Oct 21 2003
STATISTICAL MECHANICS II:
CHAOS, AND WHAT TO DO ABOUT IT
|PHYS 7123||Fall semester 2003|
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):
e-TEXTBOOK: Classical and Quantum Chaos webbook, available on www.nbi.dk/ChaosBook/
PLACE AND TIMES:
Howey S106, TR 09:35-10:55
Rytis Paskauskas, email@example.com, Phone: 404/384-9407
Please deliver solutions to problem sets by Thursday, at the
lecture, or place them in Predrag's mailbox.
9:35-10:55 Tue Aug 19 2003 in Howey S106
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.
problem set 1 (optional)
9:35-10:55 Thu Aug 21 2003 in Howey S106
Slaven Peleš: 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.
problem set 2
solutions to problem sets 1 and 2
9:35-10:55 Tue Aug 26 2003 in Howey S106
Slaven Peleš: 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.
problem set 3
solutions to problem 4.1
Lecture 4 9:35-10:55 Thu Aug 28 2003 in Howey S106
Lecture 5 9:35-10:55 Tue Sep 2 2003 in Howey S106
Billiards, Bunimovich-Sinai formula for linear stability in billiards.
Chapter 5, sects. 5.2, 5.3
Lecture 6 9:35-10:55 Thu Sep 4 2003 in Howey S106
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 7: all, except Sect 7.4.1 - Liouville opertor.
Appendix K: Infinite dimensional operators (for students with advanced exposure to quantum mechanics, and mathematically inclined, mostly)
problem set 4
solutions to problem set 4
9:35-10:55 Tue Sep 9 2003 in Howey S106
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.
problem set 5
9:35-10:55 Thu Sep 11 2003 in Howey S106
Luz V. Vela-Arevalo: Newtonian mechanics
We are going to spend some time in looking at the local behavior of flows that are invariant with respect to the symplectic structure, that is, flows of Hamiltonian systems. The ability to express mechanical systems in terms of Hamilton's equations provides us with an amazing framework to study general properties and symmetries of this type of flows. We will study the local stability conditions of equilibria and periodic orbits. We also review the theorems of local conjugacies that produce rectification of the flow. We talk about the relation between the Hamiltonian formalism of classical mechanics and the semiclassical approximation via the Hamilton-Jacobi equation.
Chapter 5: sect. 5.1 (updated, unstable version Chapter 5 contains extra material covered by Luz)
Appendix C: C.1 Symplectic invariance
Chapter 26: sect 26.1 - Hamilton-Jacobi theory (advanced: optional!)
problem set 6
Lecture 9 9:35-10:55 Tue Sep 16 2003 in Howey S106
Luz V. Vela-Arevalo: Classical helium atom
Chapter 29: sect 29.1 (excluding 29.1.3 and beyond - in updated, unstable version moved to Sect 6.3 of Chapter 6)
problem set 7
9:35-10:55 Thu Sep 18 2003 in Howey S106
Dynamics, qualitative I
We start learning how to count: qualitative dynamics of simple stretching and mixing flows is used to introduce symbolic dynamics.
Chapter 9: Qualitative dynamics for pedestrians
problem set 8
solutions to problem set 8
Lecture 11 9:35-10:55 Tue Sep 23 2003 in Howey S106
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.
Chapter 10: Qualitative dynamics for cyclists (advanced, optional! unstable version, hence chapter numbering conflicts with the the stable version)
Lecture 12 9:35-10:55 Thu Sep 25 2003 in Howey S106
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:
last day to drop course
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 11: Trace formulas
Lecture 14 9:35-10:55 Thu Oct 02 2003 in Howey S106
We derive the spectral determinants, dynamical zeta functions.
Chapter 12: Spectral determinants
problem set 10
Lecture 15 9:35-10:55 Tue Oct 07 2003 in Howey S106
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 13: Why does it work?
problem set 11
Thu Oct 09 2003:
Lecture 16 9:35-10:55 Thu Oct 09 2003 in Howey S106
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.
midterm recess, no lecture Tue Oct 14 2003
Lecture 17 9:35-10:55 Thu Oct 16 2003 in Howey S106
Lecture 18 9:35-10:55 Tue Oct 21 2003 in Howey S106
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 15: Cycle expansions
problem set 12
9:35-10:55 Thu Oct 23 2003 in Howey S106
Slaven Peleš: 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 14: Fixed points, and how to find them
Cristel Chandre's lecture notes
problem set 13
Rest of the schedule is
9:35-10:55 Tue Oct 28 2003 in Howey S106
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 16: Why cycle?
Wed Oct 29 2003: Register for Phys-7224 (Quantum Chaos) or PHYS-4421 (Continuum mechanics)
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 2003 - Predrag's office
Mon Dec 15 2003 in Howey S106
Fri Dec 26 2003: Register for Phys-7224 (Quantum Chaos) or PHYS-4421 (Continuum mechanics)
Predrag.Cvitanovic at physics.gatech.edu