From Kinetic Models to Hydrodynamics: Some Novel Results

​​From Kinetic Models to Hydrodynamics serves as an introduction to the asymptotic methods necessary to obtain hydrodynamic equations from a fundamental description using kinetic theory models and the Boltzmann equation. The work is a survey of an active research area, which aims to bridge time and length scales from the particle-like description inherent in Boltzmann equation theory to a fully established “continuum” approach typical of macroscopic laws of physics.The author sheds light on a new method—using invariant manifolds—which addresses a functional equation for the nonequilibrium single-particle distribution function. This method allows one to find exact and thermodynamically consistent expressions for: hydrodynamic modes; transport coefficient expressions for hydrodynamic modes; and transport coefficients of a fluid beyond the traditional hydrodynamic limit. The invariant manifold method paves the way to establish a needed bridge between Boltzmann equation theory and a particle-based theory of hydrodynamics. Finally, the author explores the ambitious and longstanding task of obtaining hydrodynamic constitutive equations from their kinetic counterparts.​ The work is intended for specialists in kinetic theory—or more generally statistical mechanics—and will provide a bridge between a physical and mathematical approach to solve real-world problems.​ Table of Contents Cover From Kinetic Models to Hydrodynamics - Some Novel Results ISBN 9781461463054 ISBN 9781461463061 Preface Acknowledgments Contents Chapter 1 Introduction Chapter 2 From the Phase Space to the Boltzmann Equation 2.1 The Phase Space Description 2.2 The Boltzmann Equation 2.2.1 H-Theorem and the Maxwellian Distribution 2.3 Hydrodynamic Limit and Other Scalings 2.4 Linearized Collision Integrals and Kinetic Models Chapter 3 Methods of Reduced Description 3.1 The Bogoliubov Hypothesis and Macroscopic Equations 3.2 The Hilbert and the Chapman-Enskog Methods 3.3 Grad's Moment Method 3.4 The Invariant Manifold Theory Chapter 4 Hydrodynamic Spectrum of Simple Fluids 4.1 Correlations in Space and Time 4.2 Linearized Hydrodynamics and Collective Modes Chapter 5 Hydrodynamic Fluctuations from the Boltzmann Equation 5.1 Eigenfrequencies of the Boltzmann Equation 5.2 The Invariant Manifold Technique 5.3 Coordinate Representation and Symmetries 5.4 The BGK Kinetic Model 5.5 The Maxwell Molecules Gas 5.5.1 Hydrodynamic Modes and Transport Coefficient 5.5.2 Short-Wavelength Hydrodynamics Chapter 6 Grad's 13-Moments System 6.1 Derivation of the 13-Moment System from the Boltzmann Equation 6.2 Hydrodynamics from the Linearized One-Dimensional Grad's System 6.2.1 Invariance Equations 6.2.2 Exact Hydrodynamic Solutions 6.2.3 Hyperbolic Transformation for Exact Hydrodynamics 6.3 Exact Hydrodynamics from Three-Dimensional Linearized Grad's Equations Chapter 7 Conclusions