Critical Phenomena in Natural Sciences: Chaos, Fractals, Selforganization and Disorder: Concepts and Tools

Abstract

Since the discovery of the renormalization group theory in statistical physics, the realm of applications of the concepts of scale invariance and criticality has pervaded several fields of natural and social sciences. This is the leitmotiv of Didier Sornette's book, who in Critical Phenomena in Natural Sciences reviews three decades of developments and applications of the concepts of criticality, scale invariance and power law behaviour from statistical physics, to earthquake prediction, ruptures, plate tectonics, modelling biological and economic systems and so on. This strongly interdisciplinary book addresses students and researchers in disciplines where concepts of criticality and scale invariance are appropriate: mainly geology from which most of the examples are taken, but also engineering, biology, medicine, economics, etc. A good preparation in quantitative science is assumed but the presentation of statistical physics principles, tools and models is self-contained, so that little background in this field is needed. The book is written in a simple informal style encouraging intuitive comprehension rather than stressing formal derivations. Together with the discussion of the main conceptual results of the discipline, great effort is devoted to providing applied scientists with the tools of data analysis and modelling necessary to analyse, understand, make predictions and simulate systems undergoing complex collective behaviour. The book starts from a purely descriptive approach, explaining basic probabilistic and geometrical tools to characterize power law behaviour and scale invariant sets. Probability theory is introduced by a detailed discussion of interpretative issues warning the reader on the use and misuse of probabilistic concepts when the emphasis is on prediction of low probability rare---and often catastrophic---events. Then, concepts that have proved useful in risk evaluation, extreme value statistics, large limit theorems for sums of independent variables with power law distribution, random walks, fractals and multifractal formalisms, etc, are discussed in an immediate and direct way so as to provide ready-to-use tools for analysing and representing power law behaviour in natural phenomena. The exposition then continues discussing the main developments, allowing the reader to understand theoretically and model strongly correlated behaviour. After a concise, but useful, introduction to the fundamentals of statistical physics a discussion of equilibrium critical phenomena and the renormalization group is proposed to the reader. With the centrality of the problem of non-equilibrium behaviour in mind, a discussion is devoted to tentative applications of the concept of temperature in the off-equilibrium context. Particular emphasis is given to the development of long range correlation and of precursors of phase transitions, and their role in the prediction of catastrophic events. Then, basic models such as percolation and rupture models are described. A central position in the book is occupied by a chapter on mechanisms for power laws and a subsequent one on self-organized criticality as a general paradigm for critical behaviour as proposed by P Bak and collaborators. The book concludes with a chapter on the prediction of fields generated by a random distribution of sources. The book maintains the promise of the title of providing concepts and tools to tackle criticality and self-organization. The second edition, while retaining the structure of the first edition, considerably extends the scope with new examples and applications of a research field which is constantly growing. Any scientific book has to solve the dichotomy between the depth of discussion, the pedagogical character of exposition and the quantity of material discussed. In general the book, which evolved from a graduate student course, favours these last two aspects at the expense of the first one. This makes the book very readable and means that, while complicated concepts are always explained by means of simple examples, important results are often mentioned but not derived or discussed in depth. Most of the time this style of exposition manages to successfully convey the essential information, other times unfortunately, e.g. in the case of the chapter on disordered systems, the presentation appears rather superficial. This is the price we pay for a book covering an impressively vast subject area and the huge bibliography (more than 1000 references) furnishes a necessary guide for acquiring the working knowledge of the subject covered. I would recommend it to teachers planning introductory courses on the field of complex systems and to researchers wanting to learn about an area of great contemporary interest.