Version 10, the stable version aug 2003  nov 2004
CHAOS: CLASSICAL AND QUANTUM
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Contents  
Index  
Chapter 1 
Overture

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

Jun 18 2003
70% finished 

Exercises  Jan 30 2002  
Aug 21 2002  
Chapter 3 
Maps

Jun 18 2003
70% finished 

Exercises  Jan 30 2002  
Aug 21 2002  
Chapter 4 
Local stability

Nov 20 2002
60% finished 

Exercises  Jan 30 2002  
Jan 30 2002  
Chapter 5 
Newtonian dynamics

Jun 18 2003
60% finished 

Appendix C:  Stability of Hamiltonian flows: more details, especially for the helium.  
Exercises  Jan 30 2002  
Jan 30 2002  
Chapter 6 
Get straight

Jun 18 2003
60% finished 

Exercises  Aug 30 2003  
Chapter 7 
Transporting densities

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

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

Aug 30 2003
60% finished 

Exercises  22 aug 98  
10 Feb 2000  
Chapter 11 
Trace formulas

Nov 20 2002
85% finished 

Exercises  Jan 30 2002  
Aug 10 2002  
Chapter 12 
Spectral determinants

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

4 Oct 98
70% finished 

Exercises  16 mar 98  
12 aug 2000  
Chapter 15 
Cycle expansions

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? Powerlaw 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 
Part II: Quantum chaos
Chapter 22 
Prologue

Jun 15 2003
70% finished 

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  
Chapter 24 
WKB quantization
A review of the WentzelKramersBrillouin quantization of 1dimensional systems. 
Jan 30 2002
85% finished  
Exercises  Jun 15 2003  
Chapter 25 
Relaxation for cyclists
In Chapter 14 we offered an introductory, handson guide to extraction of periodic orbits by means of the NewtonRaphson 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 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 infinitedimensional 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
Takehome problem set for the third millenium. 
6 Sept 96
10% finished 
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 threebody problem, the ergodic hypothesis, and nonlinear oscillators. 
22 Jul 97
66% finished  
Appendix B 
Infinitedimensional 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 shorttimes 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 NewtonRaphson 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 nth cummulant. 
12 aug 2000
30% finished  
Appendix K 
Infinite dimensional operators
What is the meaning of traces and determinants for infinitedimensional operators? 
9 Feb 96
95% finished  
Appendix L 
Statistical mechanics recycled
The Isinglike 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 