Scale Invariance: From Phase Transitions to Turbulence,

New York
, 2012. $99.00 (397 pp.). ISBN 978-3-642-15122-4

The remarkable scale invariance exhibited in many objects and processes has profound implications for numerous problems in modern physics and on the analytical methods used to solve those problems. In Scale Invariance: From Phase Transitions to Turbulence, Annick Lesne and Michel Laguës show how scale invariance is exhibited in such physical processes as diffusion, turbulence, and superconductivity and in such systems as polymer solutions. That undertaking is a bold one, since each example presented requires significant explanation of its own physics, even before the issue of scale invariance is discussed. The reader will be surprised by the level of breadth and depth attained in this book of less than 400 pages.

The book begins with an introduction to scale invariance in the context of critical behaviors. Chapter 2 provides an overview of fractals and introduces self-similarity concepts. In chapter 3, the authors describe scaling relations and the renormalization group and its use in predicting critical behaviors. Subsequent chapters address scale invariance in specific physical phenomena.

In addition to the high quality of its numerous diagrams and plots and the admirable readability of its equations and fonts, Scale Invariance is well written and surprisingly engaging. Particularly so is chapter 4, which contains a fine presentation of anomalous diffusion. The authors also do an excellent job of including the history of the topics they present; they make wise decisions about when to include significant historical details and when to blur over them. Occasionally, though, the writing seems overwrought. For example, relatively simple points are often delivered in an unnecessarily complicated fashion, and sentences can be much longer than necessary. Perhaps the translation of the original from French to English is partly to blame.

The discussion of turbulence, its appearance in the book’s title notwithstanding, is relegated to one section in the chapter on chaotic dynamics, which appears near the end of the book. Unusual as that seems, it is perhaps appropriate: Although turbulence is a problem of great importance, it is also a largely unsolved phenomenon, and the authors generally focus on topics in which greater understanding has been attained.

The diligent reader will benefit from Scale Invariance, but as can be expected for a book covering a broad range of topics, an exhaustive explanation of each topic is not possible. Significant overlap exists between Scale Invariance and various books on chaotic dynamics and universality. Universality in Chaos (2nd edition, Taylor & Francis, 1984), edited by Predrag Cvitanović, comes to mind. Its focus is different, but both books include such topics as fractal geometry, strange attractors, turbulence, and Lyapunov exponents.

The authors provide detail on each topic, but not enough to adequately educate the novice or to satisfy the expert. For their type of presentation to work, it is crucial that the authors focus on the book’s unifying theme of scale invariance. And, to be fair, they do return to that theme in each chapter; but at times they don’t emphasize the idea strongly enough, and some of the chapters tend toward becoming merely a survey of the topic under discussion.