From Hyperbolic Systems to Kinetic Theory
Description
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!
Reviews / Votes
From the reviews:"The book is well organized. . Tartar is excellent in bringing out the essence of ideas and methods . . The monograph is an interesting read from more than one point of view. First, the mathematically literate reader finds an extensive . review of the relevant subjects; reading the sketches of proofs offers a bird's eye view of the underlying concepts as well as the exceptional mind of the author. . All in all, a remarkable book." (Reinhard Illner, SIAM Reviews, Vol. 51 (2), June, 2009)"This is a very nice book devoted to hyperbolic systems of conservation laws. . This book can be recommended to all researchers in respective areas, especially to graduate students." (Mitsuru Yamazaki, Mathematical Reviews, Issue 2010 j)More details
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