In this chapter we introduce statistical mechanics in a very general form, and explore how the tools of statistical mechanics can be used to describe complex systems.
To illustrate what statistical mechanics is, let us consider a physical system made of a number of interacting particles. When it is just a single particle in a given potential, it is an easy problem: one can write down the solution (even if one could not calculate everything in closed form). Having two particles is equally easy, as this so-called “two-body problem” can be reduced to two modified one-body problems (one for the centre of mass, other for the relative position). However, a dramatic change occurs when the number of particles is increased to three. The study of the three-body problem started with Newton, Lagrange, Laplace and many others, but the general form of the solution is still unknown. Even relatively recently, in 1993 a new type of periodic solution has been found, where three equal mass particles interacting gravitationally chase each other in a figure-of-eight shaped orbit. This and other systems where the degrees of freedom is low belongs to the subject of dynamical systems, and is discussed in detail in Chapter 2 of this volume. When the number of interacting particles increases to very large numbers, like 1023, which is typical for the number of atoms in a macroscopic object, surprisingly it gets simpler again, as long as we are interested only in aggregate quantities. This is the subject of statistical mechanics.