Roberto Artuso. Caos e diffusione (normale e anomala)

Transcription

1 Roberto Artuso Caos e diffusione (normale e anomala) Lucia Cavallasca, Giampaolo Cristadoro, Predrag Cvitanović, Per Dahlqvist, Gregor Tanner Damiano Belluzzo, Fausto Borgonovi, Giulio Casati, Shmuel Fishman, Italo Guarneri, Laura Rebuzzini, Michele Rusconi, Dima Shepelyansky Napoli, 29 maggio 2008

2 Boris Chirikov Ed Lorenz

3 microscopic dynamics / macroscopic transport

4 microscopic dynamics / macroscopic transport ergodicity, mixing, positive Lyapunov exponents

5 microscopic dynamics / macroscopic transport ergodicity, mixing, positive Lyapunov exponents diffusive motion, anomalous transport properties, heat conductivity

6 microscopic dynamics / macroscopic transport ergodicity, mixing, positive Lyapunov exponents? diffusive motion, anomalous transport properties, heat conductivity

7 5.0 diverging heat conductivity 4.5 ln " ln N Lepri, Livi, Politi

8 stochastic transport random walk (fixed increments) Levy flight (increments from a power law distribution)

9 Computer Vision and Image Understanding 101 (2006) Tracking soccer players aiming their kinematical motion analysis Pascual J. Figueroa a, Neucimar J. Leite a, *, Ricardo M.L. Barros b

10 Epidemics Brockmann, Hufnagel, Geisel

11 I might be moving to Montana soon.. Frank Zappa

12 tracers motion in chaotic fluids Weeks, Urbach, Solomon, Swinney random walk Levy flight

13 Clay Mathematics Institute Dedicated to increasing and disseminating mathematical knowledge HOME ABOUT CMI PROGRAMS NEWS & EVENTS AWARDS SCHOLARS PUBLICATIONS Navier-Stokes Equation Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations. Although these equations were written down in the 19th Century, our understanding of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations. The Mille Official P Charles F Lecture b good answer is worth 1million $

14 Vol May 2008 doi: /nature06948 heavy tailed distribution of scattering centers A Lévy flight for light Pierre Barthelemy 1, Jacopo Bertolotti 1 & Diederik S. Wiersma 1 Lévy transport Diffusive transport

15 Ratchet effect - Parrondo paradox

16 Ratchet effect - Parrondo paradox

17 Ratchet effect - Parrondo paradox Game A - simple coin winning probability 1/2-ε

18 Ratchet effect - Parrondo paradox Game A - simple coin winning probability 1/2-ε Game B - if the capital is a multiple of 3 winning probability 1/10-ε - if not winning probability 3/4-ε

19 Ratchet effect - Parrondo paradox Game A - simple coin winning probability 1/2-ε Game B - if the capital is a multiple of 3 winning probability 1/10-ε - if not winning probability 3/4-ε If ε>0 both games are unfair

20 the paradox!

21 the paradox!

22 the paradox!

23 the paradox!

24 examples of deterministic transport normal vs anomalous the full spectrum theoretical tools

25 purely deterministic motion

26 space-periodic systems Lorentz gas Sinai billiard

27 the simplest example: 1d maps

28 the simplest example: 1d maps x 0

29 the simplest example: 1d maps x 1 x 0

30 the simplest example: 1d maps x 2 x 1 x 0

31 the simplest example: 1d maps x 3 x 2 x 1 x 0

32 the simplest example: 1d maps x 3 x 2 x 4 x 1 x 0

33 the simplest example: 1d maps x 3 x 2 x 4 x 1 x 0

34 Microscopic instability: Lyapunov exponent Transport properties: diffusion constant

35 Microscopic instability: Lyapunov exponent Transport properties: diffusion constant δx(t) δx(0)e λt (x t x 0 ) 2 D t

36 Microscopic instability: Lyapunov exponent Transport properties: diffusion constant

37 Microscopic instability: Lyapunov exponent Transport properties: diffusion constant Klages, Dorfmann, Cristadoro

38 Radons quenched disorder

39 Radons quenched disorder

40 Radons quenched disorder Golosov localization

41 Sanders, Larralde

42 Sanders, Larralde

43 non-trivial transport without chaos anomalous transport, singular continuous spectrum RA, Guarneri, Rebuzzini

44 quantum interlude p n+1 = p n + K sin(x n ) x x+1 = x n L sin(p n+1 ) kicked Harper map superdiffusion RA, Casati, Shepelyansky

45 0. Standard Map, k = 0.2 Sticking to regular regions k=0.2 p n+1 = p n + k sin(x n ) x x+1 = x n + p n+1 StdMap J.D.Meiss

46 0. Standard Map, k = 0.2 x n+1 = x n + p n+1 p n+1 = p n k 2π sin(2πx n) Sticking to regular regions k=0.2 p n+1 = p n + k sin(x n ) x x+1 = x n + p n+1 StdMap J.D.Meiss

47 Lorentz gas with infinite horizon infinite free flights are possible ( x t x 0 ) 2 t ln t logarithmic divergence of D

48 Transport exponents Anomalous behavior in moments spectrum x t x x 0 q t ν(q) = t q β(q) t x 0 q t β(q) Normal, gaussian transport yields β(q)=q/2 Different parameter values for the standard map q Castiglione, Mazzino, Muratore, Vulpiani

49 The approach Transfer matrix - Perron Frobenius operator Employ periodic orbits (families of them)

50 Chaos: Classical and Quantum Part I: Deterministic Chaos Predrag Cvitanović, Roberto Artuso, Ronnie Mainieri, Gregor Tanner and Gábor Vattay formerly of CATS

51 Unbounded vs torus dynamics forse qualcosa di meglio

52 Unbounded vs torus dynamics forse qualcosa di meglio

53 Unbounded vs torus dynamics forse qualcosa di meglio

54 Unbounded vs torus dynamics forse qualcosa di meglio Correspondence is complete once we assign jumping numbers σ

55 Unbounded vs torus dynamics forse qualcosa di meglio Correspondence is complete once we assign jumping numbers σ

56 standing mode running mode, periodic orbit for billiard

57 Statistical mechanics approach... i = s i = N 1 1 N +1 1d Ising partition function Z N (β, H) = {s i } e βe I ({s i }) = T r T N leading eigenvalue yields Gibbs free energy spectral gap rules spatial correlation decay

58 dynamical systems transfer operator Ω dxϱ(x)(f T )(x) = Ω dx(lϱ)(x)f (x) evolution on observables (Koopman) evolution on probability densities (Perron-Frobenius) spectral analysis of L 1 is the leading eigenvalue (invariant measure) the spectral gap rules temporal correlation decay

59 transfer operators and transport 1 (Lϱ)(x) = y:t y=x T (y) ϱ(y) = Ω dz ϱ(z)δ(x T (z)) the spectral problem is in general highly non trivial: even the choice of a function space is delicate (this is not a mathematical detail: ugly observables generally decay at a slower rate). we introduce a generalized transfer operator accounting for transport properties

60 (L β h) (x) = Ω dz h(z) e β(t (z) z) δ(x T (z)) what s the use? G n (β) = e β(t n (x 0 ) x 0 ) 0 λ(β) n the generating function grows asymptotically as the leading eigenvalue moments are obtained by Taylor expansion of G

61 how to compute λ(β) smallest zeroes of generalized zeta functions product over the set of unstable periodic orbits of the dynamical systems

62 ζ 1 β (z) = {p} ( 1 z n p eβ σ p Λ p )

63 ζ 1 β (z) = {p} ( 1 z n p eβ σ p Λ p ) set of unstable periodic orbits

64 ζ 1 β (z) = {p} ( 1 z n p eβ σ p Λ p )

65 ζ 1 β (z) = {p} ( 1 z n p eβ σ p Λ p ) their period

66 ζ 1 β (z) = {p} ( 1 z n p eβ σ p Λ p )

67 ζ 1 β (z) = {p} ( 1 z n p eβ σ p Λ p ) their instability

68 ζ 1 β (z) = {p} ( 1 z n p eβ σ p Λ p )

69 ζ 1 β (z) = {p} ( 1 z n p eβ σ p Λ p ) their space shift

70 transport and analytic properties (x n x 0 ) k 0 k 1 β k 2πı a+ı a ı ds e sn d ds ln [ ζ 1 β,(0) (e s )] β=0 D = lim n 1 2n d 2 dβ 2 ( 1 2πı a+ı a ı ds e sn sζ 1 ) β,(0) (e s ) ζ 1 β,(0) (e s ) β=0 analytic properties of zeta functions near their first zero give the asympotics of moments (via Tauberian theorems for Laplace transforms) RA, Cristadoro, Dahlqvist

71 Qualitative 1-d intermittency

72 Qualitative 1-d intermittency strong chaos exponential instability Λ n a n weak chaos power-law instability Λ n n a

73 Qualitative 1-d intermittency

74 Qualitative 1-d intermittency strong chaos simple zero (polynomial) weak chaos non analytic behavior

75 take-home message even low-dimensional dynamical systems can provide a rich variety of transport properties (diffusion, anomalous scaling, ratchet behavior) analysis in terms of periodic orbits (zeta functions) yields exact results for some models, in the realm of a purely deterministic approach

76 RA, G. Cristadoro: Deterministic (anomalous) transport