*Short Book Reviews*,
International Statistical Institute, Vol. 5, No. 3 – December 1985

Entropy, Large Deviations, and Statistical Mechanics,

R. S. Ellis. New York: Springer-Verlag, 1985, pp. xiv + 364 pp. DM 184.00.

This book is about the relation between the probabilistic
theory of large deviations and statistical mechanics, in particular the occurrence
of phase transitions, entropy being a connecting concept. It provides a crisp
and illuminating introduction to these fields and also develops the themes primarily
covered in great mathematical detail. It is a strength of the book that the
author, rather than aiming to give a very comprehensive account of the
extensive subject areas he addresses, has chosen to focus on a limited number
of judiciously chosen topics. Thus, the statistical mechanical part consists
mainly in a detailed discussion of three main types of models, each treated in
one chapter. It is a further advantage of the book that much of the more
technical material has been placed separately in the second part of the book.
There are interesting connections to the theory of exponential families and to
saddle-point approximations and in this relation the exposition of Varadhan’s extension of Laplace’s
method deserves special mention. The book fills a lacuna in the literature in a
splendid fashion.