Simplicity of the complex

More is different
September 20, 2002

Stephen Julian revels in a celebration of beauty beyond our imagination

This book is a collection of 21 articles based on a recent conference honouring Nobel laureate Philip W. Anderson, one of the world's leading theoretical physicists. Its title, More is Different , is from an influential article written by Anderson in 1972, in which he argued for the intellectual equality of different branches of science. Specifically, he attacked the view that sub-atomic particle physics is more "fundamental" and intellectually valid than Anderson's own field of condensed-matter physics. But his argument had much broader implications.

Anderson maintains that the traditional reductionist view of science is wrong, or at least incomplete, because complex systems show behaviours that are qualitatively new and different from those of the sub-units from which they are made. Moreover, it is not possible to predict the behaviour of a complex system starting from a knowledge of its constituents. From the behaviour of a small number of electrons it would be impossible - at least impossible in practice - to predict in advance the experimental observation of the phenomenon of superconductivity, which occurs only with very large numbers of electrons. Similarly, from a knowledge of how individual neurons in the human brain interact, one would not be able to predict a priori that consciousness emerges when a huge number of neurons are connected together.

Thus, each branch of science has its own "emergent" aspects that appear only when a certain level of complexity is attained, and all branches of science are in this sense "fundamental".

Anderson's idea is now widely accepted in the scientific community, with profound consequences. Probably the most significant is the burgeoning interdisciplinary science of complexity, but in physics there have been major shifts too, with the incorporation of condensed-matter physics into all major theoretical physics groups at universities - groups that once, particularly in the US, were the domain of sub-atomic particle physics.

For the general reader, by far the most interesting and accessible chapter will be the introductory article by Anderson himself, in which he reviews the motivation and development of his thinking and gives his impression of how his ideas have shaped the science of complexity, and vice versa. Anderson is not one to sell himself short, but his article is thought provoking and filled with deep insights.

In his 1972 article, he used his concept of "broken symmetry" to illustrate the thesis that "more is different". In modern physics, systems are categorised according to whether or not they are symmetrical under certain operations, such as right versus lefthandedness, or reversal of the direction of time. "Symmetry breaking" refers to his realisation in the 1950s that, in condensing into its low-energy states, a complex system may suddenly lose a symmetry via a catastrophic event called a phase transition.

In the present article, Anderson reiterates the importance of that choice, describing how the idea of broken symmetry has become better defined, due to progress in our understanding of the mathematical limit in which the number of particles in a system becomes infinite. He also suggests applications of his thesis to a range of fields, including physics, biology, economics and cosmology. For example in economics, he suggests that markets are an emergent phenomenon, in that they appear spontaneously when a society reaches a certain size.

Anderson's article is beautifully written: in a brief discussion of the quantum measurement problem as a possible case of emergent behaviour, he proposes that "to an electron, the properties of the (measurement) apparatus - Stern-Gerlach magnets, slits and the like - are much more mysterious than the properties of the electron are to us". Anyone interested in the whole enterprise of science will find this piece rewarding.

The 20 articles that follow are of a quite different nature, reviewing the state of understanding of subjects that in one way or another touch on Anderson's present or past interests. Each is by a different author or authors, many of whom are distinguished. Most deal with condensed-matter physics, but a few are on broader applications of ideas in complexity.

Among the latter group of chapters, John Hopfield focuses on a simple mathematical model of the olfactory system. In his model the presence of a large number of receptors in the nose is the crucial feature that allows the olfactory system to distinguish a given smell against a background of other smells. Hopfield's article is quite fun and one of the least demanding mathematically.

Per Bak, together with Kan Chen, describes a model of non-equilibrium complex systems that is based on the propagation of forest fires. They claim, not completely convincingly perhaps, that their model applies to a range of physical systems, including the distribution of luminous matter in the universe.

The related paper by Luciano Pietronero et al is by contrast based firmly in observations of galactic distributions in the universe and forcefully makes the point that the geometry of galaxy structures is fractal up to the largest scale at which observations have been made. Fractal geometries are a characteristic feature of complex systems, and the authors argue that this conclusion has important implications for cosmology.

The condensed-matter physics articles cover a wide range, from superfluid helium-3 to the theory of amorphous solids, reflecting the wide variety of problems that Anderson has been involved in solving. "Strongly correlated electron systems" are prominent, being the quantum many-body implementation of complexity and including topics such as conduction in disordered systems, high-temperature superconductivity, heavy fermion systems and other exotic superconductors. Taken together, these chapters could form the basis of a graduate seminar course, and this is about the level at which most of the articles are written.

The most accessible are those that give an overview of a field of research, notably T. V. Ramakrishnan on electron localisation in disordered systems, Yoshi Maeno on superconductivity in strontium-ruthenate, Hans Rudi Ott's review of heavy fermion physics, and the review of superconductivity in organic metals by Stuart Brown et al. Ramakrishnan's article is a particular delight.

Gabriel Kotliar gives a thorough but demanding summary of recent theoretical developments in the theory of the so-called Mott transition, which is the metal-insulator transition in pure systems (as opposed to Ramakrishnan's treatment of disordered systems).

High-temperature superconductivity has been the focus of Anderson's attention since the discovery, in 1986, of copper-oxide materials that superconduct at temperatures nearly ten times higher than anything seen in the previous 75 years of superconductivity research. In addition to useful surveys of the experimental situation in this field by Juan Carlos Campuzano and Bernhard Keimer, there are three theoretical articles, by Patrick Lee, Todari Senthil and Matthew Fisher, and Ganopathy Baskaran. These present variations of Anderson's celebrated RVB theory of 1986, and they are definitely technical; but by reading all three articles together, an uninitiated reader may just about get a feel for the state of research. For these articles are at the cutting edge, which means that their applicability is still in doubt: certainly they cannot all be correct in all of their details, because they sometimes contradict each other. They give an intriguing glimpse into what has been an extremely controversial area of research, and the fact that this problem has resisted theoretical solution for 15 years, during which the features of the phenomenon have become well defined experimentally, offers a clear demonstration that such things cannot be "predicted" in the strict sense of the word.

On the whole, this book fills a need for an accessible introduction to some problems at the forefront of condensed-matter physics research. A serious attempt has been made to present each subject in a simple way. Nevertheless, most of the articles require at least a degree in undergraduate physics or the equivalent. My one criticism is that the book's subtitle, Fifty Years of Condensed Matter Physics , is potentially misleading. In this book 50 years of research are being celebrated, not surveyed, with the exception of Anderson's own article.

But Anderson's central message - that complex systems display behaviour that is surprising and beautiful and beyond the predictive powers of our imagination, and hence they deserve autonomous treatment by the scientific world - is well conveyed. The editors have done an excellent job of assembling articles that offer a concrete and comprehensive demonstration of his original thesis.

Stephen Julian is reader in physics, Cavendish Laboratory, University of Cambridge.

More is different: Fifty years of condensed matter physics

Editor - N. Phuan Ong and Ravin N. Bhatt
ISBN - 0 691 08865 9 and 08866 7
Publisher - Princeton University Press
Price - £69.00 and £29.95
Pages - 345

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