From Hyperbolic Systems to Kinetic Theory

Equations of state are not always effective in continuum mechanics. Maxwell and Boltzmann created a kinetic theory of gases, using classical mechanics. How could they derive the irreversible Boltzmann equation from a reversible Hamiltonian framework? By using probabilities, which destroy physical reality! Forces at distance are non-physical as we know from Poincaré's theory of relativity. Yet Maxwell and Boltzmann only used trajectories like hyperbolas, reasonable for rarefied gases, but wrong without bound trajectories if the "mean free path between collisions" tends to 0. Tartar relies on his H-measures, a tool created for homogenization, to explain some of the weaknesses, e.g. from quantum mechanics: there are no "particles", so the Boltzmann equation and the second principle, can not apply. He examines modes used by energy, proves which equation governs each mode, and conjectures that the result will not look like the Boltzmann equation, and there will be more modes than those indexed by velocity!

Résumé

This fascinating book, penned by Luc Tartar of America's Carnegie Mellon University, starts from the premise that equations of state are not always effective in continuum mechanics. Tartar relies on H-measures, a tool created for homogenization, to explain some of the weaknesses in the theory. These include looking at the subject from the point of view of quantum mechanics. Here, there are no "particles", so the Boltzmann equation and the second principle, can't apply.

Contenu

Historical Perspective.- Hyperbolic Systems: Riemann Invariants, Rarefaction Waves.- Hyperbolic Systems: Contact Discontinuities, Shocks.- The Burgers Equation and the 1-D Scalar Case.- The 1-D Scalar Case: the E-Conditions of Lax and of Oleinik.- Hopf's Formulation of the E-Condition of Oleinik.- The Burgers Equation: Special Solutions.- The Burgers Equation: Small Perturbations; the Heat Equation.- Fourier Transform; the Asymptotic Behaviour for the Heat Equation.- Radon Measures; the Law of Large Numbers.- A 1-D Model with Characteristic Speed 1/?.- A 2-D Generalization; the Perron-Frobenius Theory.- A General Finite-Dimensional Model with Characteristic Speed 1/?.- Discrete Velocity Models.- The Mimura-Nishida and the Crandall-Tartar Existence Theorems.- Systems Satisfying My Condition (S).- Asymptotic Estimates for the Broadwell and the Carleman Models.- Oscillating Solutions; the 2-D Broadwell Model.- Oscillating Solutions: the Carleman Model.- The Carleman Model: Asymptotic Behaviour.- Oscillating Solutions: the Broadwell Model.- Generalized Invariant Regions; the Varadhan Estimate.- Questioning Physics; from Classical Particles to Balance Laws.- Balance Laws; What Are Forces?.- D. Bernoulli: from Masslets and Springs to the 1-D Wave Equation.- Cauchy: from Masslets and Springs to 2-D Linearized Elasticity.- The Two-Body Problem.- The Boltzmann Equation.- The Illner-Shinbrot and the Hamdache Existence Theorems.- The Hilbert Expansion.- Compactness by Integration.- Wave Front Sets; H-Measures.- H-Measures and "Idealized Particles".- Variants of H-Measures.- Biographical Information.- Abbreviations and Mathematical Notation.