Chaotic Dynamics and Transport in Classical and Quantum by Pierre Collet, M. Courbage, S. Métens, A. Neishtadt, G.

By Pierre Collet, M. Courbage, S. Métens, A. Neishtadt, G. Zaslavsky

This publication deals a contemporary up to date overview at the most vital actions in this present day dynamical platforms and statistical mechanics through the very best specialists within the area. It provides a modern and pedagogical view on theories of classical and quantum chaos and complexity in hamiltonian and ergodic platforms and their functions to anomalous shipping in fluids, plasmas, oceans and atom-optic units and to manage of chaotic shipping. The ebook is issued from lecture notes of the overseas summer season tuition on "Chaotic Dynamics and shipping in Classical and Quantum platforms" held in Cargèse (Corsica) 18th to the thirtieth August 2003. It displays the spirit of the college to supply lectures on the post-doctoral point on simple techniques and instruments. the 1st half issues ergodicity and combining, complexity and entropy services, SRB measures, fractal dimensions and bifurcations in hamiltonian structures. Then, types of dynamical evolutions of shipping strategies in classical and quantum platforms were mostly defined. the second one half matters delivery in fluids, plasmas and reacting media. nonetheless, new experiments of chilly optically trapped atoms and electrodynamics hollow space were completely awarded. ultimately, a number of papers endure on synchronism and regulate of chaos.

The target market of the court cases are physicists, mathematicians and all scientists keen on Chaos and Dynamical platforms concept and their basic purposes in Physics and within the technology of complicated and Nonlinear phenomena.

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Extra info for Chaotic Dynamics and Transport in Classical and Quantum Systems: Proceedings of the NATO Advanced Study Institute on International Summer School on Chaotic ... II: Mathematics, Physics and Chemistry)

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18) k=0 This result holds also for the absolute value. While the distance traveled by the particle along the x-direction is alway xn = nα, the distance traveled by the particle along y-direction depend on the initial condition (x, ε) and seems as a random walk. 19) 24 The mean value of Sn over the initial conditions is zero and the problem is to estimate the asymptotic behavior of the variance of Sn . We shall come back to this question in the next section. Second example: G = S 1 A particle moves on an infinite plane among periodically distributed obstacles with spatial period equal to 1 along both (q1 , q2 )-directions.

Space with a distance d. 1 1 Given > 0, a set Y ⊆ X is -separated iff for any different x, y ∈ Y one has d(x, y) ≥ . 2 The number C (X, d) = C := max{|Y |, Y is an -separated set}, where | · | denotes the cardinality of a set, is called the -complexity of X. 3 An -separated set Y is optimal iff |Y | = C . Let us show the following natural inequality. 1 Given D1 , D2 ⊆ X and ε > 0 one has Cε (D1 ∪ D2 ) ≤ Cε (D1 ) + Cε (D2 ). 1 Let Y ⊆ D1 ∪ D2 be an optimal ε-separated set in D1 ∪ D2 . Then Yi = Y ∩ Di is an ε-separated set in Di and |Y | ≤ |Y1 | + |Y2 | ≤ Cε (D1 ) + Cε (D2 ).

1) Notes on Spectral Theory, Mixing and Transport. 17 In order to stress on the dependence of U with respect to T, we should use the notation UT , but we shall omit it and use it only when necessary. The spectral properties of this operator are called the spectral properties of the DS. The first important spectral property of a DS is a characterization of its ergodicity. e. T A = A, eventually modulo a subset of zero measure) has either zero measure or full measure. 2 (X, T, µ) is ergodic if and only if the only eigenfunction of U for the eigenvalue 1 is the constant function.

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