Basic Types of Coarse-Graining

We consider two basic types of coarse-graining: the Ehrenfest’s coarsegraining and its extension to a general principle of non-equilibrium thermodynamics, and the coarse-graining based on uncertainty of dynamical models and ε-motions (orbits). Non-technical discussion of basic notions and main coarse-graining theorems are presented: the theorem about entropy overproduction for the Ehrenfest’s coarse-graining and its generalizations, both for conservative and for dissipative systems, and the theorems about stable properties and the Smale order for ε-motions of general dynamical systems including structurally unstable systems. A brief discussion of two other types, coarse-graining by rounding and by small noise, is also presented. Computational kinetic models of macroscopic dynamics are considered. We construct a theoretical basis for these kinetic models using generalizations of the Ehrenfest’s coarse-graining.