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Reviews of Modern Physics

Scaling, universality, and renormalization: Three pillars of modern critical phenomena

Eugene Stanley

Published Mar 1, 1999• 940 citations
Publisher

Abstract

This brief overview is designed to introduce some of the advances that have occurred in our understanding of phase transitions and critical phenomena. The presentation is organized around three simple questions: (i) What are the basic phenomena under consideration? (ii) Why do we care? (iii) What do we actually do? To answer the third question, the author shall briefly review scaling, universality, and renormalization, three of the many important themes which have served to provide the framework of much of our current understanding of critical phenomena. The style is that of a colloquium, not that of a mini-review article.

DOI
10.1103/revmodphys.71.s358

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This brief overview is designed to introduce some of the advances that have occurred in our understanding of phase transitions and critical phenomena. The presentation is organized around three simple questions: (i) What are the basic phenomena under consideration? (ii) Why do we care? (iii) What do we actually do? To answer the third question, the author shall briefly review scaling, universality, and renormalization, three of the many important themes which have served to provide the framework of much of our current understanding of critical phenomena. The style is that of a colloquium, not that of a mini-review article.
This paper is a brief summary of a talk that was designed to address the question of whether two of the pillars of the field of phase transitions and critical phenomena – scale invariance and universality – can be useful in guiding research on a broad class of complex phenomena. We shall see that while scale invariance has been tested for many years, universality is relatively more rarely discussed. In particular, we shall develop a heuristic argument that serves to make more plausible the universality hypothesis in both thermal critical phenomena and percolation phenomena, and suggest that this argument could be developed into a possible coherent approach to understanding the ubiquity of scale invariance and universality in a wide range of complex systems.
In the past two decades or so, we have learned how to understand crackling noise in a wide variety of systems. We review here the basic ideas and methods we use to understand crackling noise - critical phenomena, universality, the renormalization group, power laws, and universal scaling functions. These methods and tools were originally developed to understand continuous phase transitions in thermal and disordered systems, and we also introduce these more traditional applications as illustrations of the basic ideas and phenomena.
This book provides a comprehensive review of the theory of phase transitions and its modern applications, based on the five pillars of the modern theory of phase transitions i.e. the Ising model, mean field, scaling, renormalization group and universality. This expanded second edition includes, along with a description of vortices and high temperature superconductivity, a discussion of phase transitions in chemical reaction and moving systems. The book covers a close connection between phase transitions and small world phenomena as well as scale-free systems such as the stock market and the Internet.
A theory for tricritical phenomena proposed by Wegner and the author is described on an elementary level. The emphasis of the discussion is on (i) the differences between critical and tricritical phase transitions, (ii) the crossover phenomena that occur near tricritical points, and (iii) the determination of tricritical exponents, scaling fields and scaling functions. The results have been obtained by renormalization‐group techniques and a scaling field approach to crossover phenomena. The elements of these techniques are briefly explained.
An introductory review of the central ideas in the modern theory of dynamic critical phenomena is followed by a more detailed account of recent developments in the field. The concepts of the conventional theory, mode-coupling, scaling, universality, and the renormalization group are introduced and are illustrated in the context of a simple example---the phase separation of a symmetric binary fluid. The renormalization group is then developed in some detail, and applied to a variety of systems. The main dynamic universality classes are identified and characterized. It is found that the mode-coupling and renormalization group theories successfully explain available experimental data at the critical point of pure fluids, and binary mixtures, and at many magnetic phase transitions, but that a number of discrepancies exist with data at the superfluid transition of 4He^{4}\mathrm{He}4He.