Statistical fluid dynamics equation and its reduction to the Navier–Stokes equations in macroscopic form

In this work, the statistical fluid dynamics equation for the molecule number density function is derived by using the Taylor-series expansion as an alternative theoretical modeling of the collision integral in the Boltzmann equation. Formally, its macroscopic forms for the fluid density and velocity are reduced to the continuity equation and the Navier–Stokes equations. In principle, the statistical dynamics equation can be solved with the linear response theory, and then the macroscopic fluid velocity is calculated as the particle velocity moments of the molecule number density function. To elucidate this theory, as an example, the approximate analytical solutions for the molecule number density function and the fluid velocity are obtained for a flat plate suddenly moving at a constant velocity in the direction normal to the front surface in still fluid, demonstrating the formation of a boundary layer in the fluid compression process.