From Newton to Boltzmann: Hard Spheres and Short-Range Potentials
            by  I. Gallagher, L. Saint-Raymond, and B. Texier
            EMS Zurich Lectures in Advanced Mathematics, Volume 18
            American Mathematical Society
150 pp. (2014)
ISBN: : 978-3-03719-129-3



Reviewed by: Leonid Bunimovich
Department of Mathematics
Georgia Institute of Technology
E-mail: bunimovh (at) math.gatech.edu

The book From Newton to Boltzmann contains a detailed derivation—on short time intervals—of the Boltzmann equation, in the Boltzmann–Grad limit, from the Newtonian dynamics of a gas of pairwise interacting particles via a compactly supported potential satisfying a convexity assumption. Recall that in the Boltzmann–Grad limit the number of particles tends to infinity while the density of the gas tends to zero and the mean path between collisions remains constant. 

The exposition essentially follows the same approach as Lanford’s celebrated proof of this result for the hard-sphere gas
(see also Lanford’s review “The Hard Sphere Gas in the Boltzmann–Grad Limit”). The major problem arises because of recollisions of the particles and therefore appearance of memory in the system. However, as Lanford showed, the dynamics can be decomposed via collision trees, and, with a probability convergent to 1, these are finite and contain no recollisions.    

The book is self-contained and well and clearly written. It could be used for a special-topics course for graduate students.