updated June 4, 1997
GEOMETRY OF CHAOS
Spring quarter 1997
Physics D60-0: NU Graduate Chaos Course
Lecture notes and problem sets: http://www.nbi.dk/ChaosBook/.
Problem with the problems is that numbers might differ from the lecture notes you got off the web - simplest to go through problems with underlined titles. I would like to receive solutions on Mondays.
Place and times:TTh 1:30 - 3:00 in Tech L386;
go up the main Tech building staircase to the third floor, turn left, go past the Applied Math to the end of the large cross hallway; the classroom is to your right across the hall.
Lecture 1 1:30
- 3:00 tuesday, April 1, 1997 in Tech L386
A brief historical introduction to the subject; Chapter 1 of the Cyclist Treatise is recommended reading prior to the lecture, in order to open up the discussion.
Lecture 2 1:30
- 3:00 thursday, April 3, 1997 in Tech L386
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 2, sects. 2.1-2.2,
Problems: at least 1.1, 1.4, 1.5
Lecture 3 1:30
- 3:00 tuesday, April 8, 1997 in Tech L386
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.
Reading: chapter 2, sects. 2.2-2.4,
Problems: at least 2.1, 2.2, 2.4, 2.6, 2.8
1:30 - 3:00 thursday, April 10, 1997 in Tech L386
Dynamics, qualitative III
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.
Lecture 5 1:30
- 3:00 tuesday, April 15, 1997 in Tech L386
Local dynamics, quantitative I
We review the basic concepts of local dynamics required to move beyond mere counting in order to assign relative weights to topologically distinct parts of the phase space. In this lecture we define flows and maps, and discuss local properties of flows: linear stability, types of eigenvalues for linear maps, stable/unstable manifolds, numerical methods.
Reading: Chapter 3
Lecture 6 1:30
- 3:00 thursday, April 17, 1997 in Tech L386
Local dynamics, quantitative II
We continue the discussion of local properties of flows and maps: Henon map, Bunimovich-Sinai formula for linear stability in billiards.
Reading: Chapter 3
1) Use your pinball program to numerically investigate the Poincare section; strips of surviving initial conditions (exer. 3.6), phase space density for closed system (exr. 3.7).
2) figure out how to compute stability of billiard trajectories, include this in your numerical billiard dynamics program.
3) implement numerical integration (perhaps the Runge-Kutta of exer. 3.8) for a continuos time dynamical system of your choice, plot some typical trajectories.
Lecture 7 1:30
- 3:00 tuesday, April 22, 1997 in Tech L386
Fixed points, and what to do about them
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. Existence of unit eigenvalues necessitates further cleverness.
Reading: Chapter 4
1) use your pinball program to numerically determine a few shortest cycles, presumably by implementing a minimization routine.
2) figure out how to compute stability of the shortest pinball cycles by hand. Compare with your numerical billiard stability program.
3) start implementing numerical Poincare sections for a continuos time dynamical system of your choice
4) start implementing a Newton-Raphson routine (only start - problems 3) and 4) might take a week or two to get in running shape).
1:30 - 3:00 thursday, April 24, 1997 in Tech L386
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, sects. 6.1-6.2
Problems: at least 6.1, 6.2
Lecture 9 1:30
- 3:00 tuesday, April 29, 1997 in Tech L386
Evolution operators and their spectra
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 6, sects. 6.3-6.5
Problems: at least 6.3, 6.6
Lecture 10 1:30
- 3:00 thursday, May 1, 1997 in Tech L386
Getting used to cycles
In last two lectures 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 rederive them in a rather different, more intuitive way, and extol their virtues. This is recommended bedtime reading, material that we will not cover in today's lecture. 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, count eigenvalues, and so on.
Reading: chapters 7 and 8
Problems: at least 8.1, 8.2, 8.3, 8.4
1:30 - 3:00 tuesday, May 6, 1997 in Tech L386
In last three lectures 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.
Reading: chapter 9, sects. 9.1-9.4,
Problems: at least 9.1,9.2, 9.5, 9.6
Lecture 12 1:30
- 3:00 thursday, May 8, 1997 in Tech L386
Applications of cycle expansions
Today we shall apply periodic orbit theory to evaluation of Lyapunov exponents, entropies and dimensions.
Reading: the (old) chapter 9 is not worth printing out yet - too messy, sorry
Problems: pause until I receive cycle expansion evaluations of escape rates for either a 1-d map or the 3-disk billiard (old problem sets)
Lecture 13 1:30
- 3:00 tuesday, May 13, 1997 in Tech L386
We continue discussing main applications of the periodic orbit theory, this time in the guise of ``thermodynamic formalism''.
Reading: chapter 10, (previously numbered 9) is still a mess, not
worth printing out on paper yet.
Problems: evaluate the Lyapunov exponent by cycle expansion for any system for which you have cycle data.
Lecture 14 1:30
- 3:00 thursday, May 15, 1997 in Tech L386
Quantum Mechanics I
We try to get from Schrödinger equation (January 1926) via Madelung (October 1926) and Hamilton (1821) to the semiclassical propagator. While the periodic orbit quantization trace formula cannot be honestly derived from this in only one lecture, we will try to give you the feeling for how it is derived in lecture 17.
Reading: chapter 11 (current homepage numbering), sects. 10.1-10.3 (as numerated in the current web version of the chapter)
Problems: at least 10.1, 10.2
Lecture 15 1:30
- 3:00 tuesday, May 20, 1997 in Tech L386
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.
Reading: current chapter 10, sects. 10.1, 10.2, 10.4,
Problems: at least 10.6, 10.9
Lecture 16 1:30
- 3:00 thursday, May 22, 1997 in Tech L386
Nonequilibrium statistical mechanics from deterministic chaos?
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.
Lecture 17 1:30
- 3:00 tuesday, May 27, 1997 in Tech L386
Quantum Mechanics II
We continue the semiclassical quantization of chaotic mechanics commenced in lecture 15. This seems to require an orgy of partial derivatives which we shall try to avoid - the lecture will be essentailly a guided tour to the Gutzwiller trace formula which is much simpler and more intuitive than the derivation that leads to it; it has the same structure as the classical trace formulas we are familiar with, except that now the cycle weights are complex. Time permiting we shall sketch out how this is used to quantize Helium.
Reading: browse chapters 11, 13
Problems: 2/3 of the registered students are way behind - we now concentrate on the term paper projects
Lecture 18 1:30
- 3:00 thursday, May 29, 1997 in Tech L386
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 15, sects. *.1, *.2 and *.3
Problems: we concentrate on the term paper projects
Lecture 19 1:30
- 3:00 tuesday, June 3, 1997 in Tech L386
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.
Reading: chapter 15, sects. *.1, *.2 and *.3
Problems: we concentrate on the term paper projects
Lecture 20 (the last of the course) 1:30
- 3:00 thursday, June 5, 1997 in Tech L386
What's 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. Other people have other visions of what ``quantum chaos'' might be, and 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.
4:30 PM friday, June 13, 1997 - Predrag's office
return to the Department of Physics & Astronomy.