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Deterministic walks on a square lattice

Ra´ ul Rechtman

Instituto de Energ´ ıas Renovables, Universidad Nacional Aut´

• noma de M´

exico, Temixco, Mor., Mexico rrs@cie.unam.mx

Advances in Nonequilibrium Statistical Mechanics: large deviations and long-range correlations, extreme value statistics, anomalous transport, and long-range interactions, The Galileo Galilei Institute for Theoretical Physics, Arcetri, Florence, July 1, 2014

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 1 / 53

Contents

1

Introduction

2

A walker on an initially ordered flipping rotor landscape

3

A walker on a partially ordered flipping rotor landscape

4

Two walkers on an initially ordered flipping rotor landscape

5

A walker on an initially disordered flipping rotor landscape

6

Concluding remarks

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 2 / 53

Introduction

Deterministic walks

Lorentz gas. A walker on a landscape. The walker interacts with the landscape during the walk. Landscape = 2d square lattice with obstacles. Complex system. Simple model of anomalous transport.

• H. A. Lorentz, Proc. Amst. Acad. 7 438 (1905).
• E. G. D. Cohen, L. Bunimovich, J. P. Boon, X. P. Kong, P. M. Binder, H-F. Meng.

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 3 / 53

Introduction

The Ehrenfest’s wind-tree model (1911) dfi dt = k(fi+1 + fi−1 − 2fi), i = 0, . . . 3 f0(0) = 1, f1(0) = f2(0) = f3(0) = 0,

• i

fi = 1, f0 = 1 4

• 1 + e(−2kt)2 ,

f1 = f3 = 1 4

• 1 + e(−2kt)

1 − e(−2kt) , f2 = 1 4

• 1 − e(−2kt)2 .

v0 v1 v2 v3

0.25 0.5 0.75 1

t

f0 f1, f3 f2

• P. Ehenfest, T. Ehrenfest, Begriffliche Grundlagen der Statistische Auffassung in der Mechanik, Encyklop¨

adie der Mathematische Wissenschaften vol. 4 pt 32 (Leipzig: Teubner), 1911. Engl. Trans. M. J. Moravcsik, The Conceptual Foundations of the Statistical Approach in Mechanics, Ithaca, Cornell University Press, 1959. R. Rechtman, A. Salcido, A. Calles, EPL 12 27 (1991). Ra´ ul Rechtman (IER-UNAM) Deterministic walks 4 / 53

Introduction

a

• X. P. Kong, E. G. D. Cohen, Phys. Rev. B 40, 4838 (1989). Th. Ruijgrok, E. G. D. Cohen, Phys. Lett. A, 133 415 (1988). H-F. Meng, E. G. D. Cohen,
• Phys. Rev. E 50 2482 (1994).

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 5 / 53

Introduction

a

• X. P. Kong, E. G. D. Cohen, Phys. Rev. B 40, 4838 (1989). Th. Ruijgrok, E. G. D. Cohen, Phys. Lett. A, 133 415 (1988). H-F. Meng, E. G. D. Cohen,
• Phys. Rev. E 50 2482 (1994).

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 5 / 53

Introduction

flipping mirror landscape right mirror left mirror flipping rotor landscape right rotor left rotor

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Introduction

flipping mirror landscape flipping rotor landscape a a

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Introduction

flipping mirror landscape flipping rotor landscape a a

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Introduction

flipping mirror landscape flipping rotor landscape a a

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 7 / 53

Introduction

flipping mirror landscape flipping rotor landscape a a

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 7 / 53

Introduction

flipping mirror landscape flipping rotor landscape a a

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 7 / 53

Introduction

flipping mirror landscape flipping rotor landscape a a

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 7 / 53

Introduction

flipping mirror landscape flipping rotor landscape a a

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 7 / 53

Introduction

flipping mirror landscape flipping rotor landscape a a

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 7 / 53

Introduction

A walker moves on a 2D square lattice, the landscape, in discrete time steps to a nearest neighbor site according to the landscape. In so doing, he alters the landscape locally. At time t the walker is at (x, y) with

• ne of four velocities v0 = (1, 0), v1 = (0, 1), v2 = (−1, 0), or v3 = (0, −1). The state of the landscape,

m(x, y), is either 1 or -1 and after the walker passes, m changes sign. The landscape is made of flipping rotors

• r flipping mirrors. In the first case, the particle turns right or left according to m(x, y), and in the second one,

the particle is reflected by a “mirror” with an inclination of 45◦ or 135◦.

v0 v1 v2 v3

flipping mirror landscape m(x, y) =

• 1

walker reflects from a mirror at 45◦ −1 walker reflects from a mirror at 135◦ flipping rotor landscape m(x, y) =

• 1

walker rotates 90◦ to the right −1 walker rotates 90◦ to the left

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 8 / 53

Introduction

flipping mirror landscape v ′

x = mvy

v ′

y = +mvx

m′ = −m x′ = x + v ′

x

y ′ = y + v ′

y

flipping rotor landscape v ′

x = mvy

v ′

y = −mvx

m′ = −m x′ = x + v ′

x

y ′ = y + v ′

y

The primed (unprimed) quantities refer to t + 1 (t).

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 9 / 53

Introduction

At t = 0, m(x, y) = 1 ∀ x, y and the walker is in the center of the lattice with v = v1.

flipping mirror landscape

40 80 40 80

flipping rotor landscape

40 80 40 80 5 10 15 20 25 30 35

The walker moves alternatively one step vertically, one horizontally. The walker has moved during 9,000 time steps. The colors show the number of times each site has been visited.

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 10 / 53

Contents

1

Introduction

2

A walker on an initially ordered flipping rotor landscape

3

A walker on a partially ordered flipping rotor landscape

4

Two walkers on an initially ordered flipping rotor landscape

5

A walker on an initially disordered flipping rotor landscape

6

Concluding remarks

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 11 / 53

Initially ordered flipping rotor landscape

40 80 40 80 t = 100 1 2 3 4 5 6 7

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 12 / 53

Initially ordered flipping rotor landscape

40 80 40 80 t = 1000 3 6 9 12 15 18

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 12 / 53

Initially ordered flipping rotor landscape

40 80 40 80 t = 3000 5 10 15 20 25

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 12 / 53

Initially ordered flipping rotor landscape

40 80 40 80 t = 5000 6 12 18 24 30

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 12 / 53

Initially ordered flipping rotor landscape

40 80 40 80 t = 7000 6 12 18 24 30

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 12 / 53

Initially ordered flipping rotor landscape

40 80 40 80 t = 9000 6 12 18 24 30

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 12 / 53

Initially ordered flipping rotor landscape

40 80 40 80 t = 11000 6 12 18 24 30

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 12 / 53

Initially ordered flipping rotor landscape

At t = 0, m(x, y) = 1 ∀(x, y) and the walker is in the center of the lattice with v = v1.

40 80 40 80 t = 11000 6 12 18 24 30

v ′

x = mvy

v ′

y = −mvx

m′ = −m x′ = x + v ′

x

y ′ = y + v ′

y

v′ = vk−m m′ = −m r′ = r + v′ After almost 10,000 time steps, T0, the walker begins to move periodically. Every 100 or so time steps, T1, it moves 2 sites horizontally and 2 vertically.

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 13 / 53

Initially ordered flipping rotor landscape

T0 = 9, 977. For t > T0 the particle moves periodically with period T1 = 104.

15 30 45 60 75 4,000 8,000 T0 12,000

t

x y

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 14 / 53

Initially ordered flipping rotor landscape

For t > T0, the walker moves periodically with period T1 = 104 and x and y diminish by 2 with a speed u = 2 √ 2/104.

15 30 45 60 T0 T0 + T1 T0 + 2T1 T0 + 3T1

t

x y

The two straight lines have slope −2/104.

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 15 / 53

Initially ordered flipping rotor landscape

t = T0 t = T0 + T1

41 43 45 47 49 51 20 22 24 26 39 41 43 45 47 49 18 20 22 24

At t = T0 the walker is at the site marked by the red circle (left Fig.) with v = v2. At t = T0 + T1 the walker is at the site marked by the red circle (right Fig.) with v = v2. The walker moved two sites to the left and two

• down. In doing so the walker prepared the landscape in such a way that its motion becomes periodic. The state
• f the rotors of the two Figs. are the same, except on the top row and the right column, but these sites are not

visited by the particle as shown in the next Fig.

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 16 / 53

Initially ordered flipping rotor landscape

41 43 45 47 49 51 20 22 24 26

Trajectory of the walker between t = T0 and t = T1 to be compared with the previous Figs. At t = T0 the walker is in (25, 50), the upper right red circle, and at t = T0 + T1, the walker is in (23, 48), the lower left circle.

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 17 / 53

Initially ordered flipping rotor landscape with periodic boundary conditions

200 200 After T0 time steps the walker moves periodically along a diagonal,reaches a border, enters

• n the opposite one. It eventually

goes back to the central part of the lattice and after some time it again moves periodically. This goes on and on. The total time is T = 80, 000. This behavior suggests that the walker will move periodically if there is a sufficiently large region with ordered rotors.

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 18 / 53

Contents

1

Introduction

2

A walker on an initially ordered flipping rotor landscape

3

A walker on a partially ordered flipping rotor landscape

4

Two walkers on an initially ordered flipping rotor landscape

5

A walker on an initially disordered flipping rotor landscape

6

Concluding remarks

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 19 / 53

Partially ordered flipping rotor landscape

At t = 0, m(x, y) = 1 (right rotor), with 0 ≤ x < 80, 0 ≤ y < 80. Inside the small box, 20 ≤ x < 60, 20 ≤ y < 60, m(x, y) = −1 (left rotor) with probability q. The landscape is initially disordered inside the small box and ordered outside of it. q = 0.01 q = 0.2

20 60 80 20 60 80 5 10 15 20 20 60 80 20 60 80 5 10 15 20 Ra´ ul Rechtman (IER-UNAM) Deterministic walks 20 / 53

Partially ordered flipping rotor landscape

q = 0.5 q = 0.8

20 60 80 20 60 80 5 10 15 20 20 60 80 20 60 80 5 10 15 20 Ra´ ul Rechtman (IER-UNAM) Deterministic walks 21 / 53

Partially ordered flipping rotor landscape

q = 0.99 q = 1.0

20 60 80 20 60 80 5 10 15 20 20 60 80 20 60 80 5 10 15 20

As long as the walker finds an ordered landscape he will move periodically with period T1.

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 22 / 53

Partially ordered flipping rotor landscape

p = q = 0.5

10 20 30 40 50 60 70 80 5000 10000 15000 20000 25000 30000

x t

10 20 30 40 50 60 70 80 5000 10000 15000 20000 25000 30000

y t

The escape time tesc is the time when x or y cross one of the red lines.

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 23 / 53

Partially ordered flipping rotor landscape

Distribution of escape times, φ for diffferent values of p. For every value of q, φ is the result of 10,000 simulations. q = 0.01 q = 0.20

0.015 0.03 5000 10000 15000 20000

φ tesc

0.01 0.02 0.03 10000 20000 30000 40000

φ tesc φ

M(tesc) = 6, 954, D(tesc) = 3, 922 M(tesc) = 7, 200, D(tesc) = 3, 876 M is the median and D the average absolute deviation.

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 24 / 53

Partially ordered flipping rotor landscape

Distribution of escape times, φ for diffferent values of p. For every value of p, φ is the result of 10,000 simulations. q = 0.50 q = 0.80

0.015 0.03 0.045 5000 10000 15000 20000

φ tesc

0.015 0.03 0.045 5000 10000 15000 20000

φ tesc

M(tesc) = 7, 584, D(tesc) = 4, 080 M(tesc) = 8, 874, D(tesc) = 4, 088

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 25 / 53

Contents

1

Introduction

2

A walker on an initially ordered flipping rotor landscape

3

A walker on a partially ordered flipping rotor landscape

4

Two walkers on an initially ordered flipping rotor landscape

5

A walker on an initially disordered flipping rotor landscape

6

Concluding remarks

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 26 / 53

Two walkers on an initially ordered flipping rotor landscape

Antonio Prohias, Spy vs Spy, Mad magazine, January 1960 to March 1987. Ra´ ul Rechtman (IER-UNAM) Deterministic walks 27 / 53

Two walkers on an initially ordered flipping rotor landscape

Walker Red is chased by walker Blue. Initially they are a distance d apart, both with the same velocity, v0. The initial positions of the walkers are marked by the filled circles. d = 1 d = 2

2 4 6 8 10 2 4 6 8 10 1 2 3 4 5 6 7 3 4 5 6 7

Blue follows Red in square At t = 8 Blue catches Red spirals in the red circle

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Two walkers on an initially ordered flipping rotor landscape

d = 3 d = 4

10 15 20 25 30 35 5 10 15 20 25 30 35 40 45 5 10 15 20 25 30 35 40 10 15 20 25 30 35 40 45 50 55

Blue never catches Red Blue forgets about Red

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 29 / 53

Two walkers on an initially ordered flipping rotor landscape

d = 5 d = 6

10 15 20 25 30 35 16 20 24 28 32 40 42 44 46 48 50 52 54 46 47 4849 50 51 52 53 54

Blue never catches Red At t = 84 Blue catches Red in the black circle If Blue is after Red, his best strategy is to be two sites away, the next best one is to be 6 sites away. For d odd Blue “never” catches Red. The patterns have some symmetry.

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 30 / 53

Contents

1

Introduction

2

A walker on an initially ordered flipping rotor landscape

3

A walker on a partially ordered flipping rotor landscape

4

Two walkers on an initially ordered flipping rotor landscape

5

A walker on an initially disordered flipping rotor landscape

6

Concluding remarks

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 31 / 53

Iinitially disordered flipping mirror landscape

Initially m(r) = 1 (right mirror) with probability p and m(r) = −1 (left mirror) with probability q = 1 − p, with 0 ≤ x < L, and 0 ≤ y < L. p = 0.9, T = 334 p = 0.8, T = 659

40 80 40 80 0.5 1 1.5 2 2.5 3 3.5 4 40 80 40 80 2 4 6 8 10

p = 0.7, T = 1, 047 p = 0.5, T = 2, 319

40 80 40 80 1 2 3 4 5 6 7 8 9 40 80 40 80 1 2 3 4 5 6 7 8 9

At t = 0, x = y = 40, v = v0 . Ra´ ul Rechtman (IER-UNAM) Deterministic walks 32 / 53

Iinitially disordered flipping rotor landscape

A landscape of side L. Initially m(r) = 1 (right rotor) with probability p and m(r) = −1 (left rotor) with probability 0 ≤ x < L, and 0 ≤ y < L, with r = (x, y), x, y ∈ N, p = 0.9, T = 7, 675 p = 0.8, T = 7, 174

40 80 40 80 2 4 6 8 10 12 14 16 18 40 80 40 80 2 4 6 8 10

p = 0.7, T = 3, 011 p = 0.5, T = 2, 238

40 80 40 80 2 4 6 8 10 40 80 40 80 2 4 6 8 10

At t = 0, x = y = 40, v = v0 . Ra´ ul Rechtman (IER-UNAM) Deterministic walks 33 / 53

Initially disordered landscape

• (r − r0)2

p = 2dDtα,

d = 2 (1)

• (r − r0)2

p =

• (r − r0)2

1−p

flipping mirror landscape flipping rotor landscape

1000 10000 100000 1e+06 1e+07 1e+08 1e+09 1000 10000 100000

• (r − r0)2

t

p = 0.05 p = 0.10 p = 0.30 p = 0.40 100 1000 10000 100000 1e+06 1000 10000 100000

• (r − r0)2

t

p = 0.01 p = 0.06 p = 0.10 p = 0.20 p = 0.40

· p is the average over N samples of initial landscapes with a fraction p of right mirrors. N = 100, 000, T = 100, 000, and L suficiently large. Two exceptions: in the flipping mirror landscape, for p = 0.05 and p = 0.10, N = 1, 000. The fit of Eq. (1) to the data is for 10, 000 ≤ t ≤ 100, 000.

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 34 / 53

Initially disordered landscape

For comparison we also consider a random walker that at every site turns right with probability p and left with probability q.

• (r − r0)2

p = 2dDtα,

d = 2

0.01 0.1 1 10 100 0.1 0.2 0.3 0.4 0.5

D p

flipping rotor flipping mirror random walk 0.7 0.8 0.9 1 1.1 0.1 0.2 0.3 0.4 0.5

α p

flipping rotor flipping mirror random walk

D and α are taken from the best fits for t > 10, 000. Note logarithmic vertical scale for D. Subdiffusion (α < 1) on the flipping rotor landscape for 0 < p < 0.3 and 0.7 p < 1.0. Superdiffusion (α > 1) on the flipping mirror landscape for 0 < p 0.15 and 0.85 p < 1.

The error of the fits is smaller than the size of the points of the graphs Ra´ ul Rechtman (IER-UNAM) Deterministic walks 35 / 53

Initially disordered landscape

w(t) =

t

• s=0

m(x(s), y(s)) = nl(t) − nr (t) w(t)p = Btβ (2) w(t)p = −w(t)1−p nl(t) (nr(t)) are the number of left (right) turns of the walker after t time steps. flipping mirror landscape flipping rotor landscape

100 1000 10000 100000 1000 10000 100000

w

t

p = 0.05 p = 0.10 p = 0.30 p = 0.40 10 100 1000 10000 1000 10000 100000

w

t

p = 0.01 p = 0.10 p = 0.20 p = 0.40

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 36 / 53

Initially disordered landscape

w(t)p = Btβ w(t)p = −w(t)1−p

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.1 0.2 0.3 0.4 0.5

B p

flipping rotor flipping mirror random walk 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5

β p

flipping rotor flipping mirror random walk

For random walks B = (1 − 2p)/4, shown in black in the Fig. on the left, and β = 1, Fig. on the right. B → 0 as p → 1/2 due to the symmetry of w. T = 100, 000, N = 100, 000, and L sufficiently large. The fit

• f Eq. (2) to the data is for 0 ≤ t ≤ 100, 000.

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 37 / 53

Initially disordered landscape

At time t, a walker has visited Ns sites. Ns(t)p = Ctγ (3) Ns(t)p = Ns(t)1−p

0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5

C p

flipping rotor flipping mirror random walk 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5

γ p

flipping rotor flipping mirror random walk

T = 100, 000, N = 100, 000, and L sufficiently large. The fit of Eq. (4) to the data is for 10, 000 ≤ t ≤ 100, 000.

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 38 / 53

Initially disordered landscape

The one dimensional kurtosis Kx is defined by Kx =

• (x − x0)4

− 3

• (x − x0)2

(x − x0)22 flipping mirror landscape flipping rotor landscape

• 2
• 1.5
• 1
• 0.5

0.5 20000 40000 60000 80000 100000

Kx

t

p = 0.05 p = 0.06 p = 0.08 p = 0.10 p = 0.30

• 1.5
• 1
• 0.5

0.5 1 25000 50000 75000 100000

Kx t

p = 0.01 p = 0.02 p = 0.03 p = 0.20

T = 100, 000, N = 100, 000, and L sufficiently large. The fit of Eq. (4) to the data is for 10, 000 ≤ t ≤ 100, 000.

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 39 / 53

Initially disordered landscape

φ(x, y, t)∆x∆y is the probability of finding a walker at (X, Y ) with x < X < x + ∆x and y < Y < y + ∆y at time t. flipping mirror landscape flipping rotor landscape

x y φ x y φ

random walk

x y φ

p = 0.20, T = 10, 000, N = 20, 000 and L suficiently large.

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 40 / 53

Initially disordered flipping rotor landscape

φ = φ(x, L/2, T) p = 0.01 p = 0.02

1e-05 0.0001 0.001 0.01 0.1 1200 1300 1400 1500 1600 1700 1800

φ x

1e-05 0.0001 0.001 0.01 0.1 1200 1300 1400 1500 1600 1700 1800

φ x

p = 0.04 p = 0.05

1e-05 0.0001 0.001 0.01 0.1 1200 1300 1400 1500 1600 1700 1800

φ x

1e-05 0.0001 0.001 0.01 0.1 1100 1200 1300 1400 1500 1600 1700 1800 1900

φ x

T = 200, 000, N = 100, 000, L = 3, 000, and ∆x = 9. Normal distribution φ(x) =

1 σ √ 2π exp

• −(x−x)2

2σ2

• in

black.

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 41 / 53

Initially disordered landscape

From the previous results, Kx is the average of Kx after a transient that is taken as one half the final time. flipping rotor landscape

• 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.1 0.2 0.3 0.4 0.5

Kx p

T = 100, 000, N = 100, 000, and L sufficiently large.

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 42 / 53

Initially disordered landscape

The “observed probability” pobs is the average fraction of right obstacles the wallkers encounter. flipping mirror landscape flipping rotor landscape

0.1 0.15 0.2 0.25 0.3 0.35 20000 40000 60000 80000 100000

pobs

t

p = 0.05 p = 0.06 p = 0.07 p = 0.08 p = 0.09 p = 0.10 0.39 0.4 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 20000 40000 60000 80000 100000

pobs

t

p = 0.03 p = 0.05 p = 0.07 p = 0.09

N = 1, 000 N = 100, 000

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 43 / 53

Initially disordered landscape

The “observed probability” pobs is the average fraction of right obstacles the wallkers encounter. flipping mirror landscape flipping rotor landscape pobs(t = 0) pobs(t = 105) pobs(t = 0) pobs(t = 105) 0.30 0.458106 0.01 0.474162 0.40 0.481395 0.10 0.468232 0.20 0.464660 0.30 0.471276 0.40 0.484226

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 44 / 53

Initially disordered landscape

flipping mirror landscape p = 0.06 p = 0.07

1000 10000 100000 1e+06 1e+07 1e+08 1e+09 100 1000 10000 100000

• (r − r0)2

t

1000 10000 100000 1e+06 1e+07 1e+08 1e+09 100 1000 10000 100000

• (r − r0)2

t

p t0 D α t1 D α 0.06 10,000 0.193 1.784 10,000 97.669 1.042 0.07 1,000 0.273 1.702 10,000 49.559 1.06 Two scalings, one for t < t0, the other one for t1 < t. T = 10, 000, N = 1, 000. For p = 0.06, L = 40, 000 and for p = 0.07, L = 35, 000.

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 45 / 53

Initially disordered landscape

flipping rotor landscape p = 0.03 p = 0.07

10 100 1000 10000 100 1000 10000 100000

• (r − r0)2

t

10 100 1000 10000 100 1000 10000 100000

• (r − r0)2

t

p t0 D α t1 D α 0.03 3,000 0.102 0.838 10,000 0.311 0.714 0.05 3,000 0.159 0.812 10,000 0.311 0.741 Two scalings, one for t < t0, the other one for t1 < t. T = 10, 000, N = 100, 000, and L = 5, 000.

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 46 / 53

Contents

1

Introduction

2

A walker on an initially ordered flipping rotor landscape

3

A walker on a partially ordered flipping rotor landscape

4

Two walkers on an initially ordered flipping rotor landscape

5

A walker on an initially disordered flipping rotor landscape

6

Concluding remarks

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 47 / 53

Concluding remarks

A simple example of a complex system.

A walk on an initial ordered rotor landscape. A walk on a partially ordered rotor landscape. Two walkers on an initially ordered landscape.

A model for anomalous transport

• (r − r0)2

= 2dDtα Crowded biological media. Polymeric networks. Porous materials. Cytoskeletal fibers and molecular motors.

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 48 / 53

• E. G. D. “Eddie” Cohen, The Rockefeller University

Centro de Investigaci´

• n en Energ´

ıa, UNAM, January 2002.

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 49 / 53

Acknowledgments

The Galileo Galilei Institute of Theoretical Physics, Arcetri, Italy. Istituto Nazionale di Fisica Nucleare, Sezione Firenze. Direcci´

• n General de Asuntos del Personal Ac´

ademico, Universidad Nacional Aut´

• noma de

M´ exico. Instituto de Energ´ ıas Renovables, Universidad Nacional Aut´

• noma de M´

exico.

Ra´ ul Rechtman (IER-UNAM) Deterministic walks 50 / 53

Thanks!!

Instituto de Energ´ ıas Renovables, UNAM, Temixco, Morelos, Mexico

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