We consider two basic types of coarse-graining: the Ehrenfests'
coarse-graining and its extension to a general principle of non-equilibrium
thermodynamics, and the coarse-graining based on uncertainty of dynamical
models and Epsilon-motions (orbits). Non-technical discussion of basic notions
and main coarse-graining theorems are presented: the theorem about entropy
overproduction for the Ehrenfests' coarse-graining and its generalizations,
both for conservative and for dissipative systems, and the theorems about
stable properties and the Smale order for Epsilon-motions of general dynamical
systems including structurally unstable systems. Computational kinetic models
of macroscopic dynamics are considered. We construct a theoretical basis for
these kinetic models using generalizations of the Ehrenfests' coarse-graining.
General theory of reversible regularization and filtering semigroups in
kinetics is presented, both for linear and non-linear filters. We obtain
explicit expressions and entropic stability conditions for filtered equations.
A brief discussion of coarse-graining by rounding and by small noise is also
presented.Comment: 60 pgs, 11 figs., includes new analysis of coarse-graining by
filtering. A talk given at the research workshop: "Model Reduction and
Coarse-Graining Approaches for Multiscale Phenomena," University of
Leicester, UK, August 24-26, 200
