  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.
