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Basic Types of Coarse-Graining

By Alexander N. Gorban


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 ε-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 ε-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 semi-groups 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

Publisher: Dept. of Mathematics, University of Leicester
Year: 2006
OAI identifier: oai:lra.le.ac.uk:2381/4266

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  1. (1954). A model for collision processes in gases. I. Small amplitude processes in charged and neutral one–component systems, doi
  2. (1967). A unifying principle in statistical mechanics, doi
  3. (1970). About one class of invariant sets of smooth dynamical systems,
  4. (2003). An encyclopaedia of cubature formulas, doi
  5. (1912). Begru¨ndung der kinetischen Gastheorie, doi
  6. (2002). Canonical distribution functions in polymer dynamics: I. Dilute solutions of flexible polymers, doi
  7. (2003). Canonical distribution functions in polymer dynamics: II Liquid–crystalline polymers, doi
  8. (1978). Chain transitivity and the domain of influence of an invariant set, doi
  9. (1995). Collapsing effect in numerical simulation of a class of chaotic dynamical systems and random mappings with a single attracting centre, doi
  10. (2004). Combined micro–macro integration scheme from an invariance principle: application to ferrofluid dynamics, doi
  11. (2004). Constructive methods of invariant manifolds for kinetic problems, doi
  12. (2001). Corrections and enhancements of quasi–equilibrium states, doi
  13. (1965). Derivation of Grad–type equations and study of their properties by the method of entropy maximization, doi
  14. (1986). Dissipation in many–body systems: A geometric approach based on information theory, doi
  15. (1999). Dynamic inverse modeling and its testing in large-eddy simulations of the mixing layer. doi
  16. (1927). Dynamical systems, AMS Colloquium Publications, doi
  17. (1987). Effects of phase space discretization on the long–time behavior of dynamical systems, doi
  18. (2001). Ehrenfests’ argument extended to a formalism of nonequilibrium thermodynamics, doi
  19. (2005). Elastic Principal Graphs and Manifolds and their Practical Applications, doi
  20. (1960). Elementary Principles of Statistical Mechanics, doi
  21. (2002). Entropy function approach to the lattice Boltzmann method, doi
  22. (2003). Equation-free, coarse-grained multiscale computation: enabling microscopic simulators to perform system-level analysis, doi
  23. (1966). Equations of motion in nonequilibrium statistical mechanics, doi
  24. (1984). Equilibrium encircling. Equations of chemical kinetics and their thermodynamic analysis,
  25. (2003). Family of additive entropy functions out of thermodynamic limit, doi
  26. (1985). Finite precision and transient behavior, doi
  27. (2000). Finite–precision stationary states at and away from equilibrium, doi
  28. (2002). Galerkin Proper Orthogonal Decomposition Methods for a General Equation in Fluid Dynamics, doi
  29. (2003). Galilean-invariant latticeBoltzmann models with H theorem, doi
  30. (1994). General approach to constructing models of the Boltzmann equation, doi
  31. (1963). General Circulation Experiments with the Primitive Equations: I. The Basic Equations, doi
  32. (1973). Genericity of the dynamical systems almost all orbits of which are stable under sustained perturbations,
  33. (2002). Geometry of irreversibility, in: doi
  34. (2003). Geometry of irreversibility: The film of nonequilibrium states, doi
  35. (1972). Gibbs measures in ergodic theory, doi
  36. (1998). Hamiltonian description of the ideal fluid, doi
  37. (1977). Hyperbolicity and chain recurrence, doi
  38. (1969). Identifying perturbations which preserved asymptotic stability, doi
  39. (1985). Influence of damping on quantum interference: An exactly soluble model, doi
  40. (1963). Information theory and statistical mechanics, in: Statistical Physics. Brandeis Lectures, doi
  41. (1959). Information theory and statistics, doi
  42. (1995). Introduction to the Modern Theory of Dynamical Systems, doi
  43. (2005). Invariant manifolds for physical and chemical kinetics, doi
  44. (1998). Invertibility of dynamical systems in granular phase space, doi
  45. (2003). Irreversibility in the short memory approximation, doi
  46. (2002). Kinetic Boundary condition for the lattice Boltzmann method, doi
  47. (2002). Kinetic theory and fluid dynamics, doi
  48. (2004). Kinetic Theory of Turbulence Modeling: Smallness Parameter, Scaling and Microscopic Derivation of Smagorinsky Model, doi
  49. (2001). Lagrangian methods for the tensor-diffusivity subgrid model, doi
  50. (2005). Large-Eddy Simulations of Turbulence, doi
  51. (1992). Lattice BGK models for Navier–Stokes equation, doi
  52. (1989). Lattice gas-dynamics with enhanced collisions, doi
  53. (2002). Macroscopic dynamics through coarse–graining: A solvable example, doi
  54. (2002). Magnetization dynamics, rheology, and an effective description of ferromagnetic units in dilute suspension, doi
  55. (1970). Mathematical theory of non-uniform gases, Third edition, doi
  56. (1984). Maximization of entropy, kinetic equations, and irreversible thermodynamics Phys. doi
  57. (1998). Maximum entropy principle for lattice kinetic equations, doi
  58. (1999). Metric structures for Riemannian and non-Riemannian spaces. doi
  59. (1996). Moment Closure Hierarchies for Kinetic Theories, doi
  60. (2003). Monotonically equivalent entropies and solution of additivity equation, doi
  61. Nonextensive statistical mechanics and its applications, doi
  62. (1989). Nonlinear Galerkin methods, doi
  63. (1989). Nonlinear reciprocity and the maximum entropy formalism, doi
  64. (2003). Nonlinear regularization for large-eddy simulation, doi
  65. (2004). Numerical Comparison of Bhatnagar–Gross–Krook models with proper Prandtl number, doi
  66. (2003). On exponentially attracting invariant manifolds of ODEs, doi
  67. (2000). On riddling and weak attractors. doi
  68. (2004). On the (Boltzmann) Entropy of Nonequilibrium Systems, doi
  69. (1988). On the computation of inertial manifolds, doi
  70. (1985). On the concept of attractor, doi
  71. (1949). On the kinetic theory of rarefied gases, doi
  72. (1964). On the macroscopic description of kinetic processes,
  73. (2001). On the modelling of the subgrid-scale and filtered-scale stress tensors in large-eddy simulation, doi
  74. (1978). On the pseudoorbit tracing property and its relationship to stability,
  75. (1932). On the quantum correction for thermodynamic equilibrium, doi
  76. (1957). On the stability under sustained perturbations,
  77. (1944). On the stability under uniformly influencing perturbations,
  78. (2002). Optimal prediction with memory, doi
  79. (1999). Perfect entropy functions of the Lattice Boltzmann method, doi
  80. (1986). Principal component analysis, doi
  81. (2005). Projecting to a slow manifold: singularly perturbed systems and legacy codes, doi
  82. Projection operator techniques in nonequilibrium statistical mechanics, doi
  83. (1996). Proof of the Uniqueness of the Solution of the Equations of the Law of Mass Action, In: doi
  84. (2003). Quantum moment hydrodynamics and the entropy principle, doi
  85. Quasi–equilibrium approximation and non–standard expansions in the theory of the Boltzmann kinetic equation, in:
  86. (2006). Quasi–equilibrium closure hierarchies for the Boltzmann equation, doi
  87. (2002). Random Dynamical Systems, doi
  88. Random Graphs (Cambridge Studies in Advanced Mathematics), doi
  89. (2002). Reconstruction lemma and fluctuation–dissipation theorem,
  90. (1992). Recovery of the Navier–Stokes equation using a lattice–gas Boltzmann Method Phys. doi
  91. (2002). Renormalization group method applied to kinetic equations: roles of initial values and time, Annals Phys. doi
  92. (2002). Role of the H theorem in lattice Boltzmann hydrodynamic simulations, doi
  93. (1988). Roundoff–induced periodicity and the correlation dimension of chaotic attractors, doi
  94. (1996). Rounoff–induced coalescence of chaotic trajectories, doi
  95. (1999). Sampling Theory in Fourier and Signal Analysis: Advanced Topics, doi
  96. (1996). Sampling Theory in Fourier and Signal Analysis: Foundations,
  97. (2004). Singularities of Transition Processes in Dynamical Systems: Qualitative Theory of Critical Delays,
  98. (1980). Slow relaxations and bifurcations of omega-limit sets of dynamical systems, doi
  99. (1999). Smooth Dynamics and New Theoretical Ideas in Nonequilibrium Statistical Mechanics, doi
  100. (1996). Statistical mechanics of nonequilibrium processes,
  101. (1999). Statistical Mechanics: A Selective Review of Two Central Issues, doi
  102. (1966). Structurally stable systems are not dense, doi
  103. (1934). Sur les movements dun fluide visqueux remplaissant lespace,
  104. (2004). The adjoint filter operator in large-eddy simulation of turbulent flow, doi
  105. (1988). The Boltzmann equation and its applications, doi
  106. (2000). The Boltzmann equation is a renormalization group equation, doi
  107. (1911). The Conceptual Foundations of the Statistical Approach in Mechanics, In: doi
  108. (1996). The foundations of informational statistical thermodynamics revisited, doi
  109. (1992). The general problem of the stability of motion, doi
  110. (2001). The lattice Boltzmann equation for fluid dynamics and beyond, doi
  111. (1984). The macrodynamics of open systems and the variational principle of the local potential, doi
  112. (1966). The motion stability theory.
  113. (1993). The proper orthogonal decomposition in the analysis of turbulent flows, doi
  114. (1974). The renormalization group and the ǫ-expansion, Phys. Reports 12C
  115. (2006). Thermodynamic Equilibria and Extrema: Analysis of Attainability Regions and Partial Equilibrium,
  116. (1980). Thermodynamics of nonequilibrium processes far from equilibrium, in:
  117. (1992). Turbulence: the filtering approach, doi
  118. (2004). Uniqueness of thermodynamic projector and kinetic basis of molecular individualism, doi
  119. (1991). Universal Expansion of the Triplet Distribution Function, Teoret. i Matem. doi

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