Wave Chaos Theory

What is ‘Chaos’?

 

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Henry Poincare

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Heinrich Bruns

Chaos is a branch of mathematical physics which studies behavior of dynamical systems with a high sensitivity to initial conditions. The importance of understanding chaos dates back to the 19th century when serious efforts where made to predict trajectories of three celestial bodies which are subject to mutual gravitational interactions (link to Oscar II ‘Is the solar system stable’ challenge). In 1887 Heinrich Bruns and Henri Poincaré showed that this physical problem does not have a general analytical solution. They showed furthermore that the trajectories of planets follow chaotic trajectories in general, expect when trapped in long lived regular islands.

 

A pinball game is a good example of a chaotic system. Let us consider in two initial conditions of the pinball being slightly separated from each other. After a short time, the trajectories become completely different, even though the system is deterministic. So, if we are interested in calculating future positions of the pinball, we can do it only, if we have detailed information about the initial conditions of the ball which may be impossible due to the practical limitations. Therefore, these systems are considered to be chaotic systems. A popular other deterministic chaotic system is the weather.

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Pinball-sensitivity to initial conditions [1]

Mushroom billiard-chaotic

 

What is ‘Wave Chaos’?

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Martin Gutzwiller

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Michael Berry

‘Wave Chaos’, ‘Quantum Chaos’ or sometimes referred as ‘Quantum Chaology’ point to the same thing. This subject area has been part of Physics for over last 30 years. The Wave Chaos studies behavior of Quantum systems, whose correpsonding classical system is chaotic [2]. Basically, Wave Chaos tries to descibe the relation between the Quantum World and the Classical chaos.

 

 

 

 

 

References
[1] Predrag Cvitanovic, Roberto Artuso, Ronnie Mainieri, Gregor Tanner and Gabor Vattay, Chaos: Classical and Quantum

[2] Michael Berry, Quantum Chaology