Skip to main content Accessibility help
×
Home
  • Print publication year: 2013
  • Online publication date: June 2014

4 - Statistical mechanics of complex systems

Summary

Abstract

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.

Related content

Powered by UNSILO
[1] E. T., Jaynes, Probability Theory: The logic of science. Cambridge University Press (2003).
[2] J. P., Sethna, Statistical Mechanics: Entropy, order parameters, and complexity. Oxford University Press (2006).
[3] A.-L., Barabási, H. E., Stanley, Fractal Concepts in Surface Growth. Cambridge University Press (1995).
[4] T., Vicsek, A., Czirók, E., Ben-Jacob, I., Cohen, O., Shocket, Phys. Rev. Lett. 75, 1226 (1995)
[5] A., Czirók, H. E., Stanley, T., Vicsek; J. Phys. A: Math. Gen. 30, 1375 (1997)