30 jan 2000, version 6.0.1

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CLASSICAL AND QUANTUM CHAOS

Contents
Index

Chapter 1 Overture
overhead An overview of the main themes of the book. Recommended reading before you decide to download anything else. [23 jan 2000, 90% finished]
  • Appendix A: you might also want to read about the history of the subject.
    Exercises [23 jan 2000]
    Chapter 2 Dynamics
    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. [23 jan 2000, 70% finished]
  • Exercises [23 jan 2000]

  • Chapter 3 Local stability
    Review of basic concepts of local dynamics: local linear stability and stable/unstable manifolds for flows, billiards and maps. [23 jan 2000, 45% finished]
  • Appendix B Stability of Hamiltonian flows: more details, especially for the Helium.
  • Exercises [23 jan 2000]
  • Extras [13 Sept 97]

  • Chapter 4 Transporting densities
    A first attempt to move the whole phase space around - we introduce fancy operators, discuss natural measure. [23 jan 2000, 60% finished]
  • Exercises [23 jan 2000]

  • Chapter 5 Averaging
    overheadoverheadoverheadoverheadoverhead The necessity of studying the averages of observables in chaotic dynamics; the formulas for averages are cast in a multiplicative form that motivates the introduction of evolution operators. [23 jan 2000, 90% finished]
  • Exercises [23 jan 2000]

  • Chapter 6 Trace formulas
    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. The duality is made precise by means of trace formulas. [23 jan 2000, 85% finished]
  • Exercises [23 jan 2000]

  • Chapter 7 Qualitative dynamics
    Qualitative dynamics of simple stretching and mixing flows; Smale horseshoes and symbolic dynamics. The topological dynamics is incoded by means of transition matrices/Markov graphs. [8 aug 99, 60% finished]
  • Exercises [16 mar 98]

  • Chapter 9 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 8 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, 80% finished]
  • Exercises [16 mar 98]
  • Extras [13 Sept 97]

  • Chapter 10 Spectral determinants
    We derive the spectral determinants, dynamical zeta functions. [23 jan 2000, 85% finished]

    Chapter 11 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 periodic orbits. The chapter explains in detail how to carry them out. [30 Aug 98, 90% finished]
  • Exercises [23 jan 2000]
  • Extras [17 July 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 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 14 Thermodynamic formalism
    Generalized dimensions, entropies and such. [16 mar 98, 20% finished]
  • Exercises [16 mar 98]

  • Chapter 15 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 16 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 17 Semiclassical evolution
    Semiclassical propagator and Green's function. [04 oct 99, 85% finished]
  • Exercises [4 oct 99]

  • Chapter 18 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 19 Semiclassical chaotic scattering
    Scattering off N disks, exact and semiclassical. [16 sep 98, 20% finished]
  • Exercises [16 sep 98]

  • Chapter 20 Helium atom
    overheadoverhead The helium atom spectrum computed via spectral determinants. [new version 13 Apr 97, 85% finished]
  • Exercises [3 sep 99]
  • Extras [17 July 98]

  • Chapter 21 Diffraction distraction
    Diffraction effects of scattering off wedges, eavesdropping around corners incorporated into periodic orbit theory. [25 Sep 98, 95% finished]

    Chapter 22 Irrationally winding
    Circle maps and their thermodynamics analyzed in detail. [Dec 96, 85% finished]

    Chapter 23 Border of order
    We face up to the cycles of marginal stability and the anomalous diffusion. [Dec 96, 45% finished]

    Chapter 24 Statistical mechanics
    The Ising-like spin systems recycled. 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]

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    Appendices

    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 Helium stability
    Helium stability worked out in detail. [10 Jan 99 80% 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 Transporting densities
    Transporting densities worked out in detail. [10 Jan 99 80% finished]
    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 Infinite dimensional operators
    What is the meaning of traces and determinants for infinite-dimensional operators? [9 Feb 96, 95% finished]
    Appendix F Difraction distraction
    What is the meaning of difraction? [9 Feb 96, 95% finished]
    Appendix G Solutions
    Solutions to selected problems - often more instructive than the text itself. Recommended. [9 Feb 96, 95% finished]

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    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]
  • Help Extras ProjectsCourses | ThesesWhy webbook? ]


    Predrag Cvitanovic', predrag@nbi.dk