guarantees mass-density incompressibility.

MMKE ensures the validity of Navier-Stokes equations.

Initial stochastic PDFs decay to Maxwellian equilibrium.

Abstract

Based on the recently-established Master kinetic equation and related Master constant H-theorem which describe the statistical behavior of the Boltzmann-Sinai classical dynamical system for smooth and hard spherical particles, the problem is posed of determining a microscopic statistical description holding for an incompressible Navier-Stokes fluid. The goal is reached by introducing a suitable mean-field interaction in the Master kinetic equation. The resulting Modified Master Kinetic Equation (MMKE) is proved to warrant at the same time the condition of mass-density incompressibility and the validity of the Navier-Stokes fluid equation. In addition, it is shown that the conservation of the Boltzmann-Shannon entropy can similarly be warranted. Applications to the plane Couette and Poiseuille flows are considered showing that they can be regarded as final decaying states for suitable non-stationary flows. As a result, it is shown that an arbitrary initial stochastic $1-$body PDF evolving in time by means of MMKE necessarily exhibits the phenomenon of Decay to Kinetic Equilibrium (DKE), whereby the $1-$body PDF asymptotically relaxes to a stationary and spatially-uniform Maxwellian PDF.