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4 - Statistical mechanics of complex systems

Published online by Cambridge University Press:  05 June 2014

Ellák Somfai
Affiliation:
Wigner RCP SZFI, H-1121 Budapest, Konkoly-Thege M. u. 29-33, Hungary
Robin Ball
Affiliation:
University of Warwick
Vassili Kolokoltsov
Affiliation:
University of Warwick
Robert S. MacKay
Affiliation:
University of Warwick
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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.

Type
Chapter
Information
Complexity Science
The Warwick Master's Course
, pp. 210 - 245
Publisher: Cambridge University Press
Print publication year: 2013

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References

[1] E. T., Jaynes, Probability Theory: The logic of science. Cambridge University Press (2003).Google Scholar
[2] J. P., Sethna, Statistical Mechanics: Entropy, order parameters, and complexity. Oxford University Press (2006).Google Scholar
[3] A.-L., Barabási, H. E., Stanley, Fractal Concepts in Surface Growth. Cambridge University Press (1995).Google Scholar
[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)

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