1. Transport in Chaotic Systems and Fingerprints of Pseudorandomness Shmuel Fishman Paris, November 2007 Dedicated to the memory of Daniel Grempel

4. Fingerprints of randomness P.W. Anderson – Electrons, Classical Optics. Demonstration for transparencies – M.V. Berry & S. Klein Classical  diffusion (A. Einstein, 1905) Quantum  suppression of classical diffusion by quantum interference; Anderson Localization (P.W. Anderson, 1958)

5. classical quantum Anderson localization Classical diffusion Localization for pseudorandom Systems Kicked Rotor G. Casati, B.V. Chirikov, F.M. Izrailev & J. Ford B.V. Chirikov, F.M. Izrailev & D.L. Shepelyansky S. Fishman, D.R. Grempel & R.E. Prange

6. What is Pseudorandomness? Sensitive dependence on initial conditions. For coin toss, dice or roulette, the attractor depends on initial conditions with high sensitivity, but this sensitivity is limited (Vulovic and Prange). The attractors are not riddled as is the case in some model systems (Grebogi, Ott, Yorke).

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9. How Does it Work? parameterization Iterations scan the digits of an irrational number Example:

10. folding Generation of pseudorandom numbers (D. Knuth). all integers exponential growthfolding reduces to [0,1] Ideal pseudo- randomness exponential growth

11. For kicked rotor sequence slow scanning of the digits of  Is such a process sufficiently random for Anderson localization?

12. Outline Lecture 1: The kicked rotor model. Diffusion for chaotic systems and its meaning. Classical Accelerator Modes. Decay of correlations. Ruelle-Pollicott resonances and a possible analog for these in mixed systems. Lecture 2: Anderson localization in 1D: the Anderson model, exponential localization of eigenstates and absence of diffusion, models (Lloyd, white noise,Anderson), the relation between the density of states and the localization length. The scaling theory for localization. Point spectrum. Lecture 3: The mapping of the Kicked Rotor on the Anderson model and its implications. Quasienergies and quasienergy states. The relation between the localization length and the classical diffusion coefficient. The effect of accelerator modes on transport. Lecture 4: Pseudorandomness and localization: what are the properties of the pseudorandom sequence that are essential for localization? What is the relation of the quality of pseudorandom number generators and localization? Lecture 5: Relation to experiments: finite width (in time) of kicks, driving of neutral atoms, conservation of quasimomentum, Raizen’s experiments on kicked cold atoms and observation of Anderson localization in momentum, the effect of gravity in the corresponding Oxford experiments and the accelerator modes. The ε – classical mechanics and the Farey tree organization of resonances.