Chaos: Classical and Quantum
This work, culminating in the book Chaos: Classical and Quantum, represents a comprehensive graduate-level exploration of deterministic chaos. It bridges the gap between the study of simple, integrable systems (like the harmonic oscillator) and the complex, turbulent reality of nature.
The central thesis is that chaotic dynamics is generated by the interplay of locally unstable motions and the interweaving of their global stable and unstable manifolds. This topology is robust, creating a rigid skeleton upon which the chaos hangs.
The Pinball Paradigm
We use the physicist's game of pinball—three equidistantly placed reflecting disks—to motivate the theory. It is a system that is locally unstable (positive Lyapunov exponent) yet globally mixing.
Hidden in the apparent chaos is a self-similar tree of cycles (periodic orbits). Our work demonstrates that one can compute exact averages of observables—such as escape rates and transport coefficients—by expanding them in terms of these fundamental cycles.
Quantum Chaos
Perhaps the most profound insight is the connection between classical and quantum mechanics in the chaotic regime. We show that the semi-classical quantum mechanics of classically chaotic systems is described by Zeta functions and cycle expansions of the exact same form as their classical counterparts.
The spectrum of a quantum system, like the Helium atom, can thus be understood through the topology of the classical periodic orbits that shadow the quantum waves.
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