Deterministic walks on a square lattice
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1 Deterministic wlks on squre lttice Rúl Rechtmn Instituto de Energís Renovbles, Universidd Ncionl Autónom de México, Temixco, Mor., Mexico Advnces in Nonequilibrium Sttisticl Mechnics: lrge devitions nd long-rnge correltions, extreme vlue sttistics, nomlous trnsport, nd long-rnge interctions, The Glileo Glilei Institute for Theoreticl Physics, Arcetri, Florence, July 1, 214 Rúl Rechtmn (IER-UNAM) Deterministic wlks 1 / 53
2 Contents 1 Introduction 2 A wlker on n initilly ordered flipping rotor lndscpe 3 A wlker on prtilly ordered flipping rotor lndscpe 4 Two wlkers on n initilly ordered flipping rotor lndscpe 5 A wlker on n initilly disordered flipping rotor lndscpe 6 Concluding remrks Rúl Rechtmn (IER-UNAM) Deterministic wlks 2 / 53
3 Introduction Deterministic wlks Lorentz gs. A wlker on lndscpe. The wlker intercts with the lndscpe during the wlk. Lndscpe = 2d squre lttice with obstcles. Complex system. Simple model of nomlous trnsport. H. A. Lorentz, Proc. Amst. Acd (195). E. G. D. Cohen, L. Bunimovich, J. P. Boon, X. P. Kong, P. M. Binder, H-F. Meng. Rúl Rechtmn (IER-UNAM) Deterministic wlks 3 / 53
4 Introduction The Ehrenfest s wind-tree model (1911) f = 1 4 df i dt = k(f i+1 + f i 1 2f i ), i =,... 3 f () = 1, f 1() = f 2() = f 3() =, f i = 1, i [ 1 + e ( 2kt)] 2, f1 = f 3 = 1 [ 1 + e ( 2kt)] [ 1 e ( 2kt)], 4 f 2 = 1 [ 1 e ( 2kt)] 2. 4 v f f1, f3 f2 v2 v.5.25 v3 t P. Ehenfest, T. Ehrenfest, Begriffliche Grundlgen der Sttistische Auffssung in der Mechnik, Encyklopädie der Mthemtische Wissenschften vol. 4 pt 32 (Leipzig: Teubner), Engl. Trns. M. J. Morvcsik, The Conceptul Foundtions of the Sttisticl Approch in Mechnics, Ithc, Cornell University Press, R. Rechtmn, A. Slcido, A. Clles, EPL (1991). Rúl Rechtmn (IER-UNAM) Deterministic wlks 4 / 53
5 Introduction X. P. Kong, E. G. D. Cohen, Phys. Rev. B 4, 4838 (1989). Th. Ruijgrok, E. G. D. Cohen, Phys. Lett. A, (1988). H-F. Meng, E. G. D. Cohen, Phys. Rev. E (1994). Rúl Rechtmn (IER-UNAM) Deterministic wlks 5 / 53
6 Introduction X. P. Kong, E. G. D. Cohen, Phys. Rev. B 4, 4838 (1989). Th. Ruijgrok, E. G. D. Cohen, Phys. Lett. A, (1988). H-F. Meng, E. G. D. Cohen, Phys. Rev. E (1994). Rúl Rechtmn (IER-UNAM) Deterministic wlks 5 / 53
7 Introduction flipping mirror lndscpe flipping rotor lndscpe right mirror left mirror right rotor left rotor Rúl Rechtmn (IER-UNAM) Deterministic wlks 6 / 53
8 Introduction flipping mirror lndscpe flipping rotor lndscpe Rúl Rechtmn (IER-UNAM) Deterministic wlks 7 / 53
9 Introduction flipping mirror lndscpe flipping rotor lndscpe Rúl Rechtmn (IER-UNAM) Deterministic wlks 7 / 53
10 Introduction flipping mirror lndscpe flipping rotor lndscpe Rúl Rechtmn (IER-UNAM) Deterministic wlks 7 / 53
11 Introduction flipping mirror lndscpe flipping rotor lndscpe Rúl Rechtmn (IER-UNAM) Deterministic wlks 7 / 53
12 Introduction flipping mirror lndscpe flipping rotor lndscpe Rúl Rechtmn (IER-UNAM) Deterministic wlks 7 / 53
13 Introduction flipping mirror lndscpe flipping rotor lndscpe Rúl Rechtmn (IER-UNAM) Deterministic wlks 7 / 53
14 Introduction flipping mirror lndscpe flipping rotor lndscpe Rúl Rechtmn (IER-UNAM) Deterministic wlks 7 / 53
15 Introduction flipping mirror lndscpe flipping rotor lndscpe Rúl Rechtmn (IER-UNAM) Deterministic wlks 7 / 53
16 Introduction A wlker moves on 2D squre lttice, the lndscpe, in discrete time steps to nerest neighbor site ccording to the lndscpe. In so doing, he lters the lndscpe loclly. At time t the wlker is t (x, y) with one of four velocities v = (1, ), v 1 = (, 1), v 2 = ( 1, ), or v 3 = (, 1). The stte of the lndscpe, m(x, y), is either 1 or -1 nd fter the wlker psses, m chnges sign. The lndscpe is mde of flipping rotors or flipping mirrors. In the first cse, the prticle turns right or left ccording to m(x, y), nd in the second one, the prticle is reflected by mirror with n inclintion of 45 or 135. v 1 flipping mirror lndscpe { 1 wlker reflects from mirror t 45 m(x, y) = 1 wlker reflects from mirror t 135 v 2 v 3 v flipping rotor lndscpe { 1 wlker rottes 9 to the right m(x, y) = 1 wlker rottes 9 to the left Rúl Rechtmn (IER-UNAM) Deterministic wlks 8 / 53
17 Introduction flipping mirror lndscpe flipping rotor lndscpe v x = mvy v y = +mvx m = m x = x + v x y = y + v y v x = mvy v y = mvx m = m x = x + v x y = y + v y The primed (unprimed) quntities refer to t + 1 (t). Rúl Rechtmn (IER-UNAM) Deterministic wlks 9 / 53
18 Introduction At t =, m(x, y) = 1 x, y nd the wlker is in the center of the lttice with v = v 1. flipping mirror lndscpe flipping rotor lndscpe The wlker moves lterntively one step verticlly, one horizontlly. The wlker hs moved during 9, time steps. The colors show the number of times ech site hs been visited. Rúl Rechtmn (IER-UNAM) Deterministic wlks 1 / 53
19 Contents 1 Introduction 2 A wlker on n initilly ordered flipping rotor lndscpe 3 A wlker on prtilly ordered flipping rotor lndscpe 4 Two wlkers on n initilly ordered flipping rotor lndscpe 5 A wlker on n initilly disordered flipping rotor lndscpe 6 Concluding remrks Rúl Rechtmn (IER-UNAM) Deterministic wlks 11 / 53
20 Initilly ordered flipping rotor lndscpe 8 t = Rúl Rechtmn (IER-UNAM) Deterministic wlks 12 / 53
21 Initilly ordered flipping rotor lndscpe 8 t = Rúl Rechtmn (IER-UNAM) Deterministic wlks 12 / 53
22 Initilly ordered flipping rotor lndscpe 8 t = Rúl Rechtmn (IER-UNAM) Deterministic wlks 12 / 53
23 Initilly ordered flipping rotor lndscpe 8 t = Rúl Rechtmn (IER-UNAM) Deterministic wlks 12 / 53
24 Initilly ordered flipping rotor lndscpe 8 t = Rúl Rechtmn (IER-UNAM) Deterministic wlks 12 / 53
25 Initilly ordered flipping rotor lndscpe 8 t = Rúl Rechtmn (IER-UNAM) Deterministic wlks 12 / 53
26 Initilly ordered flipping rotor lndscpe 8 t = Rúl Rechtmn (IER-UNAM) Deterministic wlks 12 / 53
27 Initilly ordered flipping rotor lndscpe At t =, m(x, y) = 1 (x, y) nd the wlker is in the center of the lttice with v = v 1. 8 t = 11 3 v x = mvy v y = mvx 24 m = m x = x + v x 4 18 y = y + v y v = v k m m = m r = r + v After lmost 1, time steps, T, the wlker begins to move periodiclly. Every 1 or so time steps, T 1, it moves 2 sites horizontlly nd 2 verticlly. Rúl Rechtmn (IER-UNAM) Deterministic wlks 13 / 53
28 Initilly ordered flipping rotor lndscpe T = 9, 977. For t > T the prticle moves periodiclly with period T 1 = x y , 8, T 12, t Rúl Rechtmn (IER-UNAM) Deterministic wlks 14 / 53
29 Initilly ordered flipping rotor lndscpe For t > T, the wlker moves periodiclly with period T 1 = 14 nd x nd y diminish by 2 with speed u = 2 2/14. 6 x y T T + T 1 T + 2T 1 T + 3T 1 t The two stright lines hve slope 2/14. Rúl Rechtmn (IER-UNAM) Deterministic wlks 15 / 53
30 Initilly ordered flipping rotor lndscpe t = T t = T + T At t = T the wlker is t the site mrked by the red circle (left Fig.) with v = v 2. At t = T + T 1 the wlker is t the site mrked by the red circle (right Fig.) with v = v 2. The wlker moved two sites to the left nd two down. In doing so the wlker prepred the lndscpe in such wy tht its motion becomes periodic. The stte of the rotors of the two Figs. re the sme, except on the top row nd the right column, but these sites re not visited by the prticle s shown in the next Fig. Rúl Rechtmn (IER-UNAM) Deterministic wlks 16 / 53
31 Initilly ordered flipping rotor lndscpe Trjectory of the wlker between t = T nd t = T 1 to be compred with the previous Figs. At t = T the wlker is in (25, 5), the upper right red circle, nd t t = T + T 1, the wlker is in (23, 48), the lower left circle. Rúl Rechtmn (IER-UNAM) Deterministic wlks 17 / 53
32 Initilly ordered flipping rotor lndscpe with periodic boundry conditions 2 2 After T time steps the wlker moves periodiclly long digonl,reches border, enters on the opposite one. It eventully goes bck to the centrl prt of the lttice nd fter some time it gin moves periodiclly. This goes on nd on. The totl time is T = 8,. This behvior suggests tht the wlker will move periodiclly if there is sufficiently lrge region with ordered rotors. Rúl Rechtmn (IER-UNAM) Deterministic wlks 18 / 53
33 Contents 1 Introduction 2 A wlker on n initilly ordered flipping rotor lndscpe 3 A wlker on prtilly ordered flipping rotor lndscpe 4 Two wlkers on n initilly ordered flipping rotor lndscpe 5 A wlker on n initilly disordered flipping rotor lndscpe 6 Concluding remrks Rúl Rechtmn (IER-UNAM) Deterministic wlks 19 / 53
34 Prtilly ordered flipping rotor lndscpe At t =, m(x, y) = 1 (right rotor), with x < 8, y < 8. Inside the smll box, 2 x < 6, 2 y < 6, m(x, y) = 1 (left rotor) with probbility q. The lndscpe is initilly disordered inside the smll box nd ordered outside of it. 8 q =.1 q = Rúl Rechtmn (IER-UNAM) Deterministic wlks 2 / 53
35 Prtilly ordered flipping rotor lndscpe 8 q =.5 q = Rúl Rechtmn (IER-UNAM) Deterministic wlks 21 / 53
36 Prtilly ordered flipping rotor lndscpe 8 q =.99 q = As long s the wlker finds n ordered lndscpe he will move periodiclly with period T 1. Rúl Rechtmn (IER-UNAM) Deterministic wlks 22 / 53
37 Prtilly ordered flipping rotor lndscpe p = q = x 4 y t t The escpe time t esc is the time when x or y cross one of the red lines. Rúl Rechtmn (IER-UNAM) Deterministic wlks 23 / 53
38 Prtilly ordered flipping rotor lndscpe Distribution of escpe times, φ for diffferent vlues of p. For every vlue of q, φ is the result of 1, simultions. q =.1 q =.2.3 φ.3.15 φ.2.1 φ t esc t esc M(t esc ) = 6, 954, D(t esc ) = 3, 922 M(t esc ) = 7, 2, D(t esc ) = 3, 876 M is the medin nd D the verge bsolute devition. Rúl Rechtmn (IER-UNAM) Deterministic wlks 24 / 53
39 Prtilly ordered flipping rotor lndscpe Distribution of escpe times, φ for diffferent vlues of p. For every vlue of p, φ is the result of 1, simultions..45 q =.5 q = φ φ t esc t esc M(t esc ) = 7, 584, D(t esc ) = 4, 8 M(t esc ) = 8, 874, D(t esc ) = 4, 88 Rúl Rechtmn (IER-UNAM) Deterministic wlks 25 / 53
40 Contents 1 Introduction 2 A wlker on n initilly ordered flipping rotor lndscpe 3 A wlker on prtilly ordered flipping rotor lndscpe 4 Two wlkers on n initilly ordered flipping rotor lndscpe 5 A wlker on n initilly disordered flipping rotor lndscpe 6 Concluding remrks Rúl Rechtmn (IER-UNAM) Deterministic wlks 26 / 53
41 Two wlkers on n initilly ordered flipping rotor lndscpe Antonio Prohis, Spy vs Spy, Md mgzine, Jnury 196 to Mrch Rúl Rechtmn (IER-UNAM) Deterministic wlks 27 / 53
42 Two wlkers on n initilly ordered flipping rotor lndscpe Wlker Red is chsed by wlker Blue. Initilly they re distnce d prt, both with the sme velocity, v. The initil positions of the wlkers re mrked by the filled circles. d = 1 d = Blue follows Red in squre spirls At t = 8 Blue ctches Red in the red circle Rúl Rechtmn (IER-UNAM) Deterministic wlks 28 / 53
43 Two wlkers on n initilly ordered flipping rotor lndscpe d = 3 d = Blue never ctches Red Blue forgets bout Red Rúl Rechtmn (IER-UNAM) Deterministic wlks 29 / 53
44 Two wlkers on n initilly ordered flipping rotor lndscpe d = 5 d = Blue never ctches Red At t = 84 Blue ctches Red in the blck circle If Blue is fter Red, his best strtegy is to be two sites wy, the next best one is to be 6 sites wy. For d odd Blue never ctches Red. The ptterns hve some symmetry. Rúl Rechtmn (IER-UNAM) Deterministic wlks 3 / 53
45 Contents 1 Introduction 2 A wlker on n initilly ordered flipping rotor lndscpe 3 A wlker on prtilly ordered flipping rotor lndscpe 4 Two wlkers on n initilly ordered flipping rotor lndscpe 5 A wlker on n initilly disordered flipping rotor lndscpe 6 Concluding remrks Rúl Rechtmn (IER-UNAM) Deterministic wlks 31 / 53
46 At t =, Rúl x = Rechtmn y = 4, v (IER-UNAM) = v. Deterministic wlks 32 / 53 Iinitilly disordered flipping mirror lndscpe Initilly m(r) = 1 (right mirror) with probbility p nd m(r) = 1 (left mirror) with probbility q = 1 p, with x < L, nd y < L. 8 p =.9, T = 334 p =.8, T = p =.7, T = 1, 47 p =.5, T = 2,
47 At t =, Rúl x = Rechtmn y = 4, v (IER-UNAM) = v. Deterministic wlks 33 / 53 Iinitilly disordered flipping rotor lndscpe A lndscpe of side L. Initilly m(r) = 1 (right rotor) with probbility p nd m(r) = 1 (left rotor) with probbility x < L, nd y < L, with r = (x, y), x, y N, 8 p =.9, T = 7, 675 p =.8, T = 7, p =.7, T = 3, 11 p =.5, T = 2,
48 Initilly disordered lndscpe (r r 2 ) = 2dDt α, d = 2 (1) p (r r 2 ) = (r r 2 ) p 1 p flipping mirror lndscpe flipping rotor lndscpe 1e+9 1e+6 1e+8 1 (r r) 2 1e+7 1e+6 1 (r r) 2 1 p =.5 1 p =.1 p =.3 p = t 1 p =.1 p =.6 p =.1 p =.2 p = t p is the verge over N smples of initil lndscpes with frction p of right mirrors. N = 1,, T = 1,, nd L suficiently lrge. Two exceptions: in the flipping mirror lndscpe, for p =.5 nd p =.1, N = 1,. The fit of Eq. (1) to the dt is for 1, t 1,. Rúl Rechtmn (IER-UNAM) Deterministic wlks 34 / 53
49 Initilly disordered lndscpe For comprison we lso consider rndom wlker tht t every site turns right with probbility p nd left with probbility q. (r r 2 ) = 2dDt α, d = 2 p 1 1 flipping rotor flipping mirror rndom wlk D 1 α p.8 flipping rotor flipping mirror rndom wlk p D nd α re tken from the best fits for t > 1,. Note logrithmic verticl scle for D. Subdiffusion (α < 1) on the flipping rotor lndscpe for < p <.3 nd.7 p < 1.. Superdiffusion (α > 1) on the flipping mirror lndscpe for < p.15 nd.85 p < 1. The error of the fits is smller thn the size of the points of the grphs Rúl Rechtmn (IER-UNAM) Deterministic wlks 35 / 53
50 Initilly disordered lndscpe t w(t) = m(x(s), y(s)) = n l (t) n r (t) s= w(t) p = Bt β (2) w(t) p = w(t) 1 p n l (t) (n r (t)) re the number of left (right) turns of the wlker fter t time steps. flipping mirror lndscpe flipping rotor lndscpe w w 1 p =.5 p =.1 p =.3 p = t 1 p =.1 p =.1 p =.2 p = t Rúl Rechtmn (IER-UNAM) Deterministic wlks 36 / 53
51 Initilly disordered lndscpe w(t) p = Bt β w(t) p = w(t) 1 p flipping rotor flipping mirror rndom wlk 1 B.4 β p.8 flipping rotor flipping mirror rndom wlk p For rndom wlks B = (1 2p)/4, shown in blck in the Fig. on the left, nd β = 1, Fig. on the right. B s p 1/2 due to the symmetry of w. T = 1,, N = 1,, nd L sufficiently lrge. The fit of Eq. (2) to the dt is for t 1,. Rúl Rechtmn (IER-UNAM) Deterministic wlks 37 / 53
52 Initilly disordered lndscpe At time t, wlker hs visited N s sites. N s (t) p = Ct γ (3) N s (t) p = N s (t) 1 p 1.8 flipping rotor flipping mirror rndom wlk 1 C.6 γ p.8 flipping rotor flipping mirror rndom wlk p T = 1,, N = 1,, nd L sufficiently lrge. The fit of Eq. (4) to the dt is for 1, t 1,. Rúl Rechtmn (IER-UNAM) Deterministic wlks 38 / 53
53 Initilly disordered lndscpe The one dimensionl kurtosis K x is defined by (x x) 4 3 (x x ) 2 K x = (x x ) 2 2 flipping mirror lndscpe flipping rotor lndscpe K x K x -.5 p = p =.6 p =.8 p =.1 p = t -1 p =.1 p =.2 p =.3 p = t T = 1,, N = 1,, nd L sufficiently lrge. The fit of Eq. (4) to the dt is for 1, t 1,. Rúl Rechtmn (IER-UNAM) Deterministic wlks 39 / 53
54 Initilly disordered lndscpe φ(x, y, t) x y is the probbility of finding wlker t (X, Y ) with x < X < x + x nd y < Y < y + y t time t. flipping mirror lndscpe flipping rotor lndscpe φ φ x y x y rndom wlk φ x y p =.2, T = 1,, N = 2, nd L suficiently lrge. Rúl Rechtmn (IER-UNAM) Deterministic wlks 4 / 53
55 Initilly disordered flipping rotor lndscpe.1 φ = φ(x, L/2, T ) p =.1 p = φ.1 φ e-5 1e x x p =.4 p = φ.1 φ e x 1e x ( ) T = 2,, N = 1,, L = 3,, nd x = 9. Norml distribution φ(x) = 1 σ exp (x x ) 2 2π 2σ 2 in blck. Rúl Rechtmn (IER-UNAM) Deterministic wlks 41 / 53
56 Initilly disordered lndscpe From the previous results, K x is the verge of K x fter trnsient tht is tken s one hlf the finl time. flipping rotor lndscpe K x p T = 1,, N = 1,, nd L sufficiently lrge. Rúl Rechtmn (IER-UNAM) Deterministic wlks 42 / 53
57 Initilly disordered lndscpe The observed probbility p obs is the verge frction of right obstcles the wllkers encounter. flipping mirror lndscpe flipping rotor lndscpe p obs.25.2 p =.5 p =.6.15 p =.7 p =.8 p =.9 p = t p obs p =.3 p =.5.4 p =.7 p = t N = 1, N = 1, Rúl Rechtmn (IER-UNAM) Deterministic wlks 43 / 53
58 Initilly disordered lndscpe The observed probbility p obs is the verge frction of right obstcles the wllkers encounter. flipping mirror lndscpe flipping rotor lndscpe p obs (t = ) p obs (t = 1 5 ) p obs (t = ) p obs (t = 1 5 ) Rúl Rechtmn (IER-UNAM) Deterministic wlks 44 / 53
59 Initilly disordered lndscpe flipping mirror lndscpe p =.6 p =.7 1e+9 1e+9 1e+8 1e+8 (r r) 2 1e+7 1e+6 1 (r r) 2 1e+7 1e t t p t D α t 1 D α.6 1, , , , Two sclings, one for t < t, the other one for t 1 < t. T = 1,, N = 1,. For p =.6, L = 4, nd for p =.7, L = 35,. Rúl Rechtmn (IER-UNAM) Deterministic wlks 45 / 53
60 Initilly disordered lndscpe flipping rotor lndscpe p =.3 p = (r r) (r r) t t p t D α t 1 D α.3 3, , , , Two sclings, one for t < t, the other one for t 1 < t. T = 1,, N = 1,, nd L = 5,. Rúl Rechtmn (IER-UNAM) Deterministic wlks 46 / 53
61 Contents 1 Introduction 2 A wlker on n initilly ordered flipping rotor lndscpe 3 A wlker on prtilly ordered flipping rotor lndscpe 4 Two wlkers on n initilly ordered flipping rotor lndscpe 5 A wlker on n initilly disordered flipping rotor lndscpe 6 Concluding remrks Rúl Rechtmn (IER-UNAM) Deterministic wlks 47 / 53
62 Concluding remrks A simple exmple of complex system. A wlk on n initil ordered rotor lndscpe. A wlk on prtilly ordered rotor lndscpe. Two wlkers on n initilly ordered lndscpe. A model for nomlous trnsport (r r ) 2 = 2dDt α Crowded biologicl medi. Polymeric networks. Porous mterils. Cytoskeletl fibers nd moleculr motors. Rúl Rechtmn (IER-UNAM) Deterministic wlks 48 / 53
63 E. G. D. Eddie Cohen, The Rockefeller University Centro de Investigción en Energí, UNAM, Jnury 22. Rúl Rechtmn (IER-UNAM) Deterministic wlks 49 / 53
64 Acknowledgments The Glileo Glilei Institute of Theoreticl Physics, Arcetri, Itly. Istituto Nzionle di Fisic Nuclere, Sezione Firenze. Dirección Generl de Asuntos del Personl Acádemico, Universidd Ncionl Autónom de México. Instituto de Energís Renovbles, Universidd Ncionl Autónom de México. Rúl Rechtmn (IER-UNAM) Deterministic wlks 5 / 53
65 Thnks!! Instituto de Energís Renovbles, UNAM, Temixco, Morelos, Mexico Rúl Rechtmn (IER-UNAM) Deterministic wlks 51 / 53
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