  The Euler and Navier-Stokes fluid mechanics equations are derived using a
modified statistical mechanical approach using theory taken from the
Chapman-Enskog perturbation analysis used to support the lattice Boltzmann
method. Additional distributions such as the velocity vector and total scalar
energy distributions are established. A complete discretization in velocity
space is provided. Benchmark problems are established for simple cases modeling
isothermal compressible inviscid flows. Thermal viscous flows will be the focus
of future work. Overall, a more fundamental description of mass, momentum, and
energy transport is uncovered and provides insights into the mathematical
nature of the continuum transport equations such as the incorporation of
viscosity and thermal conductivity into space and time dependence.
