Physical insight into superdiffusive dynamics of Sinai billiard through collision statistics

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Physica A 360 (2006) 197–214 www.elsevier.com/locate/physa

Physical insight into superdiffusive dynamics of Sinai billiard through collision statistics Valery B. Koksheneva,, Eduardo Vicentinib a

Departamento de Fisica, Universidade Federal de Minas Gerais, Caixa Postal 702, 30123-970, Belo Horizonte, Minas Gerais, Brazil b Departamento de Fı´sica, Universidade Estadual do Centro, Oeste, Caixa, Postal 730, CEP 85010-990, Guarapuara, PR, Brazil Received 21 December 2004; received in revised form 20 May 2005 Available online 9 August 2005

Abstract We report on distinct steady-motion dynamic regimes in chaotic Sinai billiard (SB). A numerical study on elastic reflections from the SB boundary (square wall of length L and circle obstacle of radius R) is carried out for different R=L. The research is based on the exploration of the generalized diffusion equation and on the analysis of wall-collision and the circlecollision distributions observed at late times. The asymptotes for the diffusion coefficient DR and the corresponding diffusion exponent zR are established for all geometries. The universal (R-independent) diffusion with D1 Ht1=3 and z1 ¼ 1:5 replaces the ballistic motion regime (z0 ¼ 1) attributed to square billiard (R ¼ 0).pGeometrically, this superdiffusive regime is ffiffi bounded by small radii 0oRoR1 (R1 =L ¼ 42), when both diagonal and non-diagonal Bleher’s corridors are open in the correspondent square lattice of Lorentz gas (LG) model. The relaxation dynamics observed is ensured by the universal diffusive propagation of the regulartype and the bouncing-ball-type orbits. Within the random walk scheme, this superdiffusive regime is due to the Le´vy flights between the long-distant scatterers. With the increase in circle radius (R1 pRoR2 , R2 ¼ L=2), the diagonal corridor closes, but the arc-touching effects continue to bring the long-living bouncing-ball orbits back to the non-diagonal infinite corridors. This transient non-universal dynamics with 1:5ozR o2, also associated with the trapping of regular orbits, seems to be characteristic of non-fully hyperbolic billiard systems. In SB with finite horizon (RXR2 ), all the principal corridors are closed and the interplay between square and circle boundaries generates the known chaotic dynamics, attributed to the Corresponding author. Tel.: +55 31 3499 5681; fax: +55 31 3499 5600.

E-mail addresses: valery@fisica.ufmg.br (V.B. Kokshenev), [email protected] (E. Vicentini). 0378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.06.093

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fully hyperbolic systems. This is also observed through the normal Brownian diffusion (z2 ¼ 2) and the Gaussian statistics, proved for both kinds of collisions. r 2005 Elsevier B.V. All rights reserved. Keywords: Anomaly diffusion; Collision statistics; Chaotic billiard dynamics

1. Introduction Sinai billiard (SB) is one of a few mathematical models, which exhibits rich chaotic behavior and is studied with mathematical rigor. Its dynamic properties such as hyperbolicity, ergodicity and K-mixing, established by Sinai [1] initially in planar periodic Lorentz gas (LG), were then generalized for the physically interesting case of three dimensions [2]. The Bernoulli-scheme isomorphism, proved by Gallavotti and Ornstein [3], was shown [4] to be also valid in any dimension. A mathematical exploration of the statistical properties of LG was initiated by Bunimovich and Sinai, who applied the Markov partitions to dispersive billiards [5] and planar LG [6,7] with finite horizon. Numerical researches of billiard dynamics were mostly focused on the entropy, the Lyapunov exponents, the rate of correlation decay, and on the diffusion coefficients [8–11]. In numerical experiments carried out in Ref. [9], the particle mean square displacements, generated by circle obstacles, were considered as a matter of great importance in the establishment of the diffusion dynamics in SB. This study was carried out regardless of the particle collisions with periodic SB wall boundary, which consists of the regular (piecewise-linear) and irregular (vertexangle) parts. The latter, being related to the zero-measure singularities, nevertheless, can give rise to the vertex-splitting effects, which are shown [12] well pronounced in the late-time chaotic dynamics of billiards with polygonal tables. The study by Garrido and Gallavotti [9] motivated our numerical research [13] of the billiard statistics by taking into account the particle wall collisions. As result, the wall collision statistics of non-escape particles was investigated through their survival probability in weakly open chaotic SB [14] and non-chaotic classical planar systems, such as circle and square billiards [13]. Remarkably, both the wall-collision and obstacle-collision statistics provided new insights into the delicate mathematical problem of the interplay between regular and irregular segments, which constitute the total billiard boundary. In a more general context, this problem is ultimately related to the apparent controversy between the causality and randomness that can be seemingly solved through the improving of alternative deterministic and stochastic frameworks. In the case of the almost-integrable systems, presented by open [12] and closed [15] rational polygons, it was demonstrated that the vertex-splitting events are dual with respect to the vertexordering and vertex-disordering effects in establishing chaotic-like dynamics. It was also communicated [15] that the arc-touching effects in disk-dispersing SBs play the role of vertex-splitting effects. In the present paper we report on universal and non-universal chaotic relaxation regimes in SBs, which are due to the diffusive

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propagation of the regular and ‘‘irregular’’ orbits, respectively, controlled by underlying arc-touching mechanism. The validity of the central limit theorem for the particle displacement vectors has been mathematically proved for periodic LG with finite horizon, in two-dimensional [6,7] and multidimensional [16] cases. This ensures the Gaussian distribution for displacements and the normal (Brownian) particle diffusive dynamics in fully hyperbolic systems. Bleher extended this approach to the planar LG with infinite horizon and proposed the universal (independent of geometry) superdiffusive regime, described by late-time logarithmically diverging diffusion coefficient [17]. The same enhanced diffusive behavior was later approximated by Dahlqvist through zeta function techniques [18]. On the other hand, one can show [16] that in case of infinite horizon geometry the second moment of the underlying timescale is not finite and the application of the central limit theorem is questioned. As noted by Chernov [16], the pseudo-Gaussian approximation, proposed for the displacement distribution in Ref. [17], cannot be treated as a conclusive proof. Moreover, unlike the case of Brownian diffusion, numerical observation of particle-obstacle collisions in SBs with infinite horizon [9] did not support the analytical predictions made in Refs. [17,18]. We therefore perform new experimental tests of diffusive regimes in SBs by reformulating the problem of random particle displacements into that for random collisions. The analysis is carried out through the billiard-wall and the scatterer-disk description for collision-statistic observables introduced on the basis of the geometric and dynamic correspondence, which exists between the SB boundary and the LG lattice. The late-time stationary dynamics in SBs is studied through the diffusion exponent, which is found to be continuous with the scatterer disk radius. The paper is organized as follows. In Section 2 we provide numerical experiments on relaxation dynamics in SBs driven by the fixed boundary geometry. The variation of the latter is specified in terms of Bleher’s corridors, employed in Refs. [17,18]. Also, we introduce simplified boundary-collision descriptions developed on the physically alternative (deterministic and stochastic) approaches to the problem. As outcome, the collision distribution function is parametrized in terms of the dynamic observables available from the simulation data in SBs with different boundaries. In Section 3 we give a general discussion and draw a conclusion.

2. Billiard collision statistics 2.1. Billiard geometry Dynamic behavior of particles (of unit mass and unit velocity), moving in a twodimensional closed region (billiard table) and elastically dispersed by obstacles and billiard walls, is governed solely by initial conditions and by geometry. The SB with the square-wall table of length L and the disk with radius R can be introduced as the elementary cell of a periodic LG, which is the two-dimensional periodic crystal formed by a regular set of circular scatterers (of radius R) centered at distances L. This implies that SB and LG can be treated as dynamically equivalent classical

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systems, as far as only regular-orbit motion is concerned. This statement is based on the step-to-step correspondence that takes place between non-splitting orbits [15] in the SB and the corresponding trajectory in the LG. In the case of the LG formed by the disks with RoL=2, this correspondence is clarified in inset A in Fig. 1. Between two consequent collisions with disks, particles move freely in LG along the piecewise-linear trajectories. Additionally, there exist directions in which particles ‘‘never’’ collide with scatterers. Those particles, which move in such directions, have therefore an infinite horizon and such free-motion trajectories belong to infinite corridors [17]. As shown for the first time by Bleher [17], the principal pffiffi corridors are open for small disk radii limited by 0oRoL 42. This is illustrated by the diagonal (vertical and horizontal) and non-diagonal principal corridors in inset B in Fig. 1. Noteworthy, that not only unbounded trajectories can move in free-motion corridors. The randomly injected unit-velocity particles have uniformly distributed spatial positions [13] that provide the sets of bounded long-distance trajectories,

5 6 2

3 2

3

R

4 5

a

4

6

b L

1

(A)

(B) L

R

L

y

θ (C)

(D)

x

Fig. 1. Sinai billiard versus periodic Lorentz Gas model for different boundary geometries. A: reduction of a trajectory in the LG simple square lattice to the corresponding orbit in SB is shown through the billiard wall-to-wall and wall-to-disk pffiffi collision points, respectively, 1; 2; 3; 5 and 4; 6. B: infinite horizon geometry for the disk radii 0oRoL 42. Examples of the principal free-motion corridors by Bleher [17]: the non-diagonal (a) and diagonal (b) directions. Inset: the bouncing-ball orbit formed by the SB boundary, chosen through pffiffiffi the geometrically equivalent unit cell. C: the SB with finite horizon geometry, L=2pRoL= 2. D: reduction of the unbounded trajectory of launching angle y to the corresponding orbit in square billiard (R ¼ 0).

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which also move in the corridors. Example is the finite-time living bouncing-ball pffiffi orbit, shown in Fig. 1B. When the scatterer radius reaches the magnitude L 42, the last of the non-diagonal corridors closes. The more the radius increases, the last principal corridors disappear at L=2, when scatterers start to overlap one another and lattice cells. The particle trajectories become thus bounded having a finite horizon, as shown in inset C in Fig. 1. We recall that originally H. Lorentz introduced his model one century ago to describe an electronic gas in solid metals. According to solid state physics, crystals allow for more than a single choice of the elementary cell. The symmetrically equivalent choices of the SB boundary in the simple square lattice, shown in Figs. 1A and B, provides the dynamic correspondence between the two SBs, established through the observation of dynamic characteristics and ensured by the ergodicity property. This implies that the mathematical appearance (through the analytical description) of the modified (by the disk reflections) regular and irregular orbits will be different in the two considered cases, but their physical (observed) characteristics remain the same. The choice shown in Fig. 1B was employed in numerical studies of SBs in Refs. [9–11]. A more complex boundary was introduced through the square (of side L) and pffiffiffi two disks (of radii R and R0 ) with the distance L= 2 between their centers. Such a two-scatterer SB, which corresponds to the elementary cell of the face-centered square lattice of LG. suggests distinct cases: 0oRoL=2
R0 , pffiffiffi This pffiffiffi the following 0 0 0 L=2
R pRoL= 2
R and L= 2
R pRoL=2, corresponding, respectively, to the two-scatterer SB with infinite horizon, finite horizon, and diamond geometries (see, e.g., Fig. 1 in Ref. [11]). A crossover from the face-centered lattice to the simple square lattice in periodic planar LG, used in our case, implies that R0 ¼ 0, which maintains the definition pffiffiffi [17] of the SB infinite-horizon (0oRoL=2) and finitehorizon (L=2pRoL= 2) geometries. As seen, the diamond geometry, characterized by the non-diffusive regime in chaotic dynamics [9], cannot be achieved in the onescatterer SB. 2.2. Collision distributions A continuous-time evolution of the orbits (or trajectories) of classical particles of unit mass moving with unit velocities in SB (or in LG) preserves the Liouville measure [19], namely 1 dx dy dy , (1) 2pA where the billiard table area AðRÞ is a smooth function of the variable scatterer radius. This is introduced in the three-dimensional phase space through the coordinate set x ¼ ðx; y; yÞ, which includes the particle position (x; y) and the velocity launching angle y ¼ ½0; 2p counted of the x-axis of the square billiard table. In case of the infinite-horizon geometry, the accessible area is L2
pR2 . The billiard collision distribution function C R ðn; tÞ is a probability of a particle to collide n times with the fixed billiard boundary within a time t. Employing the dmðxÞ ¼

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asymptotic (tbtc ) ergodicity property [1], we define the observable (measurable) mean collision-number, Z 1 Z t nc ðtÞ  hnðx; tÞic ¼ nC R ðn; tÞ dn ¼ nðx; tÞ dmðxÞ ¼ , (2) tc 0 numerically tested in Ref. [9]. Here nðx; tÞ stands for the number of billiard collisions during time t, for a given particle of position set x, defined in Eq. (1). The mean collision time pA (3) P is due to the two consequent elastic random collisions with the boundary, specified by the billiard area and perimeter P [19]. Eq. (2) results in the (wall and scatterer) collision distribution   dm½xðn; tÞ , C R ðn; tÞ ¼  (4)  dn tc ðRÞ ¼

defined through the Liouville measure, given in Eq. (1) by the function m½xðn; tÞ inverse to nðx; tÞ. Noteworthy, that the collision description, formally introduced for continuous dynamic system, can be deduced from the late time behavior of the associated collision subsystem (for details see, e.g., Refs. [9,13,19]). A study of the collision distributions provides a rich information on the boundarymemory effects in chaotic SB with a fixed geometry. The distribution (4) can be described through the higher-order central moments, defined as Z 1 Dm nc ðtÞ ¼ ½n
nc ðtÞm C R ðn; tÞ dn , (5) 0

with m ¼ 2; 4; . . ., which describe the m-order deviations from the mean collision number nc ðtÞ given in Eq. (2). Basically, we focus on low-order dynamic correlation effects presented by the variance of the random collision numbers n D2 nc ðtÞ ¼

1 hD2 rit 2‘2c

for tbtc

(6)

related to the variance of particle displacements hD2 rit realized during time t, through the mean free path ‘c ¼ tc . This fundamental model relation will be deduced below using the orbit-trajectory correspondence visualized in inset A in Fig. 1. Also, Eq. (6) allows one to employ the well-known model-independent equation  2=z 2 2 t hD rit H‘c for tbtc , (7) tc which generalizes the Brownian diffusion (for discussion, see e.g. Ref. [20]). In this way, we introduce a description for distinct stationary regimes in billiards with different R, through the diffusion exponent zðRÞ. Taking into consideration that the wall and scatterer collisions are independent events, the collision distributions can be specified by the wall-collision and scatterer-collision

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statistics, introduced through the corresponding distribution functions ðsÞ C R ðn; tÞ ¼ C ðwÞ R ðn; tÞ þ C R ðn; tÞ ,

tðwÞ c

tðsÞ c .

(8)

and the mean collision and The latter can be observed through ¼ t=tðwÞ c ðsÞ ðsÞ ðwÞ ðwÞ and nc ðtÞ ¼ t=tc , with the help of Eqs. (2) and (8). Furthermore, one has tc nc ¼ ðsÞ ðwÞ ðsÞ tðsÞ c nc ¼ t that, taking into account nc ðtÞ ¼ nc þ nc and tc nc ¼ t, results in the total boundary collision frequency 1 1 1 Pw þ Ps ¼ ðwÞ þ ðsÞ ¼ . tc ðRÞ tc pA tc

ncðwÞ ðtÞ

(9)

One can see that Eq. (9) specifies the known Eq. (3) through the wall and scatterer perimeters Pw ¼ 4L and Ps ¼ 2pR, respectively. 2.3. Wall collisions in square billiard Let us consider the case of square billiard R ¼ 0. If one ignores the splitting effects caused by p=2-angle vertices [12], particles exhibit the ordered motion driven by billiard walls. This implies that the velocity launching angle y (and the normal-to-wall [12] collision angle j ¼ p=2
y) is the integral of motion for a given particle and its trajectory in the corresponding LG is a straight line (see the inset D in Fig. 1). A whole number of intersections of this line with the unit-cell boundaries, encountered in x and y directions in a time t, corresponds to the wallcollision number [21] nðwÞ 0 ðy; tÞ ¼

t ðcos y þ sin yÞ . L

(10)

In view of the point symmetry of square lattice, the angle domain is reduced to 0pypp=4. For the wall-ordered motion, the characteristic time tðwÞ c ðyÞ follows from ðwÞ nðwÞ ðy; tÞ ¼ t=t ðyÞ. This results in c 0 1 tðwÞ c ðyÞ

¼

cos y þ sin y L

(11)

obtained with the help of Eq. (10). Consequently, the mean collision number is Z A Z 4t p=4 cos y þ sin y 4t t ðwÞ nc0 ðtÞ  hn0 ðy; tÞic ¼ dy ¼ ðwÞ . dx dy ¼ (12) pA 0 L pL tc0 0 In turn, Eq. (12) defines the mean collision time tc0 , which agrees with Eq. (3), where A ¼ L2 and Pw ¼ 4L. The variance for random wall-collision numbers is  2  p p ðwÞ 2 2 þ
1 n2c0 , (13) D nc0 ðtÞ ¼ h½n0 ðy; tÞ
nc0 ðtÞ ic0 ¼ 16 8 where nc0 is presented in Eq. (12). The function yðn; tÞ ¼ 0:5 arcsin½ð4tc0 n=ptÞ2
1, inverse to nðy; tÞ given in Eq. (10), leads to the wall-collision distribution C 0ðwÞ ðn; tÞ ¼

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0.12

15

0.08

∆2nc0(t)

Collision distribution function, D0(n,t)

20

10 5

0.04

0 0

50

100

150

200

t/τc0

t=100τc 0

80

90 100 Reduced collison number, n

110

Fig. 2. Analyses of the wall-collision dynamics in square billiard. The points correspond to the simulation data on the collision function at times t ¼ 100tc0 presented for collision numbers reduced by the factor pt=4tc0 . The solid line is predicted by Eq. (14). Inset: the evolution of the root-mean-square collisionnumber deviation with time, reduced by tc0 ¼ p=4. Points are experimental data and the solid line is given in Eq. (13).

4p
1 jqyðn; tÞ=qnj defined in Eqs. (4) and (8). A straightforward estimation yields " #!  16 1 4 n 2 ðwÞ
1 p pffiffiffi sin
arcsin
1 C 0 ðn; tÞ ¼ 4 2 p nc0 p2 nc0 2 pffiffiffi for p=4on=nc0 op 2=4; otherwise C ðwÞ ð14Þ 0 ðn; tÞ ¼ 0 . One can verify that Eqs. (12), (13) and (14) are self-consistent because they satisfy Eqs. (2) and (5). As can be deduced from Eqs. (7) and (13), the dynamic diffusion exponent z0 ¼ 1, that implies that the dynamic regime in the square billiard is ballistic. This result is expected, because the only ballistic trajectories were employed to derive the square-billiard distribution (14). In Fig. 2 the wall-collision distribution, predicted in Eq. (14), is compared with that simulated [22] in square billiard at observation time tðexpÞ ¼ 100tc0 . As seen in qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the inset, the simulated standard deviation D2 nc0 ðtÞ matches well the stationary prediction (13), starting with times tðexpÞ \30tc0 . 2.4. Random walk approach 2.4.1. Infinite horizon geometry In the case of the disk radii 0oRoL=2, all particles, initially randomly distributed, are dispersed by disks and thereby are involved in a diffusive process,

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at least at late times. The latter is referred to the long-living, bouncing-ball-type orbits, corresponding to the almost free motion along the long-distance trajectories in the LG lattice, rarely disturbed by tangential touch of the disks. Estimate for collision variance. Within the framework of random walks (RWs), the mean number of collisions nc ðtÞ, which occur with the SB boundary of fixed geometry during time t, can be associated with the averaged number of random steps nx and ny made by a walker in the x-axis and y-axis positive and negative directions
in the planar SB table. More precisely, if the walker does sþ x steps to the right, sx
þ steps to the left, sy steps to the down, and sy steps to the up, we define
þ
n ¼ ðnx þ ny Þ=2, with nx ¼ sþ x þ sx and ny ¼ sy þ sy . Moreover, the collision number deviation Dn ¼ ðDnx þ Dny Þ=2 can be introduced through the corresponding
þ
random step deviations Dnx ¼ sþ x
sx and Dny ¼ sy
sy . In this way, the proposed random-walk description for the orbit in SB with nc ¼ 6 (shown in inset A in Fig. 1
þ through the five complete collisions) is specified by hsþ x it ¼ 4, hsx it ¼ 2, hsy it ¼ 6,
and by hsy it ¼ 0. Here the symbolh   it indicates the average procedure carried out over the period t ¼ 6tc . Furthermore, for the mean deviations one has hDnx it ¼ 2 and hDny it ¼ 6. As can be deduced from Fig. 1, both the deviations tend to zero in the limit t=tc ! 1. This trend, characteristic of chaotic dynamics, allows one to introduce the SB collision statistics through the late-time RW description, namely nc ¼ 2hsþ x it ;

2 þ
D2 nc ¼ hDn2 it ¼ hðsþ x Þ it
hsx sx it

for tbtc .

(15)

In this estimate all directions are treated to be dynamically equivalent in the average.
þ
    It has been therefore adopted that hsþ x sx it ¼ hsy sy it and hsx sy it ¼ hsx sy it . In the corresponding LG lattice, the random motion of a given gas-like particle is described by the model particle displacement vector pffiffiffi r ¼ nx ‘cx ex þ ny ‘cy ey with ‘cx ¼ ‘cy ¼ ‘c = 2 (16) introduced through the random steps nx and ny in the horizontal ex and vertical directions ey . We recall that in the unit-velocity system the mean RW path ‘c ¼ tc . A possible realization of rc ðtÞ ¼ hrit can be visualized through the LG trajectory (see the inset A in Fig. 1) and described through the specification of RW steps. Furthermore, we introduce the displacement-vector deviation through the RW step
þ
deviations Dnx ¼ sþ x
sx and Dny ¼ sy
sy , namely Dr ¼ Dnx ‘cx ex þ Dny ‘cy ey .

(17)

For the late times, one must expect hDrit ¼ 0 and therefore the variance hDrDrit can be obtained as 2 þ
hD2 rit ¼ 2‘2c ½hðsþ x Þ it
hsx sx it 

(18)

with the help of Eq. (17). Finally, Eqs. (15) and (18) lead to Eq. (6), which provides the desired relation  2=z t 2 D nc ðtÞH , (19) tc ðRÞ where Eq. (7) is also taken into consideration.

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Observation of collision statistics. On the basis of Eq. (19), we have elaborated a numerical statistical analysis [22] for the wall collisions in SB with infinite horizon. As shown in the upper inset in Fig. 3, the variance of collision numbers indicates two superdiffusive motion regimes, which are well distinguished through the dynamic exponent zðRÞo2. For small scatterers, the R-independent regime is manifested by the observed zðexpÞ ðRÞ ¼ 1:50  0:05, shown by the closed squares in Fig. 3. This universal regime 1 emerges drastically at Ra0, replacing the ballistic motion (indicated by the open square at R ¼ 0 in Fig. 3). Starting with the crossover radius  0:35L, the diffusion exponent continuously grows with R from 1.5 to 2. This is deduced from the experimental data, analyzed in the upper inset in Fig. 3 and shown by the open squares in Fig. 3. pffiffiAffi crossover through the transient R-dependent dynamics starts at the point R1 ¼ 2L=4 ¼ 0:354L. As discussed in the description of the inset B in Fig. 1, R1 is the point a rearrangement of the LG lattice, which results in closing of diagonal Bleher’s corridors.

2.5 1000 ∆Zh

I CR=Z

(t)

CR

R/L=0.495

Diffusion dynamic exponent, z

100

10

2

0 20

0.45 0.35 0.25 0.05 0.15

ICR = 1.49 R < L/2

t/τCR 100

300

1.5 30

∆Zh

R/L=0.525

(t)

CR

0.65 0.55 0.60

R > L/2

10 8 6 4 2 40

1 0

ZCR = 2 t/τCR 100

0.25 0.5 Scatter radius, /L

200

0.75

Fig. 3. Diffusion dynamic exponent against the reduced radius of the dispersing disk in Sinai billiard. The points are simulation data derived from the billiard-wall collisions observed through their variance (19) and the estimated characteristic time tcðwÞ ðRÞ given in Eq. (3). Insets: temporal evolution of the wallcollision-number deviation for geometries with infinite (RoL=2, upper inset) and finite (R4L=2, lower inset) horizons, in the log-log coordinates. The solid lines are the best linear fit of the simulation data. The closed squares and circles correspond to the radius-independent diffusive regimes, for which the diffusion exponent is 1:5  0:05 and 2  0:05, respectively. The transient regime is shown by open squares.

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Taking into consideration that all statistic properties have the asymptotic character, we focus on the late-time behavior and re-estimate the collision distributions in the reduced coordinates, proposed in Ref. [9], namely qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n
nc ðtÞ eR ðe C nÞ ¼ D2 nc ðtÞC R ½nðe n; tÞ; t and neðn; tÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . (20) D2 nc ðtÞ Here nðe n; tÞ stands for the inverse function to neðn; tÞ. The collision distribution C R ðn; tÞ, the mean nc ðtÞ, and the variance D2 nc ðtÞ are defined, respectively, in Eqs. (4), (2) and (19). As follows from our temporal analysis, shown in the insets in Fig. 3, the superdiffusive dynamic regime becomes steady starting with the observation times tðexpÞ \50tc . The same refers to the collision distributions exemplified for both the wall and disk collision statistics in Fig. 4. As seen in Fig. 4, both kinds of collision distributions follow similar pseudoGaussian behavior. With the increased scatterer radius, the distribution functions smoothly change their shapes from the characteristic of ordered motion (illustrated

0.6 σ

Reduced collision distribution

Wall

0.5

Scatterer

0.2

0.1

z=1.50

z=1.50 R/L = 0.150

0.4

0.1

0.01

0.3

R/L = 0.350 R/L = 0.475

-1

10

-5

-4

-3

-2

0

Gaussian

-1

0.2

0.4

R/L

0.2 0.1 0 -4

-3

-2

-1

0

1

2

3

4 -4

-3

-2

-1

0

1

2

3

4

∼ Reduced collision numbers, n

Fig. 4. Reduced collision distributions against reduced collision numbers in Sinai Billiards with infinite horizon geometry (RoL=2). The points are simulation data for the scatterer (left plot) and the wall (right plot) collisions reduced through Eq. (20). The solid lines are the Gaussian distribution pffiffiffiffiffiffi eðwÞ ðe e2 ðe nÞ ¼ ð1= 2pÞ expð
e n2 =2Þ. The dashed line is the given by C nÞ ¼ ½1Gð1Þ expð
je nj3 =2Þ, following C 1

2

3

from Eq. (24) taken at z ¼ 32. Left inset: analysis of the late-time scatterer-collision function. The points are the reduced-distribution simulation data for the case R ¼ 0:25L and the observation time t ¼ 200tcðwÞ , given in the semi-log coordinates. The solid line is the best fit with the prediction given in Eq. (22), with eðsÞ ðe nÞ ¼ 0:5je nj
11=3 . The arrow indicates the analyzed experimental points. Right inset: the collision-angle C 1

dispersive parameter s against reduced scatterer radius. The points are the numerical asymptotic data [29] observed in the weakly open SB. The dashed line is the theoretical prediction (26) and the solid line is that ¼ 0:16. presented in Eq. (27), when normalized by sðexpÞ 0

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in Fig. 2) to that, which tends to the Gaussian form, shown by solid lines in Fig. 4. Furthermore, according to our simulation data, carried out within the domain 0:05oR=Lo0:35, the late-time distribution functions fall in the coinciding curves eðexpÞ ðe nÞ, attributed to the universal superdiffusive regime established by the C 1 exponent z1 ðRÞ ¼ 1:50  0:05 and ensured by the open diagonal corridors. This observation makes plausible to adopt that the mean collision time (9) is caused ðsÞ mostly by the wall mirror reflections. This means that tc1  tðwÞ c1 otc1 and thus ðsÞ nðwÞ c1 4nc1 , when 0oRoR1 . Within the continuous time RW scheme, the particle motion along the open corridors, rarely and slightly dispersed by the touching of disks, can be associated with Le´vy flights, which connect two regular orbits. For the waiting-time density probability of the flight between scatterers of distance rs , we therefore employ the function [23] Cðrs ; tÞ ¼ Lðrs Þdðrs
tÞ with Lðrs Þ / rsð2=zÞ
5 ;

1ozo2

(21)

given through long-distance asymptote (rs b‘c ) of the flight-length distribution function Lðrs Þ and the delta function. Eq. (21) represents Eq. (33) in Ref. [23], obtained with the help of Eq. (39) in Ref. [23] juxtaposed with Eq. (7). This provides the corresponding scatterer-collision distribution function, C ðsÞ 1 ðnÞ ¼ L½rs ðnÞ

drs 1 / dn n5
2=z

with rs ðnÞHn‘c ,

(22)

introduced for the trajectories of p almost free ffiffiffi pffiffiffi motion in the horizontal (rx  rs ), vertical (ry  rs ), and, diagonal (rx = 2  ry = 2  rs ) corridors. The later relation in Eq. (22) follows from Eq. (16), adapted for such kind of trajectories. Treating the diffusion exponent z in Eq. (22) as adjustable parameter, best fit of the simulated scatterer-collision distribution (shown in the left inset in Fig. 4) results in zðexpÞ ¼ 1:5, 1 which coincides with that, obtained through the variance-temporal analysis (shown in Fig. 3) of the same data. Unlike the case of the almost collisionless motion, revealed in SB through the algebraic decay of the propagator function (21), one could also expect a pure exponential decay for all correlation functions, in the frequent-collision regimes (nbnc ). This statement follows from the general mixing property attributed to smooth hyperbolic systems (for further discussion, see Ref. [9, p. 553]). In fact, the SB with infinite horizon is not fully hyperbolic system (see e.g. Refs. [24,25]) and the stretched-exponential (pseudo-Gaussian) behavior for C ðexpÞ ðnÞ is observed in Fig. 4 1 for both kind of collisions. In Ref. [26], Zumofen, Klafter and Blumen re-examined, analytically and numerically, the known theoretical schemes developed for description of restrictive diffusive processes. As a result, the model-independent behavior has been established

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for the asymptote of the propagator distribution function [27] 2 !z=ðz
1Þ 3 r 5 for tbtc W ðr; tÞ / exp4
1=2 hD2 rit

209

(23)

which is the probability for a particle to have displacement r at time t; the displacement variance hD2 rit is described in Eq. (7). As seen in Eq. (23), the pure exponential behavior occurs only in the case of z ¼ 2, corresponding to the Brownian motion regime. The relations, established in Eqs. (16) and (6) enable one to estimate the collision distributions C 1 ðnÞ through W ðr; tÞ. Similar to Eq. (22), we use Eq. (23) in the case of the wall-collision statistics. This results in pseudo-Gaussian distribution f1 ðwÞ ðe C nÞ ¼

h z expð
je njh Þ with h ¼ z
1 2Gðh
1 Þ

(24)

given in the reduced and normalized form, obtained through Eq. (20) and explicit through the gamma function GðxÞ. For seeking of simplicity, we have interpolated symmetrically the high-intensity collision (nbnðwÞ c ) distribution to the domain nonðwÞ c . A fitting analysis of the prediction (24), taken at z ¼ 1:5, with the corresponding experimental data in SB with RoR1 is shown by dashed line in the right plot in Fig. 4. As can be inferred, the universal superdiffusive motion, exposed by the wall-collision distribution (24) for nXnðwÞ c , is not sensitive to details of the underlying dispersive effects. One may therefore expect that the same universal behavior can be described within different coarse-grained schemes. In Ref. [14], the late-time chaotic behavior in weakly open SBs was considered through the geometrical conditions stipulated a surviving of the piecewise linear trajectories. The wall-collision statistics was discussed in terms of the pseudoGaussian distribution function gðwÞ s ðvÞ, introduced for the variable v ¼ v? =v ¼ cos j through the normal-to-wall velocity component v? , described by the particle wallcollision angle j. At a fixed geometry, the late-time motion is controlled by the collision-angle dispersive parameter sðRÞ. The observation of the random wallcollision numbers nðvÞ ¼ 2vt=tðwÞ c , parametrized by the observation time t, was ensured by Eq. (2), which in this case reads as [14] Z 1 t ncs ðtÞ ¼ nðvÞgs ðvÞ dv ¼ ðwÞ and tc 0 1 exp½
ðv
ð1=2ÞÞ2 =2s2  p ffiffiffiffiffiffi pffiffiffi gðwÞ ðvÞ ¼ . ð25Þ s s 2p erfð1=2 2sÞ Here erfðxÞ is the standard error function. Also, the two model estimates were proposed for the dispersive parameter, namely pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1
2R=L L sðRÞ ¼ pffiffiffi (26) for RoR2 ¼ 2 12 5ðR2 =LÞ2

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obtained [14] in the simple the ‘‘weak-dispersion’’ approximation (s51) and [29] vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 1 þ ð1
2R=LÞ2 u arcsin ð1
2R=LÞ= t4 sðRÞ ¼ s0 p 1
pðR=LÞ2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1
2R=L  s0 , ð27Þ 1
pðR=LÞ2 deduced for the long-living bouncing ball orbits with the suggested s0 ¼ 0:29, numerically established in the square-billiard [29]. Similar to Eq. (22), we have found ðwÞ ðwÞ the collision distribution C ðwÞ s ðnÞ, using gs ðvÞ and the relation nðvÞ ¼ 2vt=tc , in order to fit the experimental data, shown in the right plot in Fig. 4, treating sðRÞ as free parameter. Numerical results of this analysis are displayed by points in the right inset in Fig. 4, where the theoretical predictions are shown by lines. The dashed line is drawn through Eq. (26) and the solid line is its improved version suggested in Eq. (27) and additionally adjusted through sðexpÞ ¼ 0:16. Remarkably, similar to the 0 observed diffusion exponent zðexpÞ ðRÞ, the late-time collision-angle dispersion is also 1 stabilized. The latter is characterized by the almost constant behavior of sðexpÞ ðRÞ ¼ 0:14  0:02, derived for the universal superdiffusive regime limited by 1 RoR1 . The transient diffusive regime, observed in Fig. 4 for R1 oRoL=2, exhibits almost Gaussian collision-distribution behavior. 2.4.2. Finite horizon geometry In the case of the SB with RXL=2, when all Bleher’s corridors are closed (see inset C in Fig. 1), the validity of the central limit theorem for random-walk displacements rðtÞ proves [6] their Gaussian distribution. Taking into consideration the established in Eq. (6) relation between the random displacements and collision numbers, one therefore introduces the correspondent normal collision distribution   1 n
nc ðtÞ2 C 2 ðn; tÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp
for RXL=2 , (28) 2D2 nc ðtÞ 2pD2 nc ðtÞ where nc and D2 nc are the standard mean and deviation of collision numbers. eðsÞ ðe Experimental justification of the asymptotic Gaussian distributions C 2 nÞ ¼ pffiffiffiffiffiffi ðwÞ 2 e C 2 ðe nÞ ¼ ð1= 2pÞ expð
e n =2Þ, normalized in the reduced coordinates (20) for the scatterer and wall collisions, are displayed in the left and right plots in Fig. 5, respectively. eðsÞ ðe In general, our data on C 2 nÞ are consistent with the first observation of the normal scatterer-collision distribution reported in Ref. [9]. In addition, our shorttime analysis provides evidence that the Gaussian distribution, attributed to fully hyperbolic billiard systems, becomes steady at observation times tðexpÞ \50tc . Nevertheless, no true Gaussian distribution was expected [17] for the late-time diffusion coefficient, when R tends to L=2 from below. In this context, the guide-eye fit is not a conclusive proof for the true Gaussian behavior. Bearing in mind to

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211 3.8

0.4

Scatterer

Reduced distribution

3.8

0.3

R/L = 0.525 R/L = 0.55

ζ4

3.6 3.4

3.6

Wall

R/L = 0.60

3.4

R/L = 0.65

3.2 3

3.2

0.2

2.8 0.4

3 2.8 0.4

ζ4

R/L

R/L 0.5

0.6

0.5 0.6

0.1

0.0 -4

-2

0

2 4 -4 -2 ~ Reduced collision numbers, n

0

2

4

Fig. 5. Observation of the Gaussian collision statistics in Sinai Billiard with R4L=2. The points are simulation data for the scatterer (left plot) and the wall (right plot) collisions observed at t ¼ 200tc for distinct geometries indicated in the legend. The solid lines are the same as in Fig. 4. In the left and right inserts, the analysis at t ¼ 200tc for the fourth moment (29) is shown, respectively, for the scatterer and wall collisions. The points are simulation data and the dashed line corresponds to z4 ¼ 3.

clarify this problem, we have analyzed the reduced fourth-order moment z4 ðRÞ ¼

D4 nc ðtÞ D2 nc ðtÞ

for

t b1 tc

(29)

which is defined in Eq. (5). The true Gaussian distribution prescripts z4 ¼ 3. This can be compared with the result of our temporal analysis zðexpÞ ¼ 3:00  0:01, carried out 4 for the case of R=L ¼ 0:6 and observation times tðexpÞ \70tc . The approaching towards the Gaussian distribution, by increasing the parameter R=L, is shown in the insets in Fig. 5. One can see that the ‘‘true’’ chaotic motion in the SB with R4L=2 is established by both the normal collision and diffusion. These standard universal regimes are observed with good precision, established, respectively, by the central moment zðexpÞ ¼ 3:00  0:04 (when R ! L=2 þ 0) and by the diffusion exponent 4 zðexpÞ ¼ 2:0  0:05 (when R ! L=2
0). 2

3. Discussion and conclusion We have conducted a numerical study [22] on the diffusive dynamics in chaotic SB, corresponding to the LG model with simple square lattice. At a fixed geometry, the distinct motion regimes, established by elastic reflections of particles from the square walls and the circle dispersive disk, are found to be stabilized at observation times starting with 30–50tc . The steady-motion collision distribution functions are studied through the lowest central moments, which in both wall-collision

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and the scatterer-collision statistics exhibit similar temporal and spatial behavior (see Figs. 4 and 5). Exploration of the generalized diffusion equation for the asymptote of the variance of particle displacements (7), provides new insights into the late-time diffusive dynamics, established through the observation of the variance for particle collisions D2 nc ðtÞ and analyzed through the diffusion exponent zðRÞ and coefficient DR ðtÞHD2 nc ðtÞ=t. The main outcomes of our study may be summarized as follows. When the scatterer is absent (R ¼ 0), the collision statistics, corresponding to the ballistic-motion (z0 ¼ 1), is introduced through the deterministic orbit description developed for square billiard. For regular orbits, restricted by the singular (vertex-tovertex) diagonal orbits, the wall-collision distribution C ðwÞ 0 is almost flat (see Fig. 2). Its asymmetry is due to a sharp contribution from the bouncing-ball orbits, characterized by small wall-collision angles j5p=4. In SBs with finite, but small ðsÞ scatterer radii, the wall collisions predominate (nðwÞ c bnc ), [30] that preserves the introduced classification of long-living regular and ‘‘irregular’’ orbits, dispersed by small disk. Indeed, the wall-collision distribution C 1 (see the right plot in Fig. 4) remains to be asymmetrically shifted by the bouncing-ball orbits, which lifetime is now limited by the two consequent collisions with the disk (see Fig. 1B). One can see that the longest-time living orbits, having the smallest angles j, can provide specific surviving conditions, which are ultimately related to the singular effects, generated by tangential touch of the pffiffi dispersing disk. Moreover, in SBs with the geometry limited by 0oRoR1 ¼ L 42, all the principal free-motion corridors remain open in the corresponding LG, that enables the survived trajectories to evolve freely in the both non-equivalent diagonal and non-diagonal directions (shown in Fig. 1B). It is evident that the regular and the bouncing-ball orbits diffuse along Bleher’s diagonal and non-diagonal corridors, respectively. The stabilization of the late-time diffusive motion is revealed through the universal asymptotes known for the mean-square particle displacement (7) and the particle propagator function (23), both observed with zðexpÞ ðRÞ ¼ 1:50  0:05; and through the collision-angle dispersive parameter 1 sðexpÞ ðRÞ ¼ 0:14  0:02. The almost-constant behavior of the latter suggests the 1 underlying dynamic mechanism, providing specific conditions for maintaining the surviving orbits at the same corridor or for switching them into another non-equivalent corridor. This mechanism might be considered in terms of the arc-touching events or the coherent disk-dispersive conditions, respectively, similarly to those described in Ref. [12,15] for the formation of the sliding orbits in rational polygonal billiards, which are marginal regular orbits. This suggestion is corroborated by our observation of the Le´vy flights, which rarely occur between the long-distant scatterers in both kinds of corridors (shown in the left inset in Fig. 4). The description of singular orbits, produced by the vertex splitting, is one of the major problems in the non-dispersive billiard dynamics [31]. The singular arcsplitting effects, related to the high-order divergency of the underlying timescale, are presumably countable, but they cannot be accounted for within the standard Markov partition scheme (see also the note in Ref. [10]). By virtue of this fact, instead of the Gaussian-type modification [17] of the standard scheme [6,7], the

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213

analysis proposed is based on the generalized diffusion (7), which manifests the dynamic relaxation of non-Makovian time evolution [20]. Moreover, the same equation signals the breakdown of the standard central limit theorem, which, in the case of the anomalous diffusion, must be substituted by the generalized central-limit theorem by Le´vy–Gnedenko [20]. As result, in SBs with the ‘‘infinite horizon’’ geometry limited by RoR1 , the space-time correlations are found to increase as 2=3 hD2 r1 i1=2 , that corresponds to the diffusion coefficient D1 Ht1=3 . It is worth c t noticing that the same superdiffusive dynamics, with z ¼ 32, was established in the three-dimensional Coulomb gas and the XY models [32]. On the other hand, the asymptotic diffusion coefficient DH ln t, discussed in Refs. [17,18], seems to be common in the velocity-constant dynamic systems [33], extended by (the nonstandard mapping of Le´vy walks in) Josephson junctions [34]. In view of these known results, the question on specific conditions of observation of the logarithmic and algebraic diffusion coefficient remains a challenge in the mathematical statistical billiard theory. As follows from our numerical study, the universal character of relaxation terminates, when the ‘‘infinite horizon’’ in SB is limited by R1 pRoR2 ¼ L=2. The principal diagonal corridor is closed, some of the long-living regular orbits become trapped but others survive, being safely switched to the non-diagonal corridors. The late-time diffusive dynamics of the transient bouncing ball orbits becomes strongly geometrically dependent. This is observed through the enhanced diffusion coefficient DR t2=z
1 and the transient exponent zðRÞ, with 32ozðRÞo2. The collision statistics of both types the orbits is almost-Gaussian. In part, the collision-statistic findings presented are supported by the analytical dynamic theory. Indeed, the dynamic transient superdiffusive dynamics, associated with the trapping of regular orbits was rationalized through their parabolic character [24,25] as well as through the almost Gaussian behavior of their propagation (following from Eq. (23) with zt2) in layered media [35]. In addition, if the long-living chaotic excitations (originated from the coherently dispersed bouncing-ball orbits) exist, they could be experimentally tested through their survival probability simulated in the open SB, similarly to the chaotic-like sliding orbits observed in the open rational polygons [12]. pffiffiffi In the finite horizon SB (R2 pRoL= 2), all free-motion corridors are closed and the particle-correlation length diminishes. Physically, the ordinary diffusive dynamics is recovered due to the loss of correlations between two successive RW steps. This well-known statistical behavior is established in the fully hyperbolic billiards [6,7,16]. Our study on the SB chaotic dynamics indicates the normal ðsÞ collisions in both statistics with nðwÞ c  nc (see Fig. 5) and the normal diffusion, observed through the exponent z2 ðRÞ ¼ 2  0:05. Furthermore, the earlier experimental data on the normal disk-collision [9] and wall-collision [15] distribution C 2 are improved through the analysis of the fourth-order correlations (see insets in Fig. 5). To conclude, besides the physically trivial case of the normal relaxation, our study suggests new distinct superdiffusive dynamic regimes for the chaotic nonfully hyperbolic billiards, caused by arc-touching orbit singularities, which need further analytical investigations.

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Acknowledgements The authors are grateful to Giovanni Gallavotti and to the referee for their critical comments on earlier versions of the manuscript. Financial support by CNPq is also acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]

[28] [29] [30] [31] [32] [33] [34] [35]

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