From Hyperbolic Systems to Kinetic Theory

A Personalized Quest

  • Luc Tartar

Part of the Lecture Notes of the Unione Matematica Italiana book series (UMILN, volume 6)

About this book

Introduction

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!

Keywords

Navier-Stokes equation conservation laws fluid mechanics heat equation hyperbolic systems interacting particle systems wave equation

Authors and affiliations

  • Luc Tartar
    • 1
  1. 1.Carnegie Mellon UniversityPittsburghUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-540-77562-1
  • Copyright Information Springer Berlin Heidelberg 2008
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-540-77561-4
  • Online ISBN 978-3-540-77562-1
  • Series Print ISSN 1862-9113
  • About this book