Week 12
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We had promised to teach you something profound that your professor does not know. Well, this week is the week. If your professor knows cycle formulas for dynamical averages, please send us her name, and we'll feature it in ChaosBook. They look like cumulants, but when you start to evaluate them you realize that you would not guess their form in 1007 Phys. Rev. Letters. Takes 12 weeks of hard study to start to understand them. |
- 23. Cycle expansions
- 24. Deterministic diffusion
- Homework 12
- Optional
Spectral eigenvalues and dynamical averages are computed by expanding spectral determinants into cycle expansions, expansions ordered by the topological lengths of periodic orbits. Skip sects. 23.3.1 "Newton algorithm for determining the evolution operator eigenvalues", and 23.6 "Stability ordering of cycle expansions".
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Chapter 23 Cycle expansions |
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Cycle expansions - slides |
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Pseudo-cycles |
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Evaluating a cycle expansion |
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Probability conservation: An exact sum rule for cycles |
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Leading eigenvalue of evolution operator yields dynamical averages |
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Cycle formulas for dynamical averages |
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Discrete vs continuous time averages |
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Cycle formulas for finite grammar |
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In real life, try stability ordering |
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Summary, and what's next |
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Talking Head video - Shadowing In this simulation Xiong Ding illustrates the basic principle of periodic orbit theory: the shadowing of a long periodic orbit by a pair of short ones: The theory works because long periodic orbits are shadowed by shorter ones. In this way the short periodic orbits build up accurate estimates of the infinite time behavior of a chaotic flow. |
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Chapter 20 Averaging Read sect. 20.1, including sect. 20.1.3 "Moments, cumulants" |
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Averages - a review |
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Cumulants |
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Expectation values |
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Expectation values - from eigenvalue condition |
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Cycle formula for expectation value |
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We derive exact formulas for diffusion constants transport coefficients. All from first principles, without invoking any Boltzmann-Gibbs probabilistic notions. Read sect. 28.1: Foundations of statistical mechanics illuminated by 2-dimensional Lorentz gas.
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cycle expansion Due 19 April 2022 |
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Discussion forum for week 12 |
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| How to best (and practically) evaluate cycle averaging formulas [Feb 2022 chat] |
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Yasha Sinai gets Abel prize Sinai gets Abel Prize. Unexpectedly, Cvitanoviç nabs the coveted Night Fishing Champion trophy, to chagrin of the much more deserving Howey 3rd floor pros. |
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| Chaos-theory pioneer Yakov Sinai nabs Abel Prize |
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Sinai billiard [animation by Oliver Knill, 26 March 2014] |
We are grateful Doug Eardley, KITP, Santa Barbara CA, for recording the lecture 23 videos.