Download Transport in Chaotic Systems and Fingerprints of Pseudorandomness

Transport in Chaotic Systems
and Fingerprints of
Pseudorandomness
Shmuel Fishman
Como, March 2009
Fingerprints of randomness
Classical  diffusion (A. Einstein, 1905)
Quantum  suppression of classical diffusion by
quantum interference; Anderson Localization
(P.W. Anderson, 1958)
P.W. Anderson – Electrons,
Classical Optics .
Demonstration for transparencies – M.V. Berry & S. Klein
Localization for pseudorandom Systems
1 2
H =  pˆ  k cos    (t  m)
2
m
Kicked Rotor
p 
2
Anderson localization
P( p)
classical
Classical diffusion
t
p
quantum
t
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
t
Anderson localization
What is Pseudorandomness?
Sensitive dependence on initial conditions.
For coin toss, dice or roulette,
the attractor depends on initial
conditions with high sensitivity.
How Does it Work?
Example:
 4
Logistic Map
xn1  xn (1  xn )
xn1  4 xn (1  xn )
parameterization
1
xn  1  cos 2n 
2
1
1  cos 2n1   1  cos 2n 1  cos 2n   1  cos 2 2n
2
cos 2n 1  1  2 cos 2 2n  cos 2  2n
 n  2 n  n mod 1
 n1  2 n mod 1
since xn depends on
fractional part of θn
Take 0 in a binary representa tion 0  0110101...
0 - irrational  not periodic
Iterations scan the digits
of an irrational number
θn  2 n θ0 mod 1
Ideal pseudorandomness

exponential
growth

folding
Generation of pseudorandom numbers (D. Knuth).
X n1  (aX n  c) mod m
 n
an 1 

X n   a X 0  c
a 1 

exponential growth
xn 
Xn
m
all integers
mod m
folding
reduces to [0,1]
For kicked rotor sequence
n 
2
mod 1
slow scanning of the digits of .
Is such a process sufficiently random
for Anderson localization?
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. Effects of dimensionality
and the metal insulator transition. Direct observation of Anderson Localization in
Optics and in Atom-Optics.
•
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. “High dimensional “ kicked rotors.
•
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.
•
Lecture 6: Discussion of future directions and problems.