  A recently introduced systematic approach to derivations of the macroscopic
dynamics from the underlying microscopic equations of motions in the
short-memory approximation [Gorban et al, Phys. Rev. E, 63, 066124 (2001)] is
presented in detail. The essence of this method is a consistent implementation
of Ehrenfest's idea of coarse-graining, realized via a matched expansion of
both the microscopic and the macroscopic motions. Applications of this method
to a derivation of the nonlinear Vlasov-Fokker-Planck equation, diffusion
equation and hydrodynamic equations of the fluid with a long-range mean field
interaction are presented in full detail. The advantage of the method is
illustrated by the computation of the post-Navier-Stokes approximation of the
hydrodynamics which is shown to be stable unlike the Burnett hydrodynamics.
