Week 10
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Week 10 overview |
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Think locally, act globally: A lightning review |
- 19. Transporting densities
- 20. Averaging
- Homework 10
- Optional
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Chapter 19 Transporting densities Skip sects. 19.3 and 19.6. |
| A first attempt to move the whole phase space around - natural measure and fancy operators. | |
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What is a measure? |
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Perron-Frobenius operator |
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Dirac delta function - get used to it |
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Invariant measure |
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Ergodic measure |
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What do I measure? Observables |
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Evolution in infinitesimal time |
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Chapter 20 Averaging Read sects. 20.1 and 20.2. Skip sect. 20.1.3 "Moments, cumulants" |
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Averaging - slides Jemma Fendley |
| On the necessity of studying the averages of observables in chaotic dynamics. Formulas for averages are cast in a multiplicative form that motivates the introduction of evolution operators. Discrete-time and generating functions continued and compared. Infinitesimal time evolution related to the infinite time dynamics via a Laplace transform. The stage is set for the classical trace formula. | |
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Idea: Global-local duality |
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Idea: Perron-Frobenius operator |
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Time averages of observables |
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State space averages of observables |
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A simple idea: Exponentiate! |
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Evolution operator |
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Evolution operators form a semi-group |
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Spectra of evolution operators |
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Natural measure Due 3 April 2022 |
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Discussion forum for week 10 |