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 highschool 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.
Read chapter 1 and appendix A of P. Cvitanovic, R. Artuso, R. Mainieri, G. Vattay et al., Classical and Quantum Chaos, http://www.nbi.dk/ChaosBook/.
Confronted with a potentially chaotic dynamical system, we analyze it through a sequence of three distinct stages; diagnose, count, measure. First, we determine the intrinsic dimension of the system - the minimum number of degrees of freedom necessary to capture its essential dynamics. If the system is very turbulent (its attractor is of high dimension) we are, at present, out of luck. We know only how to deal with the transitional regime between regular motions and weak turbulence. In this regime the chaotic dynamics is restricted to a space of low dimension, the number of relevant parameters is small, and we can proceed to the second step; we count and classify all possible topologically distinct trajectories of the system. If successful, we can proceed with the third step: investigate the weights of the different pieces of the system.
In this lecture qualitative dynamics of simple stretching and mixing flows is used to introduce Smale horseshoes and symbolic dynamics, and the topological dynamics is encoded by means of transition matrices/Markov graphs.
We learn how to count and describe itineraries. While computing the topological entropy from transition matrices/Markov graphs, we encounter our first zeta function.
By now we have covered for the first time the whole distance from diagnosing chaotic dynamics 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.
Read chapters 2, 3, 10 and 11 of P. Cvitanovic, R. Artuso, R. Mainieri, G. Vattay et al., Classical and Quantum Chaos, http://www.nbi.dk/ChaosBook/.
This lecture is the core of the course: we discuss the necessity of studying the averages of observables in chaotic dynamics, and cast the formulas for averages in a multiplicative form that motivates the introduction of evolution operators.
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.
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 nothing but special cases, to be worked out at your leisure.
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.
Read chapters 5, 6, 7, 8 and 9 of P. Cvitanovic, R. Artuso, R. Mainieri, G. Vattay et al., Classical and Quantum Chaos, http://www.nbi.dk/ChaosBook/.
In last lecture we have derived a plethora of periodic orbit trace formulas, spectral determinants and zeta functions. Now we learn how to expand these as cycle expansions, series ordered by increasing topological cycle length, and evaluate average quantities 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.
The course now shifts gear to recent advances in the periodic orbit theory of chaotic, non-integrable systems, and the modern generalization of the De Broglie - Bohr quantization of hydrogen atom.
Instead of quantizing by suspending standing-wave configurations on stable Keplerian orbits, one suspends the standing-wave configurations on the infinity of unstable orbits. Such unstable periodic orbits are observed experimentally in the helium atom, the hydrogen in strong external fields, and other systems.
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.
The helium atom spectrum can then be computed via spectral determinants.
Read chapters 13, 21 and 22 of P. Cvitanovic, R. Artuso, R. Mainieri, G. Vattay et al., Classical and Quantum Chaos, http://www.nbi.dk/ChaosBook/.
Intuitively, the noise inherent in any realistic system washes out
fine details and makes chaotic averages more robust. Quantum mechanical
resolution of phase space implies that in semi-classical approaches
no orbits longer than the Heisenberg time need be taken into account.
We explore these ideas in some detail by casting stochastic dynamics
into path integral form and developing perturbative and nonperturbative
methods for evaluating such integrals. In the weak noise case the
standard perturbation theory is expansion in terms of Feynman diagrams.
Now the surprise; we can compute the same corrections faster and to a
higher order in perturbation theory by integrating over the neighborhood
of a given saddlepoint exactly by means of a nonlinear change of
field variables. The new perturbative expansion appears more compact
than the standard Feynman diagram perturbation theory; whether it is
better than traditional loop expansions for computing field-theoretic
saddlepoint expansions remains to be seen, but for a simple system we study
the result is a stochastic analog of the Gutzwiller trace formula
with the corrections so far computed to five orders higher than
what has been attainable in the quantum-mechanical applications.
A motion on a strange attractor can be approximated by shadowing the orbit by a sequence of nearby periodic orbits of finite length. This notion is here made precise by approximating orbits by primitive cycles, and evaluating associated curvatures. A curvature measures the deviation of a longer cycle from its approximation by shorter cycles; the smoothness of the dynamical system implies exponential (or faster) fall-off for (almost) all curvatures. The technical prerequisite for implementing this shadowing is a good understanding of the symbolic dynamics of the classical dynamical system. The resulting cycle expansions offer an efficient method for evaluating classical and quantum periodic orbit sums; accurate estimates can be obtained by using as input the lengths and eigenvalues of a few prime cycles.
To keep exposition simple we have here illustrated the utility of cycles and their curvatures by a pinball game. Glancing back, we see that the formalism is very general, and should work for any average over any chaotic set which satisfies two conditions: 1. the weight associated with the observable under consideration is multiplicative along the trajectory; 2. the set is organized in such a way that the nearby points in the symbolic dynamics have nearby weights.
Read articles in the "Chaotic Field Theory" section of
and the take-home problem set for the next millennium in P. Cvitanovic, R. Artuso, R. Mainieri, G. Vattay et al., Classical and Quantum Chaos, http://www.nbi.dk/ChaosBook/.