Oct 28 2001
STATISTICAL MECHANICS II:
CHAOS, AND WHAT TO DO ABOUT IT
PHYS 7123 | Fall semester 2001 |
For people following the course, check the e-mail list. Please subscribe to the course e-mail distribution even if you are only interested in a subset of the topics - send e-mail with text (and no header):
To: majordomo@cns.physics.gatech.edu
Subject: <EMPTY>
subscribe chaos_course
TEXT:
Classical and Quantum Chaos
webbook, available on
ChaosBook.org
All chapter and exercise
numbers refer to this book, unless stated otherwise.
PLACE AND TIMES:
Howey N209 , TR 11:05-12:25
TEACHING ASSISTANT: Yueheng Lan, Howey W503, gte158y@prism.gatech.edu, Phone: 404/384-9407
PROBLEM SETS:
Please deliver solutions to problem sets by Thursday, at the
lecture, or place them in Predrag's mailbox.
Lecture 1
10:05-10:55 Mon Aug 20 2001 in Howey S104
Lecture 2
11:05-11:55 Tue Aug 21 2001 in Howey S104
Lecture 3
11:05-11:55 Thu Aug 23 2001 in Howey S104
Overture
That deterministic dynamics leads to chaos is no surprise to
anyone who has tried pool, billiards or snooker - that is what the game
is about - so we start our
course about what is chaos and what to do about it by a game of pinball.
This might seem a trifle trivial, but a pinball is to chaotic dynamics
what a pendulum is to integrable systems:
thinking clearly about what
is ``chaos'' in a pinball will help us tackle more difficult problems, such
as computing diffusion constants in deterministic gases, or
computing the Helium spectrum.
We all have an intuitive feeling for what a pinball does as it bounces
between the pinball machine disks, and only high school level
Euclidean geometry is needed to describe the trajectory.
Turning this intuition into calculation will lead us,
in clear physically motivated steps, to almost everything one
needs to know about deterministic chaos:
from unstable dynamical flows, Poincaré sections,
Smale horseshoes, symbolic dynamics, pruning,
discrete symmetries, periodic orbits,
averaging over chaotic sets, evolution operators,
dynamical zeta functions, Fredholm determinants,
cycle expansions, quantum trace formulas and zeta functions,
and to the semiclassical quantization of helium.
Reading:
Chapter 1:
An overview of the main themes of the course.
Recommended reading before you decide to download anything else.
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.
Exercises
e-mail description
Lecture 4
11:05-12:25 Tue Aug 28 2001 in Howey S104
Lecture 5
11:05-12:25 Thu Aug 30 2001 in Howey S104
Trajectories
We start out by a recapitulation of the basic notions of dynamics.
Our aim is narrow; keep the exposition focused on prerequsites to the
applications to be developed in this text. I assume that you
are familiar with the dynamics on the level of introductory texts
such as Strogatz, and concentrate here on developing
intuition about what a dynamical system can do.
It will be a coarse brush sketch -
the full description of all possible behaviors of dynamical systems is
anyway beyond human ken.
Reading:
Chapter 2:
Chapter 3:
Exercises
e-mail description
Lecture 6
11:05-12:25 Tue Sep 4 2001 in Howey S104
Lecture 7
11:05-12:25 Thu Sep 6 2001 in Howey S104
Local stability
We continue the discussion of local properties of flows and
maps: Henon map,
linear stability, types of
eigenvalues for linear maps, stable/unstable manifolds,
Bunimovich-Sinai formula for linear stability in billiards.
Reading:
Chapter 4:
Exercises
Chapter 11:
Print a hard copy of Sect 11.3.1 only, if you print anything.
e-mail description
Lecture 8
11:05-12:25 Tue Sep 11 2001 in Howey S104
Lecture 9
11:05-12:25 Thu Sep 13 2001 in Howey S209
Transporting densities
So far we learned how to track an individual
trajectory, and its small neighborhood.
While the trajectory of an individual representative point may be
highly convoluted, the density of these points might evolve in
a manner that is relatively smooth.
The evolution of the density of
representative points is for this reason (and other that will
emerge in due course) of great interest.
Reading:
Chapter 5:
Optional reading:
Appendix F:
Infinite dimensional operators
Exercises
e-mail description
Lecture 10
11:05-12:25 Tue Sep 18 2001 in Howey N209
Dynamical averaging
In chaotic dynamics detailed prediction is impossible, as any
finitely specified initial condition, no matter how precise, will fill
out the entire accessible phase space (similarly finitely grained) in finite
time. Hence for chaotic dynamics one does not attempt to follow individual
trajectories to asymptotic times; what is possible (and sensible) is description
of the geometry of the set of possible outcomes, and evaluation of the
asymptotic time averages. Examples of such averages are transport coefficients
for chaotic dynamical flows, such as the escape rate, mean drift and the
diffusion rate; power spectra; and a host of mathematical constructs such
as the generalized dimensions, Lyapunov exponents and the Kolmogorov entropy.
We shall now set up the formalism for evaluating such averages within the
framework of the periodic orbit theory. The key idea is to replace the
expectation values of observables by the expectation values of generating
functionals. This associates an evolution operator with a given observable,
and leads to formulas for its dynamical averages.
Reading:
Chapter 6:
Exercises
e-mail lecture
e-mail problem set
Lecture 11
11:05-12:55 Thu Sep 20 2001 in Howey N209
Trace formulas
If there is one idea that you should learn about dynamics, it
happens in this lecture(s) and it is this: there is a fundamental local
- global duality which says that (global) eigenstates are dual to the (local)
periodic geodesics. For dynamics on the circle, this is called Fourier
analysis; for dynamics on well-tiled manifolds this is called Selberg trace
formulas and zeta functions; and for generic nonlinear dynamical systems
the duality is embodied in trace formulas, zeta functions and spectral
determinants that we will now introduce. These objects are to dynamics
what partition functions are to statistical mechanics. The bold claim is
that once you understand this, classical ergodicity, wave mechanics and
stochastic mechanics are but special cases, to be worked out at your leasure.
The strategy is this: Global averages such as escape rates can be extracted from the eigenvalues of evolution operators. The eigenvalues are given by the zeros of appropriate determinants. One way to evaluate determinants is to expand them in terms of traces, log det = tr log. The traces are evaluated as integrals over Dirac delta functions, and in this way the spectra of evolution operators become related to periodic orbits.
The rest of the course is making sense out of this objects and learning
how to apply them to evaluation of physically measurable properties of
chaotic dynamical systems.
Reading:
Chapter 7: Trace formulas
Exercises
Lecture 12
11:05-12:25 Tue Sep 25 2001 in Howey N209
Lecture 13
11:05-12:55 Tue Sep 27 2001 in Howey N209
Cristel Chandre:
Fixed points, and how to get them
Periodic orbits can be determined analytically in only very exceptional
cases.
In order to proceed, we shall need data about unstable periodic
orbits, so good numerical methods for their detemination are a necessity.
We shall start by determining periodic orbits of a unimodal map, and then
proceed to Newton-Raphson method for maps and Poincare maps of flows.
Reading:
Chapter 12: Fixed points, and how to find them
Extras
Exercises
Cristel Chandre's lecture notes
e-mail description
Lecture 14
11:05-12:25 Tue Oct 2 2001 in Howey N209
Lecture 15
11:05-12:55 Thu Oct 4 2001 in Howey N209
Spectral determinants
We derive the spectral determinants, dynamical zeta functions.
Reading:
Chapter 8: Spectral determinants
Exercises
e-mail description
Lecture 16
11:05-12:25 Tue Oct 9 2001 in Howey N209
Why does it work? I
The heuristic manipulations that led to the trace formulas and
spectral determinants are potentially dangerous, as we are dealing with
infinite-dimensional vector spaces and singular integral kernels. Intuitively,
the theory should converge because long cycles are shadowed by nearby pseudo-cycles.
Actually, for clasess of not althogether too idealized smooth flows very
strong results exists.
Reading:
Chapter 9: Why does it work?
Exercises
Lecture 17
11:05-12:55 Thu Oct 11 2001 in Howey N209
Why does it work? II
For clasess of not althogether too idealized smooth flows very
strong results exists. We explain the ideas behind proofs of Ruelle and
Rugh which establish that for nice real analytic expanding or hyperbolic
flows the spectral (Fredholm) determinants are entire, and that at least
in that context the edifice constructed in this course has a mathematical
basis.
Lecture 18
11:05-12:55 Thu Oct 18 2001 in Howey N209
Dynamics, qualitative I
We start learning how to count: qualitative dynamics of simple
stretching and mixing flows is used to introduce symbolic dynamics and
Smale horseshoes.
Reading:
Chapter 10: Qualitative dynamics
Exercises
e-mail description
Lecture 19
11:05-12:25 Tue Oct 23 2001 in Howey N209
Dynamics, qualitative II
We continue learning how to count: qualitative dynamics of Smale
horseshoes is used to introduce pruning, finite subshifts, Markov Graphs
and transition matrices.
Exercises
e-mail description
Lecture 20
11:05-12:55 Thu Oct 25 2001 in Howey N209
Lecture 21
11:05-12:25 Tue Oct 30 2001 in Howey N209
Counting
We finish learning how to count: the traces of powers of the
transition matrix count admissible cycles, and the largest eigenvalue of
the transition matrix yields the topological entropy. The secular determinant
of the transition matrix - the Artin-Mazur zeta function - is expressed
in terms of the loops of a Markov diagram.
By now we have covered for the first time the whole distance from diagnosing chaotic dynamcs to computing zeta functions. Historically, These topological zeta functions were the inspiration for injecting statistical mechanics into computation of dynamical averages; Ruelle's zeta functions are a weighted generalization of the counting zeta functions.
Reading:Can one one go beyond equilibrium statistical mechanics and derive properties of systems far from equilibrium by the methods discussed in this course? This is currently a very lively research area, and we explain how the periodic orbit theory yields transport properties in models of dissipative driven systems, such as Guassian thermostatted Lorentz gas.
Term
papers
due
no later than 16:00 Tue Dec 11 2001 -
Predrag's office