30 jan 2000,
version 6.0.1
[ Help
| Extras | Projects
| Courses | Theses
| Why webbook? ]
individual chapters in gzipped PostScript:
click
to download 2 printed pages per 1 sheet of paper
click
to download 1 printed page per 1 sheet of paper
CLASSICAL AND QUANTUM CHAOS
-
Contents
-
Index
-
Chapter 1 Overture
-
An overview of the main themes of the book. Recommended reading before
you decide to download anything else.
[23 jan 2000, 90% finished]
Appendix
A: you might also want to read about the history of the subject.
Exercises
[23 jan 2000]
Chapter 2 Dynamics
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.
[23 jan 2000, 70% finished]
Exercises
[23 jan 2000]
Chapter 3 Local stability
Review of basic concepts of local dynamics:
local linear
stability and stable/unstable manifolds for flows, billiards and maps.
[23 jan 2000, 45% finished]
Appendix B Stability of Hamiltonian flows: more details, especially for the Helium.
Exercises
[23 jan 2000]
Extras [13 Sept 97]
Chapter 4 Transporting densities
A first attempt to move the
whole phase space around - we introduce fancy operators,
discuss natural measure.
[23 jan 2000, 60% finished]
Exercises
[23 jan 2000]
Chapter 5 Averaging




The necessity of studying
the averages of observables in chaotic dynamics; the formulas
for averages are cast in a multiplicative form that
motivates the introduction of evolution operators.
[23 jan 2000, 90% finished]
Exercises
[23 jan 2000]
Chapter 6 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.
[23 jan 2000, 85% finished]
Exercises
[23 jan 2000]
Chapter 7 Qualitative dynamics
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.
[8 aug 99, 60% finished]
Exercises
[16 mar 98]
Chapter 9 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 8 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, 80% finished]
Exercises
[16 mar 98]
Extras [13
Sept 97]
Chapter 10 Spectral determinants
We derive the spectral determinants, dynamical zeta functions.
[23 jan 2000, 85% finished]
Chapter 11 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 periodic orbits.
The chapter explains in detail how to carry them out.
[30 Aug 98, 90% finished]
Exercises
[23 jan 2000]
Extras [17
July 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 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 14 Thermodynamic formalism
Generalized dimensions, entropies and such.
[16 mar 98, 20% finished]
Exercises
[16 mar 98]
Chapter 15 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 16 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 17 Semiclassical evolution
Semiclassical propagator and Green's function.
[04 oct 99, 85% finished]
Exercises
[4 oct 99]
Chapter 18 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 19 Semiclassical chaotic scattering
Scattering off N disks, exact and semiclassical.
[16 sep 98, 20% finished]
Exercises
[16 sep 98]
Chapter 20 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 21 Diffraction distraction
Diffraction effects of scattering off wedges, eavesdropping around corners
incorporated into periodic orbit theory.
[25 Sep 98, 95% finished]
Chapter 22 Irrationally winding
Circle maps and their thermodynamics analyzed in detail. [Dec
96, 85% finished]
Chapter 23
Border of order
We face up to the cycles of marginal stability and the
anomalous diffusion. [Dec 96, 45% finished]
Chapter 24 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]
Summary
Take-home problem set for the next millenium
[6 Sept.
96, 10% finished]
Appendices
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 Helium stability
Helium stability worked out in detail.
[10 Jan 99 80%
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 Transporting densities
Transporting densities worked out in detail.
[10 Jan 99
80% finished]
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
Infinite dimensional operators
What is the meaning of traces and determinants for infinite-dimensional
operators? [9 Feb 96, 95% finished]
Appendix F
Difraction distraction
What is the meaning of difraction?
[9 Feb 96, 95% finished]
Appendix G
Solutions
Solutions to selected problems - often more instructive than the
text itself. Recommended.
[9 Feb 96, 95% finished]
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]
[ Help
| Extras | Projects
| Courses |
Theses
| Why webbook? ]
Predrag Cvitanovic',
predrag@nbi.dk