Emergence, Universality, Order and Disorder

We have studied some examples of dynamical (out-of-equilibrium) systems in the previous chapter, such as those with a few degrees of freedom like the simple pendulum, and also macroscopic models of love, growth and decay. In the macroscopic models, an approximate description of a complicated system with many degrees of freedom was achieved by focussing on some essential degrees of freedom. In latter chapters we will return to a study of dynamical systems with many degrees of freedom. Here however we would like to study how statistical (averaging) methods can be used to derive effective or emergent laws that describe the equilibrium properties of a large system. (Note that the radioactive decay law was actually an example of a statistical dynamical law!). A large system, even when in equilibrium, is not easy to describe and understand analytically. Nonetheless, beginning already in the nineteenth century brilliant minds applied themselves to the problem and developed the fields of thermodynamics, kinetic theory and statistical mechanics. Some of the concepts that have been developed for the description of phenomena in equilibrium systems turn out to be relevant also for the study of more realistic out-of-equilibrium complex systems that occur in nature. Indeed, the study of equilibrium systems gives a concrete and quantitative illustration of the concepts of entropy, emergent laws and universality.