Exploring complex systems through applications

There is no universally accepted definition of a complex system, but it is often characterized as a composition of many interacting components that display emergent properties, such as self-organization, power-law distributions, and phase transitions. The complex-systems community is highly interdisciplinary, and the systems studied usually originate from disciplines outside of physics.

Because complex systems share some of the characteristics of condensed-matter systems—namely, many elementary components, multiple scales, and phase transitions—it is natural to apply statistical mechanics techniques to them. In his new textbook, When Things Grow Many: Complexity, Universality and Emergence in Nature, Lawrence Schulman does just that. Beginning with an introduction to standard statistical mechanics techniques, he then explores their application to a curated set of examples from the literature on complex systems.

In the first half of the book, Schulman introduces readers to probabilistic techniques, the mean-field approximation and how it can fail because of fluctuations, bifurcations, stability analysis, critical phenomena, and master equations. He uses clearly explained examples from physics, including the ideal-gas law, ferromagnetism, and galaxy formation, and from other fields, such as epidemiology. Although those topics could occupy an entire statistical mechanics course, Schulman adeptly breezes through them while clearly highlighting their main concepts.

Schulman’s inclusion of information theory and the maximum-entropy method, covered in chapter 6, is a welcome addition. To his credit, he focuses on a technique first developed by Edwin Jaynes that is not typically taught in physics curricula but deserves to be more widely known. In the 1950s Jaynes showed that the Boltzmann distribution can be derived by using information theory and the available information to find the most unbiased probability distribution. That approach treats the prediction of a system with many particles as a statistical inference problem, avoids thorny fundamental issues like ergodicity, and applies to fields well beyond physics. Schulman uses the Jaynes approach to derive the Maxwell velocity distribution and shows how it can be applied to develop a thermodynamic theory of ecosystems and even predict how US Supreme Court justices will vote.

In the second half of the book, Schulman applies statistical mechanics to a wide range of interesting topics, including traffic flow, flocking, galaxy morphology, segregation of urban neighborhoods, and synchronization. He introduces each problem well and provides entry points to the relevant literature.

One of the most challenging aspects of modeling complex systems is that their elements are often heterogeneous—in contrast to the identical atoms in condensed matter—and those elements’ interactions are not necessarily short ranged and do not have the nearest-neighbor topology of atomic systems. I wish When Things Grow Many had more extensively covered techniques for tackling those issues, such as agent-based modeling and network theory, because easy-to-learn tools to do so are readily available.

I enjoyed reading When Things Grow Many and learned something new from each chapter. Schulman writes in a conversational style, and he peppers the book with jokes and opinions. Even though he intimates that he doesn’t have all the answers, his fun, inviting tone will make readers want to find out if he does. Scattered throughout the book are many computational and analytical exercises, some of which are open ended. The book also contains an extensive set of appendices with brief reviews of useful topics like probability and stochastic dynamics. I expect that anyone interested in complex systems and who has the requisite knowledge of elementary calculus and linear algebra will find When Things Grow Many to be a rewarding read.