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Oct 28 2001



PHYS 7123   Fall semester 2001 
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TEXT: Classical and Quantum Chaos webbook, available on
All chapter and exercise numbers refer to this book, unless stated otherwise.

Howey N209 , TR 11:05-12:25

TEACHING ASSISTANT: Yueheng Lan, Howey W503,, 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
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.

 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.
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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
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.

 Chapter 2:
 Chapter 3:
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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.
 Chapter 4:
 Chapter 11: Print a hard copy of Sect 11.3.1 only, if you print anything.
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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.

 Chapter 5:
Optional reading:
 Appendix F: Infinite dimensional operators
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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.

 Chapter 6:
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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.

  Chapter 7: Trace formulas

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.

  Chapter 12: Fixed points, and how to find them
  Cristel Chandre's lecture notes
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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.

  Chapter 8: Spectral determinants
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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.

  Chapter 9: Why does it work?

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.

  Chapter 10: Qualitative dynamics
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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.

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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
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.

  Chapter 11: Counting
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Lecture 22           11:05-12:55 Thu Nov 1 2001 in Howey N209
Cycle expansions
So far we have derived a plethora of periodic orbit trace formulas, spectral determinants and zeta functions. Now we learn how to expanded these as cycle expansions, series ordered by increasing topological cycle length, and evaluate average quantites like escape rates. These formulas are exact, and, when the winds are kind, highly convergent. The pleasant surprise is that the terms in such expansions fall off exponentially or even faster, so that a handful of shortest orbits suffices for rather accurate estimates of asymptotic averages.

  Chapter 13: Cycle expansions
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Lecture 23           11:05-12:25 Tue Nov 6 2001 in Howey N209
Getting used to cycles
So far we have moved at rather brisk pace and derived a spew of formulas. Just to make sure that the key message - the ``trace formulas'' and their ilk - have sunk in, in chapter ``Why cycle?'' we extol their virtues. Instead, we slow down to develop some fingertip feeling for the objects evaluted so far; show that this is just a fancy way to discretize integrals, evaluate the spectral determinant on a single unstable fixed point, on a piecewise linear map, and so on.

  Chapter 14: Why cycle?

Lecture 24           11:05-12:55 Thu Nov 8 2001 in Howey N209
Applications of cycle expansions  
Today we shall apply periodic orbit theory to evaluation of Lyapunov exponents, entropies and dimensions.

Lecture 25           11:05-12:25 Tue Nov 13 2001 in Howey N209
Thermodynamic formalism
We continue discussing main applications of the periodic orbit theory, this time in the guise of ``thermodynamic formalism''.

  Chapter 15: Thermodynamic formalism
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Lecture 26           11:05-12:55 Thu Nov 15 2001 in Howey N209
Lecture 27           11:05-12:25 Tue Nov 20 2001 in Howey N209
(Nonequilibrium) statistical mechanics from deterministic chaos?  
After the excursion into Lyapunov exponents, entropies, dimensions and thermodynamical formalism, we are back to the meat of the subject; can one derive statistical mechanics from deterministic chaos? We apply the periodic orbit theory to evaluation of measurable properties of a very simple model of ideal gas, periodic Lorentz gas, and derive closed-form formulas for the diffusion constant.

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.

  Chapter 18: Deterministic diffusion
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Lecture 28           11:05-12:25 Tue Nov 27 2001 in Howey N209
Boltzmann equation, H-theorem
We go through the classic - a heuristic derivation of the 1872 Boltzmann equation for density of a dilute gas. H - theorem states that even though the microscopics dynamics is reversible, the gas tends irreversibly toward the equilibrium. How can it be? hang on.
Reading: chapters 1-3, sections 3.1-3.2 of Dorfman's notes.

Lecture 29           11:05-12:55 Thu Nov 29 2001 in Howey N209
Lecture 30           11:05-12:25 Tue Dec 4 2001 in Howey N209
Walk-in clinic
The doctors from the Center for Nonlinear Science will diagnose the ailments plaguing your project, and prescribe proper medication.
Preparation: bring in the sickly projects, such as they are today.
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Lecture 31           11:05-12:55 Thu Dec 6 2001 in Howey N209 (the last of the course)
Truth in advertising: What went wrong with this course?

A moment of truth; the theory we have learned this quarter is beautiful, but it rests on romantic assumptions about dynamics that most sytems of interest falil to live up to. The problems that we would really like to tackle still mock us: they exhibit infinite symbolic dynamics, intermittency, ``phase transitions'', cycle expansions that might not converge. Higher and infinite dimensional dynamical systems still elude us. Which is the reason why this subject is so exciting - there is still so much to do and understand.

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Term papers         due no later than 16:00 Tue Dec 11 2001 - Predrag's office