Version 10, the stable version aug 2003 - nov 2004

CHAOS: CLASSICAL AND QUANTUM

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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.
Nov 20 2002
90% finished
Appendix A: you might also want to read about the history of the subject.
Exercises Jan 30 2002
Jun 10 2003
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 chapter.
Jun 18 2003
70% finished
Exercises Jan 30 2002
Aug 21 2002
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.
Jun 18 2003
70% finished
Exercises Jan 30 2002
Aug 21 2002
Chapter 4 Local stability
Review of basic concepts of local dynamics: local linear stability for flows and maps.
Nov 20 2002
60% finished
Exercises Jan 30 2002
Jan 30 2002
Chapter 5 Newtonian dynamics
Review of basic concepts of local dynamics: Hamiltonian flows, stability for flows, billiards and their stability.
Jun 18 2003
60% finished
Appendix C: Stability of Hamiltonian flows: more details, especially for the helium.
Exercises Jan 30 2002
billiards Jan 30 2002
Chapter 6 Get straight
We can make some headway on locally straightening out flows.
Jun 18 2003
60% finished
Exercises Aug 30 2003
Chapter 7 Transporting densities
A first attempt to move the whole phase space around - natural measure and fancy operators
Nov 20 2002
60% finished
Exercises Jan 30 2002
10 Feb 2000
Chapter 8 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.
Nov 20 2002
90% finished
Exercises Jan 30 2002
Aug 10 2002
Chapter 9 Qualitative dynamics, for pedestrians
quantum pinball 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.
Nov 20 2002
60% finished
Appendix E: deals with further, more advanced symbolic dynamics techniques. 8 aug 99
80% finished
Exercises Jan 30 2002
10 Feb 2000
Chapter 10 Counting, for pedestrians
One learns 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 11 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 12 Spectral determinants
We derive the spectral determinants, dynamical zeta functions.
Nov 20 2002
85% finished
Exercises Jan 30 2002
Aug 10 2002
Chapter 13 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 14 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 15 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 16 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 17 Thermodynamic formalism
Generalized dimensions, entropies and such.
25 aug 2000
50% finished
Exercises 25 aug 2000
Chapter 18 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 19 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 20 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 21 Irrationally winding
Circle maps and their thermodynamics analyzed in detail.
Dec 96
85% finished
Exercises

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Part II: Quantum chaos


Chapter 22 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.

Jun 15 2003
70% finished
book-2p Chapter 23 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.
Jan 30 2002
85% finished
book-2p Chapter 24 WKB quantization
A review of the Wentzel-Kramers-Brillouin quantization of 1-dimensional systems.
Jan 30 2002
85% finished
Exercises Jun 15 2003
book-2p Chapter 25 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
book-2p Chapter 26 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 27 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
Chapter 28 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 29 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 30 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 www: 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).
20 nov 2002
30% finished
Appendix C Stability of Hamiltonian flows
Symplectic invariance, classical collinear helium stability worked out in detail.
12 aug 2000
80% finished
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.
15 nov 2002
50% finished
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.
30 nov 2001
60% finished
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.
Jan 30 2002
60% finished
Exercises Jan 30 2002
10 Feb 2000
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


Predrag Cvitanović