04 oct 99
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CLASSICAL AND QUANTUM
CHAOS:
A CYCLIST TREATISE
This webbook is written and maintained by the
CATS cyclist team. The webbok is very
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browse a chapter of possible interest with a postscript previewer,
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cite the appropriate authors.
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Are you ready?
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Why this book?
Das Buch
-
Contents
-
Index
-
Chapter 1 Introduction
-

An overview of the main themes of the book.
Recommended reading before you decide to download anything else.
[16 sep 99, 90% finished]
Appendix A:
you might also want to read about the history of the subject.
Exercises
[16 sep 99]
-
Chapter 2 Dynamics
-
We start out by 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.
[8 aug 99, 70% finished]
-
Exercises
[8 aug 99]
-
Chapter 3 Dynamics, qualitative
Qualitative
dynamics of simple stretching and mixing flows is used to introduce Smale
horseshoes and symbolic dynamics, and
the topological dynamics is incoded by means of transition
matrices/Markov graphs.
[8 aug 99, 70% finished]
-
Exercises
[16 mar 98]
-
Chapter 4 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, 90% finished, 310 KB]
-
Exercises
[16 mar 98]
-
Extras
[13 Sept 97]
-
Chapter 5 Local stability
- We review the basic concepts of local dynamics required to move beyond
mere counting and assign relative weights to topologically distinct parts
of the phase space: local linear stability,
stable/unstable manifolds for
flows, billiards and maps. [8 aug 99, 45%
finished]
Exercises
[16 mar 98]
Extras
[13 Sept 97]
Chapter 6 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 7 Transporting densities
A first attempt to move whole phase space around - we introduce fancy
operators that we will actually not use in practice, discuss natural
measure. Merits a quick read.
[30 Aug 98, 60% finished]
Exercises
[16 mar 98]
Chapter 8 Evolution operators
This and the next are the core chapters:
we discuss the necessity of studying the averages of
observables in chaotic dynamics, and cast the formulas for averages
in a multiplicative form that motivates the introduction of
evolution operators.
[30 Aug 98, 90% finished]
Exercises
[30 Aug 98]
Chapter 9
Traces and determinants
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. We
derive the trace formulas, spectral determinants, dynamical zeta functions.
[30 Aug 98, 85% finished]
Exercises
[30 Aug 98]
Chapter 10 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 each orbit. The chapter explains in detail how to carry them
out. [30 Aug 98, 90% finished]
Exercises
[16 mar 98]
Extras
[17 July 98]
Chapter 11 Getting used to
cycles
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 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
Thermodynamic formalism
Generalized dimensions, entropies and such.
[16 mar 98, 20% finished]
Exercises
[16 mar 98]
Chapter 14 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 15 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 16 Semiclassical
evolution
Semiclassical propagator, semiclassical Green's function.
[04 oct 99, 85% finished]
Exercises
[4 oct 99]
Chapter 17 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.
[4 oct 99, 80% finished]
Chapter 18
Semiclassical chaotic scattering
Scattering and such.
[16 sep 98, 20% finished]
Exercises
[16 sep 98]
Chapter 19 Helium atom
The helium atom spectrum computed via spectral determinants.
[new version 13 Apr 97, 85% finished]
Exercises
[3 sep 99]
Extras
[17 July 98]
Chapter 20 Creeping
Creeping effects of getting behind cyclinders and such
are incorporated into periodic orbit theory. [6
Sep 98, 5% finished]
Chapter 21 Diffraction distraction
Diffraction effects of scattering off wedges, eavesdropping around
corners are incorporated into periodic orbit theory.
[25 Sep 98, 95% finished]
Chapter 22 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, as much as this is possible
without proving theorems. [27 may 97, 15% finished]
Chapter 23 Symbolic dynamics,
generic
Theory of pruning fronts for generic flows. [13 Apr
97, 15% finished]
Extras
[13 Sept 97]
Chapter 24 Quantum and noise
trace formulas
In this chapter the quantum/noise perturbative corrections formulas
are rederived using an updated version of Bohr's old quantum mechanics.
[summer 96, 50% finished]
Chapter 25 Beyond periodic orbit
theory
Dormant chapter, will contain some material from ``Beyond
periodic orbit theory'' [Nonlinearity (1998)]
Chapter 26 Irrationally winding
Circle maps and their thermodynamics are analyzed in detail.
[Dec 96, 85% finished]
Chapter 27 Border of order
In this chapter we face up to the cycles of marginal stability and the
anomalous diffusion. [Dec 96, 45% finished]
Chapter 28 Statistical mechanics
How to recycle Ising-like spin systems. Includes sections on
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]
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 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 Billiards
Dormant [30 Mar 97]
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
Statistical mechanics applications
Diffusive properties of a prototype example of chaotic
Hamiltonian maps, hyperbolic toral automorphisms. [8 Feb 98]
Appendix G
Infinite dimensional operators
What is the meaning of traces and determinants for infinite-dimensional
operators? [9 Feb 96, 95% finished]
Appendix I Quasiclassics
Dormant [22 May 96]
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]
[
Instructions |
Extras |
Projects |
Courses |
Theses |
Why this book?
]
Periodic
orbit theory sites
Predrag
Cvitanovic', predrag@nbi.dk