# From Newton to Boltzmann: Hard Spheres and Short-Range Potentials

Share this page
*Isabelle Gallagher; Laure Saint-Raymond; Benjamin Texier*

A publication of the European Mathematical Society

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, the authors
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.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Graduate students and research mathematicians interested in partial differential equations and mathematical physics.