04 oct 99

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CLASSICAL AND QUANTUM CHAOS:
A CYCLIST TREATISE

HMV

This webbook is written and maintained by the CATS cyclist team. The webbok is very preliminary and in constant flux; probably it is wisest to browse a chapter of possible interest with a postscript previewer, before deciding to print anything. If you use this material in your research, please cite the appropriate authors. We are grateful for any comments, corrections etc., sent to predrag@nbi.dk.

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Das Buch

Contents
Index
Chapter 1 Introduction
overheadoverhead An overview of the main themes of the book. Recommended reading before you decide to download anything else. [16 sep 99, 90% finished]
Appendix A: you might also want to read about the history of the subject.
Exercises [16 sep 99]
Chapter 2 Dynamics
We start out by a recapitulation of the basic notions of dynamics. The reader familiar with the dynamics on the level of an introductory graduate nonlinear dynamics course can safely skip this chapter. [8 aug 99, 70% finished]
  • Exercises [8 aug 99]
  • Chapter 3 Dynamics, qualitative
    quantum pinballQualitative dynamics of simple stretching and mixing flows is used to introduce Smale horseshoes and symbolic dynamics, and the topological dynamics is incoded by means of transition matrices/Markov graphs. [8 aug 99, 70% finished]
  • Exercises [16 mar 98]
  • Chapter 4 Counting
    One learns how to count and describe itineraries. While computing the topological entropy from transition matrices/Markov graphs, we encounter our first zeta function.
    Appendix B deals with further, more advanced symbolic dynamics techniques. [8 aug 99, 90% finished, 310 KB]
  • Exercises [16 mar 98]
  • Extras [13 Sept 97]
  • Chapter 5 Local stability
    We review the basic concepts of local dynamics required to move beyond mere counting and assign relative weights to topologically distinct parts of the phase space: local linear stability, stable/unstable manifolds for flows, billiards and maps. [8 aug 99, 45% finished]
  • Exercises [16 mar 98]
  • Extras [13 Sept 97]
  • billiards
    Chapter 6 Fixed points, and how to get them
    Periodic orbits can be determined analytically in only very exceptional cases. In this chapter we describe some of the methods for finding periodic orbits. Maps, billiards and flows are covered. There is also a neat way to find Poincare sections. [4 Oct 98, 70% finished]
  • Exercises [16 mar 98]
  • Extras [13 Sept 97]
  • Chapter 7 Transporting densities
    A first attempt to move whole phase space around - we introduce fancy operators that we will actually not use in practice, discuss natural measure. Merits a quick read. [30 Aug 98, 60% finished]
  • Exercises [16 mar 98]
  • Chapter 8 Evolution operators
    overhead overhead overhead overhead This and the next are the core chapters: we discuss the necessity of studying the averages of observables in chaotic dynamics, and cast the formulas for averages in a multiplicative form that motivates the introduction of evolution operators. [30 Aug 98, 90% finished]
  • Exercises [30 Aug 98]
  • Chapter 9 Traces and determinants
    overhead overhead overhead If there is one idea that one should learn about chaotic dynamics, it happens in this chapter: the (global) spectrum of the evolution is dual to the (local) spectrum of periodic orbits. We derive the trace formulas, spectral determinants, dynamical zeta functions. [30 Aug 98, 85% finished]
  • Exercises [30 Aug 98]
  • Chapter 10 Cycle expansions
    To compute any average one has to expand the spectral determinants into cycle expansions. These are expansions ordered by the topological length of each orbit. The chapter explains in detail how to carry them out. [30 Aug 98, 90% finished]
  • Exercises [16 mar 98]
  • Extras [17 July 98]
  • Chapter 11 Getting used to cycles
    overhead
    In the preceeding chapters we have moved at rather brisk pace and derived a spew of formulas. Here we slow down in order to develop some fingertip feeling for the objects derived so far. Just to make sure that the key message - the ``trace formulas'' and their ilk - have sunk in, we rederive them in a rather different, more intuitive way, and extol their virtues. This part is bedtime reading. A few special determinants are worked out by hand. Try them yourself and you will get used to them. [30 Aug 98, 50% finished]
  • Exercises [16 mar 98]
  • Chapter 12 Applications
    To compute an average using cycle expansions one has to find the right eigenvalue and maybe a few of its derivatives. Here we explore how to do that for all sorts of averages, some more physical than others. [16 mar 98, 30% finished]
  • Exercises [16 mar 98]
  • Chapter 13 Thermodynamic formalism
    Generalized dimensions, entropies and such. [16 mar 98, 20% finished]
  • Exercises [16 mar 98]
  • Chapter 14 Discrete symmetries
    Dynamical systems often come equipped with discrete symmetries, such as the reflection and the rotation symmetries of various potentials. Such symmetries simplify and improve the cycle expansions in a rather beautiful way. This chapter explains how symmetries factorize the cycle expansions.
    Appendix D deals with further examples of discrete symmetry (rectangles and squares). [10 Jan 99 80% finished]
  • Exercises [10 jan 99]
  • Chapter 15 Deterministic diffusion
    We look at transport coefficients and compute the diffusion constant. All done from first principles without ever invoking probability. [9 apr 98, 85% finished]
  • Exercises [16 mar 98]
  • Chapter 16 Semiclassical evolution
    Semiclassical propagator, semiclassical Green's function. [04 oct 99, 85% finished]
  • Exercises [4 oct 99]
  • Chapter 17 Semiclassical quantization
    overhead This is what could have been done with the old quantum mechanics if physicists of 1910's were as familiar with chaos as you by now are. The Gutzwiller trace formula together with the corresponding spectral determinant, the central results of the semiclassical periodic orbit theory, are derived. [4 oct 99, 80% finished]
    Chapter 18 Semiclassical chaotic scattering
    Scattering and such. [16 sep 98, 20% finished]
  • Exercises [16 sep 98]
  • Chapter 19 Helium atom
    overhead overhead The helium atom spectrum computed via spectral determinants. [new version 13 Apr 97, 85% finished]
  • Exercises [3 sep 99]
  • Extras [17 July 98]
  • Chapter 20 Creeping
    Creeping effects of getting behind cyclinders and such are incorporated into periodic orbit theory. [6 Sep 98, 5% finished]
    Chapter 21 Diffraction distraction
    Diffraction effects of scattering off wedges, eavesdropping around corners are incorporated into periodic orbit theory. [25 Sep 98, 95% finished]
    Chapter 22 Why does it work?
    This chapter faces the singular kernels, the infinite dimensional vector spaces and all those other subtleties that are needed to put the spectral determinants on more solid mathematical footing, as much as this is possible without proving theorems. [27 may 97, 15% finished]
    Chapter 23 Symbolic dynamics, generic
    Theory of pruning fronts for generic flows. [13 Apr 97, 15% finished]
  • Extras [13 Sept 97]
  • Chapter 24 Quantum and noise trace formulas
    In this chapter the quantum/noise perturbative corrections formulas are rederived using an updated version of Bohr's old quantum mechanics. [summer 96, 50% finished]
    Chapter 25 Beyond periodic orbit theory
    Dormant chapter, will contain some material from ``Beyond periodic orbit theory'' [Nonlinearity (1998)]
    Chapter 26 Irrationally winding
    Circle maps and their thermodynamics are analyzed in detail. [Dec 96, 85% finished]
    Chapter 27 Border of order
    In this chapter we face up to the cycles of marginal stability and the anomalous diffusion. [Dec 96, 45% finished]
    Chapter 28 Statistical mechanics
    How to recycle Ising-like spin systems. Includes sections on the Feigenbaum scaling function and the Fisher droplet model. [14 Nov 96, 33% finished]
  • Exercises [9 sep 98]
  • Summary
    Take-home problem set for the next millenium [6 Sept. 96, 10% finished]
    Appendix A A brief history of chaos
    Classical mechanics has not stood still since Newton. The formalism that we use today was developed by Euler and Lagrange. By the end of the 1800's the three problems that would lead to the notion of chaotic dynamics were already known: the three-body problem, the ergodic hypothesis, and nonlinear oscillators. [22 Jul 97, 66% finished]
    Appendix B Symbolic dynamics techniques
    Further, more advanced symbolic dynamics techniques. [new Dahlqvist section B.5 on infinite subshifts included 25 Jan 98; 9 March 98 60% finished]
    Appendix C Billiards
    Dormant [30 Mar 97]
    Appendix D Discrete symmetries
    Dynamical zeta functions for systems with symmetries of squares or rectangles worked out in detail. [10 Jan 99 80% finished]
    Appendix E Statistical mechanics applications
    Diffusive properties of a prototype example of chaotic Hamiltonian maps, hyperbolic toral automorphisms. [8 Feb 98]
    Appendix G Infinite dimensional operators
    What is the meaning of traces and determinants for infinite-dimensional operators? [9 Feb 96, 95% finished]
    Appendix I Quasiclassics
    Dormant [22 May 96]
    Projects
    The essence of this subject is incommunicable in print; the only way to developed intuition about chaotic dynamics is by computing, and you are urged to try to work through the essential steps in a project that combines the techniques learned in the course with some application of interest to you. [24 mar 98]
  • Deterministic diffusion, zig-zag map
    Solution [23 mar 98]
  • Deterministic diffusion, sawtooth [23 mar 98]
  • Cardioid billiard [23 mar 98]
  • Ray splitting billiard [23 mar 98]
  • list of other projects [23 mar 98]
  • [ Instructions Extras ProjectsCoursesThesesWhy this book? ]


    Periodic orbit theory sites
    Predrag Cvitanovic', predrag@nbi.dk