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The question addressed in this monograph is the relationship between the time-reversible Newton dynamics for a system of particles interacting via elastic collisions, and the irreversible Boltzmann dynamics which gives a statistical description of the collision mechanism. Two types of elastic collisions are considered: hard spheres, and compactly supported potentials. Following the steps suggested by Lanford in 1974, we describe the transition from Newton to Boltzmann by proving a rigorous convergence result in short time, as the number of particles tends to infinity and their size simultaneously goes to zero, in the Boltzmann-Grad scaling. Boltzmann’s kinetic theory rests on the assumption that particle independence is propagated by the dynamics. This assumption is central to the issue of appearance of irreversibility. For finite numbers of particles, correlations are generated by collisions. The convergence proof establishes that for initially independent configurations, independence is statistically recovered in the limit. This book is intended for mathematicians working in the fields of partial differential equations and mathematical physics, and is accessible to graduate students with a background in analysis.

Frontmatter pp. i–iv Preface pp. v–vii Contents pp. ix–xi Part I Introduction 1 The low density limit pp. 2–6 2 The Boltzmann equation pp. 7–13 3 Main results pp. 14–18 Part II The case of hard spheres 4 Microscopic dynamics and BBGKY hierarchy pp. 20–29 5 Uniform a priori estimates for the BBGKY and Boltzmann hierarchies pp. 30–38 6 Statement of the convergence result pp. 39–48 7 Strategy of the proof of convergence pp. 49–56 Part III The case of short-range potentials 8 Two-particle interactions pp. 58–67 9 Truncated marginals and the BBGKY hierarchy pp. 68–80 10 Cluster estimates and uniform a priori estimates pp. 81–90 11 Convergence result and strategy of proof pp. 91–97 Part IV Termwise convergence 12 Elimination of recollisions pp. 99–108 13 Truncated collision integrals pp. 109–116 14 Proof of convergence pp. 117–127 15 Concluding remarks pp. 128–129 Bibliography pp. 130–132 Notation Index pp. 133–135 Index p. 137

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