Dynamical and Complex Systems

Shaun Bullett, Tom Fearn, Frank Smith

• Publisher: WSPC (EUROPE)
• Year: 2016
• ISBN:
  9781786341051
• Topic: Mathematics
• Subtopic: Geometry
• Index: Mathematics

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This book leads readers from a basic foundation to an advanced level understanding of dynamical and complex systems. It is the perfect text for graduate or PhD mathematical-science students looking for support in topics such as applied dynamical systems, Lotka–Volterra dynamical systems, applied dynamical systems theory, dynamical systems in cosmology, aperiodic order, and complex systems dynamics.

Dynamical and Complex Systems is the fifth volume of the LTCC Advanced Mathematics Series. This series is the first to provide advanced introductions to mathematical science topics to advanced students of mathematics. Edited by the three joint heads of the London Taught Course Centre for PhD Students in the Mathematical Sciences (LTCC), each book supports readers in broadening their mathematical knowledge outside of their immediate research disciplines while also covering specialized key areas.

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Contents:

• Chaos in Statistical Physics (Rainer Klages)
• Aperiodic Order (Uwe Grimm)
• Complex Systems Dynamics (Hannah M Fry)
• Dynamical Systems in Cosmology (Christian G Böhmer and Nyein Chan)
• Lotka–Volterra Dynamical Systems (Stephen Baigent)
• Applied Dynamical Systems (David Arrowsmith)

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Readership: Researchers, graduate or PhD mathematical-science students who require a reference book that covers advanced techniques used in applied mathematics research.
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Chapter 1

Chaos in Statistical Physics

Rainer Klages

School of Mathematical Sciences Queen Mary University of London Mile End Road, London E1 4NS, UK

Max Planck Institute for the Physics of Complex Systems Nöthnitzer Str. 38, D-01187 Dresden, Germany [email protected]

This chapter introduces to chaos in dynamical systems and how this theory can be applied to derive fundamental laws of statistical physics from first principles. We first elaborate on the concept of deterministic chaos by defining and calculating Lyapunov exponents and dynamical entropies as fundamental quantities characterising chaos. These quantities are shown to be related to each other by Pesin’s theorem. Considering open systems where particles can escape from a set asks for a generalisation of this theorem which involves fractals, whose properties we briefly describe. We then cross-link this theory to statistical physics by discussing simple random walks on the line, their characterisation in terms of diffusion, and the relation to elementary concepts of Brownian motion. This sets the scene for considering the problem of chaotic diffusion. Here we derive a formula exactly expressing diffusion in terms of the chaos quantities mentioned above.

1. Introduction

A dynamical system is a system, represented by points in abstract space, that evolves in time. Very intuitively, one may say that the path of a point particle generated by a dynamical system looks “chaotic” if it displays “random-looking” evolution in time and space. The simplest dynamical systems that can exhibit chaotic dynamics are one-dimensional maps, as we will discuss below. Further details of such dynamics, including a mathematically rigorous definition of chaos, are given in Chapter 6.^a Surprisingly, abstract low-dimensional chaotic dynamics bears similarities to the dynamics of interacting physical many-particle systems. This is the key point that we explore in this chapter.

Over the past few decades it was found that famous statistical physical laws like Ohm’s law for electric conduction, Fourier’s law for heat conduction, and Fick’s law for the diffusive spreading of particles, which a long time ago were formulated phenomenologically, can be derived from first principles in chaotic dynamical systems. This sheds new light on the rigorous mathematical foundations of Nonequilibrium Statistical Physics, which is the theory of the dynamics of many-particle systems under external gradients or fields. The external forces induce transport of physical quantities like charge, energy, or matter. The goal of non-equilibrium statistical physics is to derive macroscopic statistical laws describing such transport starting from the microscopic dynamics for the single parts of many-particle systems. While the conventional theory puts in randomness “by hand” by using probabilistic, or stochastic, equations of motion like random walks or stochastic differential equations, recent developments in the theory of dynamical systems enable to do such derivations starting from deterministic equations of motion. Determinism means that no random variables are involved, rather, randomness is generated by chaos in the underlying dynamics. This is the field of research that will be introduced by this chapter.

This theory also illustrates the emergence of complexity in systems under non-equilibrium conditions: Due to the microscopic nonlinear interaction of the single parts in a complex many-particle system novel, non-trivial dynamics, in this case exemplified by universal transport laws, may emerge on macroscopic scales. The dynamics of a complex system, as a whole, is thus different from the sum of its single parts. In the very simplest case, this idea is illustrated by the interaction of a point particle with a scatterer, where the latter is modelled by a one-dimensional map. This is our vehicle of demonstration in the following, because this simple model can be solved rigorously analytically.

Our chapter consists of two sections: In Section 2 we introduce to two important quantities...