UK Nonlinear News, May 2002
Springer Verlag, 2000
ISBN 3 540 67621
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?|
All in all an excellent group of survey articles.
UK Nonlinear News thanks Springer-Verlag for providing a review copy of this book.
A listing of books reviewed in UK Nonlinear News is available.