CHAOS
-
CLASSICAL AND QUANTUM
Jan 30 2002,
version 9.1.1
individual chapters in gzipped PostScript:
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Book cover
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A color 1-page per sheet version, eps.gz file (3MB) or Acrobat PDF file (3.2MB) | ||
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![]() ![]() | Contents | ||
![]() ![]() | Index | ||
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Chapter 1 |
Overture
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Jan 30 2002
90% finished |
![]() ![]() | Appendix I: | you might also want to read about the history of the subject. | |
![]() ![]() | Exercises | Jan 30 2002 | |
![]() | Aug 10 2002 | ||
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Chapter 2 |
Trajectories
|
Jan 30 2002
70% finished |
![]() ![]() | Exercises | Jan 30 2002 | |
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Chapter 3 |
Maps
|
Jan 30 2002
70% finished |
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Exercises | Jan 30 2002 |
![]() | 10 Feb 2000 | ||
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Chapter 4 |
Local stability
|
Jan 30 2002
60% finished |
![]() ![]() | Appendix A: | Stability of Hamiltonian flows: more details, especially for the helium. |
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Exercises | Jan 30 2002 | |
![]() | Jan 30 2002 | ||
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Chapter 5 |
Transporting densities
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Jan 30 2002
60% finished |
![]() ![]() | Exercises | Jan 30 2002 | |
![]() | 10 Feb 2000 | ||
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Chapter 6 |
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. |
Jan 30 2002
90% finished |
![]() ![]() | Exercises | Jan 30 2002 | |
![]() ![]() | Aug 10 2002 | ||
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Chapter 7 |
Trace formulas
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Jan 30 2002
85% finished |
![]() ![]() | Exercises | Jan 30 2002 | |
![]() | Aug 10 2002 | ||
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Chapter 8 |
Spectral determinants
|
Jan 30 2002
85% finished |
![]() ![]() | Exercises | Jan 30 2002 | |
![]() | Aug 10 2002 | ||
![]() ![]() | Chapter 9 |
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. |
Jan 30 2002
76% finished |
![]() ![]() | Exercises | 12 aug 2000 | |
![]() | 16 May 2001 | ||
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Chapter 10 |
Qualitative dynamics
|
Jan 30 2002
60% finished |
![]() ![]() | Appendix B: | deals with further, more advanced symbolic dynamics techniques. |
8 aug 99
80% finished |
![]() ![]() | Exercises | Jan 30 2002 | |
![]() | 10 Feb 2000 | ||
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Chapter 11 |
Counting
|
22 aug 2000
80% finished |
![]() ![]() | Exercises | 22 aug 98 | |
![]() | 10 Feb 2000 | ||
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Chapter 12 |
Fixed points, and how to get them
|
4 Oct 98
70% finished |
![]() ![]() | Exercises | 16 mar 98 | |
![]() | 12 aug 2000 | ||
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Chapter 13 |
Cycle expansions
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30 Aug 98
90% finished |
![]() ![]() | Exercises | Jan 30 2002 | |
![]() | 10 Feb 2000 | ||
![]() ![]() | Chapter 14 |
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. |
Jan 30 2002
50% finished |
![]() ![]() | Exercises | Jan 30 2002 | |
![]() ![]() | Aug 10 2002 | ||
![]() ![]() | Chapter 15 |
Thermodynamic formalism
Generalized dimensions, entropies and such. |
25 aug 2000
50% finished |
![]() ![]() | Exercises | 25 aug 2000 | |
![]() ![]() | Chapter 16 |
Intermittency
What to do about sticky, marginally stable trajectories? Power-law rather than exponential decorrelations? |
7 jun 2001
75% finished |
![]() ![]() | Exercises | 7 jun 2000 | |
![]() | 7 jun 2000 | ||
![]() ![]() | Chapter 17 |
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. | Jan 30 2002 |
![]() ![]() | Appendix D: | deals with further examples of discrete symmetry (rectangles and squares). |
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![]() ![]() | Exercises | 10 jan 99 | |
![]() ![]() | Chapter 18 |
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. |
12 aug 2001
85% finished |
![]() ![]() | Exercises | 16 mar 98 | |
![]() | 10 feb 2000 | ||
![]() ![]() | Chapter 19 |
Irrationally winding
Circle maps and their thermodynamics analyzed in detail. |
Dec 96
85% finished |
![]() ![]() | Exercises | ||
![]() ![]() | Chapter 20 |
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 | |
![]() | 10 Feb 2000 |
Part II: Quantum chaos
Part www: Material which will be kept on the web
![]() ![]() | Appendix A |
What reviewers say
Bohr, Feynman and so on turning in their graves. Ignore this. |
12 aug 2000
1% finished |
![]() ![]() | Appendix B |
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 D |
Linear stability of Hamiltonian flows
Symplectic invariance, classical collinear helium stability worked out in detail. |
12 aug 2000
80% finished |
![]() ![]() | Appendix F |
Symbolic dynamics techniques
Further, more advanced symbolic dynamics techniques. |
9 March 98
60% finished |
![]() ![]() | Exercises | ||
![]() ![]() | Appendix I |
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 J |
Discrete symmetries
Dynamical zeta functions for systems with symmetries of squares or rectangles worked out in detail. |
10 Jan 99
80% finished |
![]() ![]() | Appendix K |
Convergence of spectral determinants
A heuristic estimate of the n-th cummulant. |
12 aug 2000
30% finished |
![]() ![]() | Appendix M |
Infinite dimensional operators
What is the meaning of traces and determinants for infinite-dimensional operators? |
9 Feb 96
95% finished |
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Appendix P |
Solutions
Solutions to selected problems - often more instructive than the text itself. Recommended. |
Jan 30 2002
55% finished |
![]() ![]() | Appendix Q |
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 |