Hard Ball Systems and the Lorentz Gas

This is volume 101 in the Springer 'Encyclopaedia of Mathematical Sciences' and is a collection of articles about the mathematics and the physics of 'Billiards' - a dynamical system with a very long history.

From the practical point of view the game of billiards goes back thousands of years. However, those seeking advice on the playing of the game are not served by these articles. The point here is to look at the mathematics of dynamical systems with impacts and the way that this has affected study of related problems in physics, for example in statistical mechanics.

Of particular importance is the way that the shape of the enclosure affects the regularity/irregularity of these phase flows with collisions. Even simple departures from symmetry in the shape of the boundary lead to non-trivial questions about existence and measure of closed trajectories, about their stability and about invariant measures in the induced bounce map.

In his Introduction to this collection, the editor points to the difficulty of constructing a monograph on billiards and hard ball systems, the latter being collections of hard billiard balls without rotational motion, interacting via elastic collisions. In consequence this collection of survey articles was constructed and it is essential source material for students and researchers who are motivated to learn this attractive topic, as well as providing a timely stimulus for interaction between mathematicians and physicists. The collection is divided broadly into two parts: (I) Mathematics and (II) Physics.

In (I) the line guiding choice of topic is that of mathematical rigour in results and the time evolution of the theory. There are papers by:

D Burago, S Ferleger and A Kononenko	A geometric approach to semi-dispersing billiards.
T J Murphy and E G D Cohen	On the sequences of collisions among hard spheres in infinite space.
N Simany	Hard ball systems and semi-dispersive billiards: hyperbolicity and ergodicity.
N Chernov and L -S Young	Decay of correlations for Lorentz gases and hard balls.
N Chernov	Entropy values and entropy bounds.
L A Bunimovich	Existence of transport coefficients.
C Liverani	Interacting particles.
J L Lebowitz, J Piasecki and Ya Sinai	Scaling dynamics of a massive piston in an ideal gas.

In (II) the choice of material is motivated by results being obtained by methods of physics, e.g. with computer support, and to compare and contrast with those obtained through the rapid mathematical developments in the theory of hyperbolicity, ergodicity and entropy bounds. There are papers by:

R van Zon, H van Beijeren and J R Dorfman	Kinetic theory estimates for the Kolmogorov-Sinai entropy, and the largest Lyapunov exponents for dilute, hard ball gases and for dilute, random Lorentz gases.
H A Posch and R Hirschl	Simulation of billiards and of hard body fluids.
C P Dettmann	The Lorentz gas: a paradigm for nonequilibrium stationary states.
T Tel and J Vollmer	Entropy balance, multibaker maps and the dynamics of the Lorentz gas

In an Appendix there is a reprint of a lecture/article by the Editor:

D Szasz

Boltzmann's ergodic hypothesis, a conjecture for centuries?

which is welcome because it is not readily accessible elsewhere and straddles the two sections of this book, in that it gives a historic account of the problem of ergodicity from the point of view of both statistical physics and mathematics.

All in all an excellent group of survey articles.