From Kinetic Models to Hydrodynamics

1. Front Matter

   Pages i-xii

   PDF

2. Introduction

   • Matteo Colangeli

   Pages 1-2

3. From the Phase Space to the Boltzmann Equation

   • Matteo Colangeli

   Pages 3-21

4. Methods of Reduced Description

   • Matteo Colangeli

   Pages 23-35

5. Hydrodynamic Spectrum of Simple Fluids

   • Matteo Colangeli

   Pages 37-47

6. Hydrodynamic Fluctuations from the Boltzmann Equation

   • Matteo Colangeli

   Pages 49-73

7. Grad’s 13-Moments System

   • Matteo Colangeli

   Pages 75-94

8. Conclusions

   • Matteo Colangeli

   Pages 95-96

​​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.​

• Boltzmann equation theory
• Grad’s moment method system
• Navier-Stokes Fourier approximation
• hydrodynamic equations, modes, fluctuations
• kinetic theory models
• the invariant manifold method

From the reviews:

“It gives an excellent introduction to this area of research to graduate students of applied mathematics, nonspecialists and the mathematical community in general and draws attention to some very interesting questions.” (Mark Thompson, Mathematical Reviews, November, 2013)

• , Department of Mathematics, Politecnico di Torino, Torino, Italy

  Matteo Colangeli

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