careful manipulation leads to a property similar to linearity. But the kinds of systems that I discuss in this book typically show much more complex behavior, and have no such simplifying properties.

Scientific Computing. The field of scientific computing has usually been concerned with taking traditional mathematical models—most often for various kinds of fluids and solids—and trying to implement them on computers using numerical approximation schemes. Typically it has been difficult to disentangle anything but fairly simple phenomena from effects associated with the approximations used. The kinds of models that I introduce in this book involve no approximations when implemented on computers, and thus readily allow one to recognize much more complex phenomena.

Self-Organization. In nature it is quite common to see systems that start disordered and featureless, but then spontaneously organize themselves to produce definite structures. The loosely knit field of self-organization has been concerned with understanding this phenomenon. But for the most part it has used traditional mathematical methods, and as a result has only been able to investigate the formation of fairly simple structures. With the ideas in this book, however, it becomes possible to understand how vastly more complex structures can be formed.

Statistical Mechanics. Since its development about a century ago, the branch of physics known as statistical mechanics has mostly concerned itself with understanding the average behavior of systems that consist of large numbers of gas molecules or other components. In any specific instance, such systems often behave in a complex way. But by looking at averages over many instances, statistical mechanics has usually managed to avoid such complexity. To make contact with real situations, however, it has often had to use the so-called Second Law of Thermodynamics, or Principle of Entropy Increase. But for more than a century there have been nagging difficulties in understanding the basis for this principle. With the ideas in this book, however, I believe that there is now a framework in which these can finally be resolved.