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Basic Types of Coarse-Graining

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Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena

Summary

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 ɛ-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 ɛ-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’ coarsegraining. 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.

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  1. University of Leicester, Leicester, LE1 7RH, UK

    A. N. Gorban

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Editors and Affiliations

  1. Department of Mathematics, University of Leicester, University Road, LE1 7RH, Leicester, UK

    Alexander N. Gorban

  2. Institute of Computational Modeling, Russian Academy of Sciences, Russia

    Alexander N. Gorban

  3. Department of Chemical Engineering, Princeton University, Engineering Quadrangle A-217, Princeton, NJ, 08544-5263, USA

    Ioannis G. Kevrekidis

  4. School of Chemical Engineering and Analytical Science, University of Manchester, PO Box 88, Sackville St., Manchester, M60 1QD, UK

    Constantinos Theodoropoulos

  5. Department of Chemical Engineering, Worcester Polytechnic Institute, Institute Road 100, Worcester, MA, 01609-2280, USA

    Nikolaos K. Kazantzis

  6. Institut für Polymere, ETH Zürich, Wolfgang Pauli-Straße 10, CH-8093, Zürich, Switzerland

    Hans Christian Öttinger

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Gorban, A.N. (2006). Basic Types of Coarse-Graining. In: Gorban, A.N., Kevrekidis, I.G., Theodoropoulos, C., Kazantzis, N.K., Öttinger, H.C. (eds) Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena. Springer, Berlin, Heidelberg . https://doi.org/10.1007/3-540-35888-9_7

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