Chaotic diffusion in randomly perturbed dynamical systems Rainer Klages Queen Mary University of London, School of Mathematical Sciences Max Planck Institute for Mathematics in the Sciences Leipzig, 30 November 00 Chaotic diffusion and random perturbations Rainer Klages (QMUL)
Outline Motivation: random walk, deterministic diffusion and fractal diffusion coefficients Deterministic diffusion and random perturbations: four different types of perturbations; computer simulations in comparison to simple analytical approximations 3 Theoretical details: derivation and discussion of the approximate formulas Chaotic diffusion and random perturbations Rainer Klages (QMUL)
The drunken sailor at a lamppost random walk in one dimension (K. Pearson, 905): steps of length s with probability p(±s) = / to the left/right 5 0 5 0 time steps position single steps uncorrelated: Markov process define diffusion coefficient as D := lim n n < (x n x 0 ) > with discrete time step n N and average over the initial density <... > := dx (x)... of positions x = x 0, x R for sailor: D = s / Chaotic diffusion and random perturbations Rainer Klages (QMUL) 3
A deterministic random walk study diffusion on the basis of dynamical systems theory piecewise linear deterministic map { ax, 0 < x M a (x) = ax + a, < x lifted onto the real line by M a (x + ) = M a (x) + with control parameter a equation of motion: x n+ = M a (x n ) Lyapunov exponent λ = ln a > 0: map is chaotic y=m a(x) 3 a 0 6 5 4 3 n 0 3 x Grossmann/Geisel/Kapral (98) Chaotic diffusion and random perturbations Rainer Klages (QMUL) 4
Fractal diffusion coefficient and random walks D(a) exists and is a fractal function of the control parameter.5 D(a) 0.3 0. 0. random walk approximation for a<3 D(a) 0..4.6.8 3 a a 0.5 random walk approximation for a>3 0 3 4 5 6 7 8 a exact results (R.K., Dorfman, 995; Groeneveld, R.K., 00) proof: D(a) is Lipschitz continuous up to quadratic logarithmic corrections (Keller, Howard, R.K., 008) cp. random walk with exact results (R.K., Dorfman, 997) Chaotic diffusion and random perturbations Rainer Klages (QMUL) 5
Deterministic diffusion and random perturbations What happens to D(a) by imposing random perturbations onto the map? Four basic types: M a (x) = (a + a(i, n)) x + b(i, n) with random shifts and random slopes in discrete space i Z and/or discrete time n N b(i,n) a(i,n) () b(i) : quenched shifts () a(i) : quenched slopes (3) a(n) : noisy slopes (4) b(n) : noisy shifts Chaotic diffusion and random perturbations Rainer Klages (QMUL) 6
Deterministic diffusion and quenched disorder (): quenched random shifts b(i); Radons (996ff): Golosov localization/sinai diffusion: no normal diffusion (): quenched random slopes a(i) random walk in disordered lattices like random barrier model transition rates Γ i,i+ = Γ i+,i Γ i ; exact diffusion coefficient d =< /Γ > Γ s Alexander et al (98), Derrida (983),... disorder average < /Γ > Γ = /N N i=0 /Γ i, distance of sites s Haus, Kehr (987) Chaotic diffusion and random perturbations Rainer Klages (QMUL) 7
Quenched slopes: theory apply formula to deterministic diffusion with quenched slopes: rewrite d =< /Γ > Γ s =< /d(s,γ) > Γ with d(s,γ) = Γs but for the map M a (x) we have d(s,γ) = d(s,γ(s)) approximation: identify d(s, Γ(s)) with D(a + a), where D( ) is the unperturbed deterministic diffusion coefficient D(a) with random variable a sampled from pdf χ da ( a) at perturbation strength da 0, da a da [ ] da χ D app (a, da) = d( a) da ( a) da D(a + a) Chaotic diffusion and random perturbations Rainer Klages (QMUL) 8
Quenched slopes: simulations D da (a) a(i) uniformly distributed on [ da, da]: continuous perturbations, da=0.,0.4,.0.5 0.3 0. 0. D da (a) D da (a) 0 3 4 a 0.5 0 3 4 5 6 7 8 a oscillatory structure persists under small perturbations dynamical phase transition for small parameters Chaotic diffusion and random perturbations Rainer Klages (QMUL) 9
Quenched slopes: simulations D a (da) a(i) uniformly and δ-distributed on [ da, da]: continuous perturbations, a=5.9997 discrete perturbations, a=5.9997 0.9 0.9 D a (da) 0.88 D a (da) 0.95 0.86 0.9 0.84 0.85 0 0. 0. 0.3 0.4 0.5 da 0 0. 0. 0.3 0.4 0.5 da.5 discrete perturbations, a=5.9997 discrete perturbations, a=7.0003.5 D a (da) D a (da) 0.5 0.5 0 0 0 3 4 0 3 4 5 da da excellent match theory and simulations for small da multiple suppression and enhancement with da Chaotic diffusion and random perturbations Rainer Klages (QMUL) 0
Deterministic diffusion and noise approximate the diffusion coefficient by the unperturbed D(a) (cp. to Geisel et al. (98), Reimann (994ff)): basic idea for slopes with iid dichotomous noise ±da: D(a+da) D(a) 0.5 D(a-da) D(a) da 0 3 4 5 6 a D app (a, da) = (D(a da) + D(a + da))/ for a general disorder distribution χ da ( a) D app (a, da) = da da d( a) χ da ( a)d(a + a) cp. to quenched diffusion formula expanded for da 0 Chaotic diffusion and random perturbations Rainer Klages (QMUL)
Noisy slopes: simulations D da (a) a(n) δ-distributed with ±da: discrete perturbations, da=0.,0.4,.0.5 D da (a) 0.5 0 3 4 5 6 7 8 a shift of fractal structure under perturbations Chaotic diffusion and random perturbations Rainer Klages (QMUL)
Noisy slopes: simulations D a (da) a(n) and b(n) δ- and uniformly distributed on [ da, da]: discrete perturbations, a=5.9997 discrete perturbations, a=7.0003 4.4 0.9.9 D a (db) 0.9 0.88. 0.86 0.84 0 0.0 0.04 0.06 0.08 0. db D a (db) D a (da) 3.5 3.5 0 0. 0. 0.3 0.4 0.5 da.8.7.6.5 D a (da).5 0 0. 0. 0.3 0.4 0.5 db 0 3 4 5 6 da. continuous perturbations, a=5.9997 continuous perturbations, a=7.0003 D a (db).05 0.95 0.9 D a (db) 0.88 0.86 0.84 D a (da).9.8.7.6 D a (da).9.8.7.6 0 0. 0. 0.3 0.4 0.5 da 0 0.05 0.05 0.075 0. 0.85.5 db 0 0.5.5.5 3 0 3 4 5 6 db da transitions from deterministic to stochastic diffusion by suppression and enhancement Chaotic diffusion and random perturbations Rainer Klages (QMUL) 3
Theoretical aspects: precise definition of D(a,da) Let n be iid random variables with pdf χ d ( n ) of perturbation strength d 0. Let (x) be the initial pdf of points x. Then with ) D(a, d) = lim (< x n n n >,χ d < x n >,χd < x k n >,χd = dx d( 0 )d( )... d( n ) (x)χ d ( 0 )χ d ( )... χ d ( n )x k n special case: noisy slopes as an example (wlog), d = da, n = a n < x n >,χd = 0 Chaotic diffusion and random perturbations Rainer Klages (QMUL) 4
Deriving approximations in terms of the exact D(a) Let a n be uniformly distributed in [ da, da] and a = a 0. It holds: a n = a + ǫ, da ǫ da notation: x n,a+ an = M a+ an (x n ) step : da ǫ a n a < xn >,χda = dx d( a)d( a )... d( a n ) = ρ 0 (x)χ da ( a)χ da ( a )... χ da ( a n )x n,a+ a n dx d( a) (x)χ da ( a)xn,a+ a step : exchange time limit with integration D app (a, da) = lim < x n n >,χda /(n) = d( a)χ da ( a) lim dx (x)x n n,a+ a /(n) = d( a)χ da ( a)d(a + a, 0) Chaotic diffusion and random perturbations Rainer Klages (QMUL) 5
Validity of this approximation note: This approximation needs to be handled with much care! does not work for quenched shifts works for noisy shifts, but only in the limit of very small perturbations, because current; cp. to numerical results works well for noisy slopes in the limit of small perturbations Chaotic diffusion and random perturbations Rainer Klages (QMUL) 6
Random walk approximation unperturbed diffusion coefficient D(a) = lim n n 0 dx a (x)(x n x) with invariant density a(x) of map M a (x) mod random walk approximation:. no memory in jumps: replace x n = x n x by single jump at time n = over distance x = x. no memory in invariant density: assume ρ a (x) D rw (a) = 0 dx x Chaotic diffusion and random perturbations Rainer Klages (QMUL) 7
Deriving random walk approximations D rw (a) = dx x 0 consider two limiting cases of jumps: random walk I: a 3; set x = 0 if 0 M a (x) and x = otherwise, D rwi (a) = / 0, x= dx = (a )/(a) (a )/4 (a ) random walk II: a 3; set x = M a (x) x exactly, D rwii (a) = / 0 dx(m a(x) x) = (a ) /4 feed back results into perturbed random walk approximation D app (a, da) = d( a)χ da ( a)d(a + a) by replacing D(a + a) D rw (a + a) Chaotic diffusion and random perturbations Rainer Klages (QMUL) 8
Perturbed random walk approximations example: Let random slopes a be δ-distributed, 0.5 0. random walk approximation I linear limit a --> D(a) D(a+da) 0.5 D(a) 0. D(a-da) 0.05 0.5.5.75 3 for a 3 random walk I yields suppression of diffusion due to concavity (Reimann, 994ff) for a random walk I yields no change at all in the diffusion coefficient due to linearity for 3 a random walk II yields enhancement of diffusion due to convexity, D rwii (a, da) = D rwii (a) + a /4 in simulations random walk II well seen but not I Chaotic diffusion and random perturbations Rainer Klages (QMUL) 9 a
Summary simple model with diffusion due to deterministic chaos under random perturbations:. in three of four cases of random perturbations the diffusion coefficient exists. the unperturbed fractal diffusion coefficient is quite stable against perturbations 3. simple approximations for the perturbed diffusion coefficient in terms of the unperturbed diffusion coefficient 4. multiple (irregular) suppression and enhancement of deterministic diffusion by stochastic perturbations 5. transitions from deterministic to stochastic diffusion via suppresion and enhancement Chaotic diffusion and random perturbations Rainer Klages (QMUL) 0
Literature spatial disorder: R.K., PRE 65, 05503(R) (00) noise: R.K., EPL 57, 796 (00) all together: see Part ; file of this talk, details and taster sections on www.maths.qmul.ac.uk/ klages note: conference on Weak Chaos, Infinite Ergodic Theory, and Anomalous Dynamics at MPIPKS Dresden in 0! Chaotic diffusion and random perturbations Rainer Klages (QMUL)