From Hyperbolic Systems to Kinetic Theory A Personalized Quest [Paperback]

This fascinating book, penned by Luc Tartar of Americas 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, cant apply.

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 Poincar?'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!

1.Historical Perspective.- 2.Hyperbolic Systems: Riemann Invariants, Rarefaction Waves.- 3.Hyperbolic Systems: Contact Discontinuities, Shocks.- 4.The Burgers Equation and the 1-D Scalar Case.- 5.The 1-D Scalar Case: the E-Conditions of Lax and of Oleinik.- 6.Hopfs Formulation of the E-Condition of Oleinik.- 7.The Burgers Equation: Special Solutions.- 8.The Burgers Equation: Small Perturbations; the Heat Equation.- 9.Fourier Transform; the Asymptotic Behaviour for thel3U