unstable version 10.2  
6 jun 2004  
- continuously updated: best available, but you might feel less perplexed reading the stable edition

  
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individual chapters in gzipped PostScript


CHAOS: CLASSICAL AND QUANTUM

Part I: Classical chaos

1 Overture
2 Flows
3 Maps
4 Local stability
5 Newtonian dynamics
6 Get straight
7 Cycle stability
8 Transporting densities
9 Averaging
10 Qualitative dynamics, for pedestrians
11 Qualitative dynamics, for cyclists
12 Counting, for pedestrians
13 Trace formulas
14 Spectral determinants
15 Why does it work?
16 Fixed points, and how to get them
17 Cycle expansions
18 Why cycle?
19 Thermodynamic formalism
20 Intermittency
21 Discrete symmetries
22 Deterministic diffusion
23 Irrationally winding
pinball

Part II: Quantum chaos

24 Prologue
25 Quantum mechanics, briefly
26 WKB quantization
27 Semiclassical evolution
28 Semiclassical quantization
29 Relaxation for cyclists
30 Quantum scattering
31 Helium atom
32 Diffraction distraction
Epilogue

Part III: Web appendices

A Brief history of chaos
B Infinite-dimensional flows
C Stability of Hamiltonian flows
D Implementing evolution
E Symbolic dynamics techniques
F Counting itineraries
G Finding cycles
H Applications
I Discrete symmetries
J Convergence of spectral determinants
K Infinite dimensional operators
L Statistical mechanics recycled
M Noise/quantum trace formulas
N What reviewers say
O Solutions
P Projects
Cardioid billiard
Ray splitting billiard
list of other projects

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Part I: Classical chaos


Contents

Index

Chapter 1 - Overture

An overview of the main themes of the book. Recommended reading before you decide to download anything else.

Appendix A

You might also want to read about the history of the subject.
     

Chapter 2 - Flows

A recapitulation of basic notions of dynamics. The reader familiar with the dynamics on the level of an introductory graduate nonlinear dynamics course can safely skip this materials, hop to chapter 7: Transporting densities.
    

Chapter 3 - Maps

Discrete time dynamics arises by considering sections of a continuous flow. There are also many settings in which dynamics is discrete, and naturally described by repeated applications of a map.
    

Chapter 4 - Local stability

Review of basic concepts of local dynamics: local linear stability for flows and maps.
     

Chapter 5 - Newtonian dynamics

Review of basic concepts of local dynamics: Hamiltonian flows, stability for flows, billiards and their stability.

Appendix C - Stability of Hamiltonian flows

more details, especially for the helium.
      billiards

Chapter 6 - Get straight

We can make some headway on locally straightening out flows.
     

Chapter 7 - Cycle stability

We can make some headway on locally straightening out flows.
  

Chapter 8 - Transporting densities

A first attempt to move the whole phase space around - natural measure and fancy operators.
     

Chapter 9 - Averaging

On the necessity of studying the averages of observables in chaotic dynamics. Formulas for averages are cast in a multiplicative form that motivates the introduction of evolution operators.
     

Chapter 10 - Qualitative dynamics, for pedestrians

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.
     

Appendix E:

Deals with further, more advanced symbolic dynamics techniques.
  

Chapter 11 - Qualitative dynamics, for cyclists

Theory of pruning fronts for generic flows.
     



Chapter 12 Counting, for pedestrians
You learn here how to count and describe itineraries. While computing the topological entropy from transition matrices/Markov graphs, we encounter our first zeta function.
Aug 30 2003
60% finished
Exercises 22 aug 98
10 Feb 2000
Chapter 13 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.
Nov 20 2002
85% finished
Exercises Jan 30 2002
Aug 10 2002
Chapter 14 Spectral determinants
We derive the spectral determinants, dynamical zeta functions.
Nov 20 2002
85% finished
Exercises Jan 30 2002
Aug 10 2002
Chapter 15 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, to the extent this can be achieved without proving theorems.
Nov 20 2002
76% finished
Exercises 12 aug 2000
16 May 2001
Chapter 16 Fixed points, and how to get them
Periodic orbits can be determined analytically in only few exceptional cases. In this chapter we describe some of the methods for finding periodic orbits for maps, billiards and flows. There is also a neat way to find Poincare sections.
4 Oct 98
70% finished
Exercises 16 mar 98
12 aug 2000
Chapter 17 Cycle expansions
Spectral eigenvalues and dynamical averages are computed by expanding spectral determinants into cycle expansions, expansions ordered by the topological lengths of periodic orbits.
30 Aug 98
90% finished
Exercises Jan 30 2002
10 Feb 2000
Chapter 18 Why cycle?
In the preceeding chapters we have moved at rather brisk pace and derived a gaggle 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.
Nov 20 2002
50% finished
Exercises Jan 30 2002
Aug 10 2002
Chapter 19 Thermodynamic formalism
Generalized dimensions, entropies and such.
25 aug 2000
50% finished
Exercises 25 aug 2000
Chapter 20 Intermittency
What to do about sticky, marginally stable trajectories? Power-law rather than exponential decorrelations?
Nov 20 2002
75% finished
Exercises 7 jun 2000
7 jun 2000
Chapter 21 Discrete symmetries
Dynamics often comes equipped with discrete symmetries, such as the reflection and the rotation symmetries. Symmetries simplify and improve the cycle expansions in a rather beautiful way. This chapter explains how symmetries factorize the cycle expansions.
Nov 20 2002
Appendix I: deals with further examples of discrete symmetry (rectangles and squares).
Exercises 10 jan 99
Chapter 22 Deterministic diffusion
We look at transport coefficients and derive exact formulas for diffusion constants when diffusion is normal, and the anomalous diffusion exponents when it is not. All done from first principles without ever invoking any probabilistic notions.
Nov 20 2002
85% finished
Exercises 16 mar 98
10 feb 2000
Chapter 23 Irrationally winding
Circle maps and their thermodynamics analyzed in detail.
Dec 96
85% finished
Exercises


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Part II: Quantum chaos


Chapter 24 - Prologue

In the Bohr - de Broglie old quantum theory one places a wave instead of a particle on a Keplerian orbit around the hydrogen nucleus. The quantization condition is that only those orbits contribute for which this wave is stationary. Here we shall show that a chaotic system can be quantized by placing a wave on each of the infinity of unstable periodic orbits.

Chapter 25 - Quantum mechanics, briefly

We first recapitulate basic notions of quantum mechanics and define the main quantum objects of interest, the quantum propagator and the Green's function.
    

Chapter 26 - WKB quantization

A review of the Wentzel-Kramers-Brillouin quantization of 1-dimensional systems.
    

book-2p Chapter 27 Semiclassical evolution
We relate the quantum propagator to the classical flow of the underlying dynamical system; the semiclassical propagator and Green's function.
Jan 30 2002
85% finished
Exercises Jan 30 2002
10 Feb 2000
Chapter 28 Semiclassical quantization
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.
Jan 30 2002
80% finished
Exercises Jan 30 2002
Aug 10 2002
book-2p Chapter 26 Relaxation for cyclists
In Chapter 14 we offered an introductory, hands-on guide to extraction of periodic orbits by means of the Newton-Raphson method. Here we take a very different tack, drawing inspiration from variational principles of classical mechanics, and path integrals of quantum mechanics.
Aug 30 2003
85% finished
Exercises Aug 30 2003
10 Feb 2000
Chapter 29 Chaotic scattering
Scattering off N disks, exact and semiclassical.
12 aug 2000
80% finished
Appendix K: What is the meaning of traces and determinants for infinite-dimensional operators?
Exercises 12 aug 2000
10 Feb 2000
Chapter 30 Helium atom
The helium atom spectrum computed via semiclassical spectral determinants.
17 june 2000
96% finished
Appendix C: Stability of Hamiltonian flows: more details, especially for the helium.
Exercises 12 aug 2000
Aug 10 2002
Chapter 31 Diffraction distraction
Diffraction effects of scattering off wedges, eavesdropping around corners incorporated into periodic orbit theory.
Jan 30 2002
95% finished
Exercises Jan 30 2002
Epilogue
Take-home problem set for the third millenium.
6 Sept 96
10% finished

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Part III: Material which will be kept on the web


Appendix 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 Infinite-dimensional flows
Flows described by partial differential equations are infinite dimensional because if one writes them down as a set of ordinary differential equations (ODEs) then one needs an infinity of the ordinary kind to represent the dynamics of one equation of the partial kind (PDE).
Appendix C Stability of Hamiltonian flows
Symplectic invariance, classical collinear helium stability worked out in detail.
Appendix D Implementing evolution
To sharpen our intuition, we outline the fluid dynamical vision, have a bout of Koopmania, and show that short-times step definition of the Koopman operator is a prescription for finite time step integration of the equations of motion.
Exercises
Appendix E Symbolic dynamics techniques
Further, more advanced symbolic dynamics techniques.
9 March 98
60% finished
Appendix F Counting itineraries
Further, more advanced cycle counting techniques.
Exercises
Appendix G Finding cycles
More on Newton-Raphson method.
9 March 98
60% finished
Appendix H 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.
Exercises
Appendix I Discrete symmetries
Dynamical zeta functions for systems with symmetries of squares or rectangles worked out in detail.
10 Jan 99
80% finished
Appendix J Convergence of spectral determinants
A heuristic estimate of the n-th cummulant.
12 aug 2000
30% finished
Appendix K Infinite dimensional operators
What is the meaning of traces and determinants for infinite-dimensional operators?
9 Feb 96
95% finished
Appendix L Statistical mechanics recycled
The Ising-like spin systems recycled. The Feigenbaum scaling function and the Fisher droplet model.
14 Nov 96
33% finished
Exercises 9 sep 98
10 Feb 2000
Appendix M Noise/quantum trace formulas
The quantum/noise perturbative corrections formulas derived as Bohr and Sommerfeld would have derived them were they cogniscenti of chaos, with some Vattayismo rumminations along the way.
5 Jun 1995
50% finished
Appendix N What reviewers say
Bohr, Feynman and so on turning in their graves. Ignore this.
12 aug 2000
1% finished
Appendix O Solutions
Solutions to selected problems - often more instructive than the text itself. Recommended.
Jan 30 2002
55% finished
Appendix P 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.
12 aug 2000
55% finished