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The maximum number of highly localized Lyapunov vectors at low density
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Physica A 375 (2007) 563–570 www.elsevier.com/locate/physa
The maximum number of highly localized Lyapunov vectors at low density Tooru Taniguchi, Gary P. Morriss School of Physics, University of New South Wales, Sydney, New South Wales 2052, Australia Received 16 November 2005; received in revised form 2 August 2006 Available online 13 October 2006
Abstract The localization spectra, which describes the magnitude of the localization of the Lyapunov vectors in many-particle systems, exhibit a characteristic bending behavior at low density. It is shown that this behavior is due to a restriction on the maximum number of the most localized Lyapunov vectors determined by the system configuration and mutual orthogonality. For a quasi-one-dimensional system, using a randomly distributed brick model, this leads to a predicted bending point at nc 0:432N for an N particle system. Numerical evidence is presented that confirms this predicted bending point as a function of the number of particles N. r 2006 Elsevier B.V. All rights reserved. Keywords: Lyapunov vector; Localization; Large Lyapunov exponents; Low density; Many-particle system; Quasi-one-dimensional system
1. Introduction The Lyapunov spectrum is an indicator of dynamical instability in the phase space of many-particle systems. It is introduced as the sorted set flðnÞ gn , lð1Þ Xlð2Þ X of Lyapunov exponents lðnÞ , which give the exponential rates of expansion or contraction of the distance between nearby trajectories (Lyapunov vector) and is defined for each orthogonal direction of the phase space. In Hamiltonian systems and some thermostated systems, a symmetric structure of the Lyapunov spectra, the so-called the conjugate pairing rule, is observed [1–4]. One of the most significant points of the Lyapunov spectrum is that each Lyapunov exponent indicates a time scale given by the inverse of the Lyapunov exponent so we can consider the Lyapunov spectrum as a spectrum of time-scales. The smallest positive Lyapunov exponent region of the spectrum is dominated by macroscopic time and length scale behavior, and here some delocalized mode-like structures (the Lyapunov modes) have been observed in the Lyapunov vectors [5–16]. On the other hand, the largest Lyapunov exponent region of Lyapunov spectrum is dominated by short time scale behavior, and in this region the Lyapunov vectors are localized (Lyapunov localization). The position of the localized region of Lyapunov vectors moves as a function of time [17]. A variety of systems with many degrees of freedom show Lyapunov localization, for Corresponding author. Tel.: +61 2 93855667; fax: +61 2 93856060.
E-mail address: [email protected] (T. Taniguchi). 0378-4371/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2006.09.017
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example, the Kuramoto–Sivashinsky model [18], a random matrix model [19], map systems [20–23], coupled non-linear oscillators [24], and many-disk systems [25,26]. The magnitude of Lyapunov localization can be measured quantitatively by the Lyapunov localization spectrum, which is defined as a set of exponential functions of entropy-like quantities for the normalized amplitudes of the Lyapunov vector particle components [26]. It is shown in Ref. [26] that at low density the Lyapunov localization spectra show a critical bending point, which limits the number of strongly localized Lyapunov vectors. The present study addresses a question that was explored but unresolved in a previous paper [26], that is, how many strongly localized Lyapunov vectors are present in the low density limit, in other words, what kind of physical characteristics of systems determine the critical bending point of Lyapunov localization spectra? While it is clear from the study of the clock model [17,27,28] that the Lyapunov vector associated with the largest Lyapunov exponent is strongly localized, and that this arises from the cumulative effect of collisions, the number of such localized Lyapunov vectors was unknown. Here we give the answer to that question for a quasi-one-dimensional system, using a simple model distributing bricks randomly in a one-dimensional region. While the mechanism of cumulative collisions remains in general, constructing an equivalent brick model for more general situations is technically difficult. To calculate Lyapunov vectors shown in this paper we use the numerical algorithm developed by Benettin et al. [30,31] and Shimada and Nagashima [32], as characterized by intermittent reorthogonalization and renormalization of Lyapunov vectors. In this algorithm, Lyapunov vectors are calculated as follows. First, the rate of growth of the difference between two nearby phase space trajectories is calculated to give the largest Lyapunov exponent and the corresponding (normalized) Lyapunov vector. Second, eliminating the direction defined by the Lyapunov vector for the largest Lyapunov exponent, we reduce the dimension of the phase space by one. In that reduced phase space we can again define a direction that corresponds to the next largest Lyapunov exponent and its associated Lyapunov vector. Continuing this procedure we define the same number of Lyapunov exponents and the corresponding Lyapunov vectors as there are independent directions in phase space. While the Lyapunov vectors associated with the largest exponents (in absolute value) are strongly localized, the vectors associated with the smallest exponents (in absolute value) have delocalized Lyapunov vectors, referred to as Lyapunov modes. The Lyapunov vectors, as defined here, have already been investigated in many chaotic systems with many degrees of freedom [5,6,9,12–16,26]. 2. Lyapunov localization spectra at low density The system, which we consider in this paper, is quasi-one-dimensional in which the radius of the particles R ¼ 1, the mass of each particle m ¼ 1, the total energy of system E ¼ N with the number N of disks, and the system width Ly ¼ 2Rð1 þ 106 Þ and length Lx ¼ NLy ð1 þ dÞ with parameter d, related to the density by r ¼ pR2 =½ð1 þ dÞL2y . Note that in this system the particles remain ordered because the vertical size Ly prohibits the exchange of particles [9,13,26], justifying the name ‘‘quasi-one-dimensional system’’. A schematic illustration of quasi-one-dimensional system is in Fig. 1. We use periodic boundary conditions in both the longitudinal and the transverse direction. (Quasi-one-dimensional systems with different boundary conditions are considered in Refs. [9,26].) We introduce the normalized Lyapunov vector for the nth Lyapunov exponent lðnÞ as dCðnÞ ðnÞ ðnÞ ðnÞ ðdCðnÞ is the contribution of the jth particle, and we construct the normalized 1 ; dC2 ; . . . ; dCN Þ where dCj ðnÞ 2 ðnÞ Lyapunov vector component amplitude for each particle j as gðnÞ j jdCj j . The normalization of dC PN ðnÞ ðnÞ imposes the condition j¼1 gj ¼ 1 for gj . Recently, a quantity to measure the strength of Lyapunov
1
2
N
Fig. 1. A quasi-one-dimensional hard-disk system with the disk radius R, the length Lx and the width Ly satisfying the inequality 2RoLy o4R. In this system, the disks remains in the same order and they can be numbered 1; 2; . . . ; N from the left to right.
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localization for each Lyapunov exponent was proposed [26]. The localization of the nth Lyapunov vector is " # N X ðnÞ ðnÞ ðnÞ hgj ln gj i . (1) W exp j¼1
P ðnÞ ðnÞ ðnÞ The bracket h i in Eq. (1) indicates the time-average. The quantity N j¼1 hgj ln gj i in the definition of W ðnÞ can be regarded as an entropy-like quantity [21,22,29] as gj is a distribution function over the particle index j. The quantity WðnÞ satisfies the inequality 1pWðnÞ pN, and can be interpreted as the effective number of particles contributing strongly to the normalized Lyapunov vector. A Hamiltonian system satisfies the conjugate relation WðDjþ1Þ ¼ WðjÞ for any j with the phase space dimension D [26], because of the symplectic structure. The set of quantities fWðnÞ gn which we call the Lyapunov localization spectrum, has been calculated previously in many-particle systems with hard-core interactions [26] and soft-core interactions [14]. In these systems, the value of WðnÞ usually increases with the Lyapunov index n, and this implies that Lyapunov vectors for the largest exponents are the most localized. The quantity WðnÞ has a minimum value of 2, when only two particles contribute strongly to the value of WðnÞ , and this value for WðnÞ is observed numerically for a number of the largest Lyapunov exponents in the limit as the density approaches zero [26]. This suggests that there is a maximum number of Lyapunov vectors of the type shown in Fig. 2 for which only two particles contribute strongly. Fig. 2 is a graph of gð1Þ j as a function of the collision number nt and the particle index j. As shown in Fig. 2, the localized region of Lyapunov vector moves with time [17], but the localized region is almost always concentrated on two nearest-neighbor disks. It was also shown that WðnÞ can detect not only the localized behavior of Lyapunov vectors, but also the de-localized behavior observed in the Lyapunov modes [26]. One of the important characteristics of the Lyapunov localization spectrum is the bending behavior at low density [26]. In Fig. 3 we show an example of such bending behavior [33] in a quasi-one-dimensional system of 50 hard disks. In Fig. 3, five different Lyapunov localization spectra are represented at different densities. In the low density limit, this figure shows a bending point in the Lyapunov localization spectra at nc . At this bending point n nc the Lyapunov localization spectra WðnÞ changes from a linear dependence upon the Lyapunov index (n) to an exponential dependence. This bending becomes sharper at lower density, and the numerical results suggest that in the low density limit the Lyapunov localization spectrum converges to the solid line given by ( 2 in nonc ; ðnÞ W ¼ (2) g aN expfbn=ð2NÞg in nXnc ;
0.5 0.4
γ j(1)
0.3 0.2 0.1 0 449100 449150 449200
nt
449250
0
45 50 35 40 30 25 15 20 5 10 j
Fig. 2. The normalized amplitude gð1Þ j of the Lyapunov vector particle component corresponding to the largest Lyapunov exponent as a function the collision number nt and the particle index j. The system is a quasi-one-dimensional hard-disk system with d ¼ 105 and N ¼ 50, and particle indices are 1; 2; . . . ; N labeled from left to right.
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0.6
0.5
0.3
(
(n)-
2) / N
0.4
0.2
d= 1 d= 10 d = 100 d = 500 d = 10000
0.1 nc/(2N) 0 0
0.1
0.2
0.3
0.4
0.5
0.6
n / (2N) ðnÞ
Fig. 3. The normalized Lyapunov localization spectra fðW 2Þ=Ngn as functions of the normalized Lyapunov index n=ð2NÞ for a quasione-dimensional 50 hard-disk system at different densities given by d ¼ 1 (diamonds), d ¼ 10 (circles), d ¼ 100 (triangles), d ¼ 500 (inverted triangles), and d ¼ 10000 (squares). The parameter d is inversely proportional to the density as r ¼ pR2 =½ð1 þ dÞL2y . The smallest Lyapunov index corresponds to the largest Lyapunov exponent. The solid line is the asymptotic form of the Lyapunov localization spectrum as the density approaches zero, and n nc is the critical bending point.
with constants a, b, and g 2 þ aN expfbnc =ð2NÞg [so that WðnÞ jn¼nc ¼ 2 in Eq. (2)]. In other words, we can estimate the critical value n nc defined as the bending point of the Lyapunov localization spectrum by the value of the fitting parameter nc in the fit of the Lyapunov localization spectrum to the function (2). The bending behavior of Lyapunov localization spectra, shown in Fig. 3 is associated with a similar bending point in the Lyapunov exponent spectra. Moreover, the existence of a linear dependence of WðnÞ on n in the region npnc , at low density, is connected with some other known kinetic properties, for example, that the mean free time is inversely proportional to density, and the Krylov relation that the largest Lyapunov exponent lð1Þ depends on the density r like lð1Þ r ln r [26]. These results suggest that the existence of the linear dependence of WðnÞ is connected to the density range where kinetic theory provides an accurate description. These points were investigated in detail in quasi-one-dimensional systems, although the bending behavior of the Lyapunov localization spectra is also observed in fully two-dimensional square systems [26]. However, no mechanism was proposed to explain this bending behavior. In this paper, we construct a mechanism that leads to the bending behavior observed in the Lyapunov localization spectra, and predicts the critical value nc for quasi-one-dimensional systems.
3. Randomly distributed brick model To explain the maximum number of the most localized Lyapunov vectors nc , and thus the bending behavior of the Lyapunov localization spectrum at low density, we begin with some properties of the Lyapunov vectors of many-particle systems. The first property expresses the fact that different Lyapunov vectors sample independent directions in phase space. [I] Lyapunov vectors with different Lyapunov indexes are orthogonal [31]. The second property of Lyapunov vectors is based on the numerical observation that particle interactions occur between a minimum of two different particles, so for quasi-one-dimensional systems Wð1Þ ! 2 as r ! 0 as shown in Fig. 3 and Ref. [26]. [IIA] In the low density limit, for hard particles, all the normalized Lyapunov vectors in the linear region 1pnonc have strong contributions from only two particles.
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The third property is justified only for quasi-one-dimensional systems, and restricts the property [IIA] further to [IIB] The two particles, whose normalized Lyapunov vector components are non-zero values in the low density limit, are nearest-neighbors. In Ref. [26] it is observed that in the largest Lyapunov exponent region, at low density, the two strong contributions to the normalized Lyapunov vector appear due to a collision. Moreover, in quasi-onedimensional systems, particle collisions occur only between nearest neighbor particles. Therefore, property [IIB] is justified. Using the properties [IIA] and [IIB], in the low density limit, all the normalized Lyapunov vectors dCðnÞ in the largest Lyapunov exponent region 1pnonc can be well approximated by ðnÞ dCðnÞ ð0; 0; . . . ; 0; dCðnÞ mn ; dCmn þ1 ; 0; . . . ; 0Þ,
(3)
where 0 is the null vector. In general, the particle number mn depends on the Lyapunov index n and on the time. Here, the particles in the quasi-one-dimensional system are numbered from left to right, and mn and mn þ 1 are nearest neighbor particles. (For periodic boundary conditions in the longitudinal direction particles N and 1 are nearest neighbors.) Now we consider the restriction imposed by condition [I] on normalized Lyapunov vectors of the form (3). To ensure that condition [I] is satisfied, it is sufficient that the particle numbers mn and mn þ 1 in one normalized Lyapunov vector are distinct from those of the other Lyapunov vector mn0 and mn0 þ 1. This ensures that the two Lyapunov vectors are orthogonal regardless of the details of the various components for particles mn and mn þ 1 in the first vector, and mn0 and mn0 þ 1 in the second. This further restricts the maximum number of independent normalized Lyapunov vectors of the form (3), and puts an upper limit on the critical value nc . This situation is explained using a simple model whose schematic illustration is given in Fig. 4. In this randomly distributed brick model, N boxes corresponding to each of the particles are arranged on a line. Each normalized Lyapunov vector represented by Eq. (3) must have strong contributions for only two particles ðnÞ dCðnÞ mn and dCmn þ1 . These are shown as a gray-filled rectangular brick filling boxes mn and ðmn þ 1Þ in Fig. 4. To satisfy condition [I], these bricks must not overlap. The critical value nc is then the average number of bricks which can be put without overlaps on the N different boxes. Notice that there is no accumulation in this model. Bricks are dropped until no further adjacent vacancies remain. There is one more important point to obtain an explicit value of nc from the above mechanism: [III] The particle number mn in Eq. (3) is chosen randomly with respect to the Lyapunov index nðonc Þ, so that there is no overlap among the non-zero Lyapunov vector components for normalized Lyapunov vectors with different Lyapunov indices. This means that the bricks shown in Fig. 4 must be randomly distributed without overlaps. Therefore, randomly constructed configurations with any number of single empty boxes at non-neighboring positions are possible. Taking an ensemble average of the values of n~ c for each possible configuration we obtain the critical value nc . Fig. 5 shows the normalized critical value nc =ð2NÞ from such a numerical simulation for different numbers of particles N. The data in this figure shows that nc =ð2NÞ is independent of N, giving an estimate of the normalized critical value as nc 0:216. 2N
(4) 1
2
3
4
5
N-5 N-4 N-3 N-2 N-1
6
N
... μn μn +1
μn' μ n' +1
Fig. 4. A schematic illustration of the randomly distributed brick model, used to explain the maximum number of independent normalized Lyapunov vectors with only two dominant particle contributions. The boxes numbered j correspond to the jth particle component of the Lyapunov vector. Gray-filled rectangular bricks occupying boxes mn and mn þ 1 represent the dominant contributions of possible normalized Lyapunov vectors for which nonc . The value nc is the average number of bricks that can be randomly arranged on the set of boxes such that no pair of adjoining spaces remain.
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0.3
0.25
nc / (2N)
0.2
0.15
0.1
0.05
0 40
50
60
70
80
90
100
N Fig. 5. The normalized critical value nc =ð2NÞ given by the randomly distributed brick model as a function of the number of particles N. The solid line is a fit of the numerical data to a constant function nc =ð2NÞ ¼ x where the parameter x 0:216.
Table 1 The values of the fitting parameters a and b given in Eq. (2) for Fig. 6 N
a
b
40 50 60 70
2:83 0:04 2:51 0:02 2:28 0:01 2:10 0:02
7:83 0:02 7:24 0:01 6:08 0:00 6:39 0:01
The accuracy of this value can be used as a check of the proposed mechanism for the bending behavior of Lyapunov localization spectra at low density. To draw the solid line shown in Fig. 3 we have already used the parameters a and b in Eq. (2) given in Table 1 for N ¼ 50. The result shown in Fig. 3, using the critical value (4) gives a very satisfactory fit. In order to check that the result (4) is satisfied for any number of particles N, we present Fig. 6. It is the ½n=ð2NÞ-dependence of ½WðnÞ 2=N in quasi-one-dimensional systems of different sizes at the same density. Notice that values of ½WðnÞ 2=N themselves decrease slightly as the number of particles increases, meaning that the fitting parameters a and b in Eq. (2) depend only slightly on N, as shown in Table 1. However, all the data for different numbers of particles in Fig. 6 are nicely fitted using the same critical value nc =ð2NÞ, given by Eq. (4). 4. Conclusion and remarks To conclude, we have shown that a model of randomly distributed bricks on a line (Fig. 4) can predict the maximum number of Lyapunov vectors which have a Lyapunov localization WðnÞ ¼ 2, in the low density limit. These are the Lyapunov vectors for which nonc in the linear region that lead to the bending behavior of Lyapunov localization spectra. We showed that this behavior comes from a restriction on the maximum number of the most localized Lyapunov vectors with dominant components for only two particles. The randomly distributed brick model was applied to quasi-one-dimensional systems, giving the specific value nc =ð2NÞ 0:216 for the critical value nc of the Lyapunov localization spectra for any number of particles N. Numerical simulations give a value of nc =ð2NÞ 0:219 0:005. Our explanation for the bending behavior is
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0.5 0.45 0.4
2) / N
0.3
(n)-
0.25
(
0.35
0.2 0.15
N = 40 N = 50 N = 60 N = 70
0.1 0.05
nc /(2N)
0 0
0.1
0.2
0.3
0.4
0.5
0.6
n / (2N) ðnÞ
Fig. 6. The normalized Lyapunov localization spectra fðW 2Þ=Ngn as functions of the normalized Lyapunov index n=ð2NÞ in quasione-dimensional systems of different size; N ¼ 40 (diamonds), N ¼ 50 (triangles), N ¼ 60 (inverted triangles), and N ¼ 70 (squares) for a density given by d ¼ 10; 000. The lines are the asymptotic forms of Lyapunov localization spectra in the low density limit for these different values of N.
independent of the system width Ly , as long as the particle order is invariant and the system remains quasione-dimensional. We have checked this using Ly ¼ 2Rð2 106 Þ and the critical value is unchanged. The critical value of the Lyapunov localization spectra depends upon the shape of the system. Numerical results show that the critical value of the Lyapunov localization spectra for a square system is smaller than that for the quasi-one-dimensional system [26]. The randomly distributed brick model is specific to the quasione-dimensional system and would need to be generalized for square systems, and property [IIB] can no longer be assumed. The calculation of the critical value nc for a general two (or three) dimensional systems remains as a future problem. Acknowledgments The authors appreciate the financial support by the Japan Society for the Promotion Science. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
U. Dressler, Phys. Rev. A 38 (1988) 2103. D.J. Evans, G.P. Morriss, Statistical Mechanics of Non-equilibrium Liquids, Academic Press, New York, 1990. C.P. Dettmann, G.P. Morriss, Phys. Rev. E 53 (1996) R5545. T. Taniguchi, G.P. Morriss, Phys. Rev. E 66 (2002) 066203. Lj. Milanovic´, H.A. Posch, Wm.G. Hoover, Mol. Phys. 95 (1998) 281. H.A. Posch, R. Hirschl, in: D. Sza´sz (Ed.), Hard Ball Systems and the Lorentz Gas, Springer, Berlin, 2000, p. 279. J.-P. Eckmann, O. Gat, J. Stat. Phys. 98 (2000) 775. T. Taniguchi, G.P. Morriss, Phys. Rev. E 65 (2002) 056202. T. Taniguchi, G.P. Morriss, Phys. Rev. E 68 (2003) 026218. M. Mareschal, S. McNamara, Physica D 187 (2004) 311. A.S. de Wijn, H. van Beijeren, Phys. Rev. E 70 (2004) 016207. H.L. Yang, G. Radons, Phys. Rev. E 71 (2005) 036211. T. Taniguchi, G.P. Morriss, Phys. Rev. E 71 (2005) 016218. C. Forster, H.A. Posch, New J. Phys. 7 (2005) 32. T. Taniguchi, G.P. Morriss, Phys. Rev. Lett. 94 (2005) 154101.
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T. Taniguchi, G.P. Morriss / Physica A 375 (2007) 563–570 T. Taniguchi, G.P. Morriss, Eur. Phys. J. B 50 (2006) 305. T. Taniguchi, G.P. Morriss, Phys. Rev. E 73 (2006) 036208. P. Manneville, Lect. Note Phys. 230 (1985) 319. R. Livi, S. Ruffo, in: G. Turchetti (Ed.), Nonlinear Dynamics, World Scientific, Singapore, 1989, p. 220. K. Kaneko, Physica D 23 (1986) 436. M. Falcioni, U.M.B. Marconi, A. Vulpiani, Phys. Rev. A 44 (1991) 2263. G. Giacomelli, A. Politi, Europhys. Lett. 15 (1991) 387. A. Pikovsky, A. Politi, Nonlinearity 11 (1998) 1049. A. Pikovsky, A. Politi, Phys. Rev. E 63 (2001) 036207. Lj. Milanovic´, H.A. Posch, J. Mol. Liq. 96–97 (2002) 221. T. Taniguchi, G.P. Morriss, Phys. Rev. E 68 (2003) 046203. R. van Zon, H. van Beijeren, Ch. Dellago, Phys. Rev. Lett. 80 (1998) 2035. R. van Zon, H. van Beijeren, J. Stat. Phys. 109 (2002) 641. G. Casati, L. Molinari, F. Izrailev, Phys. Rev. Lett. 64 (1990) 1851. G. Benettin, L. Galgani, J.-M. Strelcyn, Phys. Rev. A 14 (1976) 2338. G. Benettin, L. Galgani, A. Giorgilli, J.-M. Strelcyn, Meccanica 15 (1980) 9; 15 (1980) 21. I. Shimada, T. Nagashima, Prog. Theor. Phys. 61 (1979) 1605. In the normalized form fWðnÞ =Ngn of Lyapunov localization spectrum, its minimum value 2=N depends on N. To effectively compare Lyapunov localization spectra with different numbers of particles, we use the normalized Lyapunov localization spectra minus 2=N in Figs. 3 and 4. This is crucial in Fig. 4 where the particle number dependence of Lyapunov localization spectra is discussed.
Physica A 375 (2007) 563–570 www.elsevier.com/locate/physa
The maximum number of highly localized Lyapunov vectors at low density Tooru Taniguchi, Gary P. Morriss School of Physics, University of New South Wales, Sydney, New South Wales 2052, Australia Received 16 November 2005; received in revised form 2 August 2006 Available online 13 October 2006
Abstract The localization spectra, which describes the magnitude of the localization of the Lyapunov vectors in many-particle systems, exhibit a characteristic bending behavior at low density. It is shown that this behavior is due to a restriction on the maximum number of the most localized Lyapunov vectors determined by the system configuration and mutual orthogonality. For a quasi-one-dimensional system, using a randomly distributed brick model, this leads to a predicted bending point at nc 0:432N for an N particle system. Numerical evidence is presented that confirms this predicted bending point as a function of the number of particles N. r 2006 Elsevier B.V. All rights reserved. Keywords: Lyapunov vector; Localization; Large Lyapunov exponents; Low density; Many-particle system; Quasi-one-dimensional system
1. Introduction The Lyapunov spectrum is an indicator of dynamical instability in the phase space of many-particle systems. It is introduced as the sorted set flðnÞ gn , lð1Þ Xlð2Þ X of Lyapunov exponents lðnÞ , which give the exponential rates of expansion or contraction of the distance between nearby trajectories (Lyapunov vector) and is defined for each orthogonal direction of the phase space. In Hamiltonian systems and some thermostated systems, a symmetric structure of the Lyapunov spectra, the so-called the conjugate pairing rule, is observed [1–4]. One of the most significant points of the Lyapunov spectrum is that each Lyapunov exponent indicates a time scale given by the inverse of the Lyapunov exponent so we can consider the Lyapunov spectrum as a spectrum of time-scales. The smallest positive Lyapunov exponent region of the spectrum is dominated by macroscopic time and length scale behavior, and here some delocalized mode-like structures (the Lyapunov modes) have been observed in the Lyapunov vectors [5–16]. On the other hand, the largest Lyapunov exponent region of Lyapunov spectrum is dominated by short time scale behavior, and in this region the Lyapunov vectors are localized (Lyapunov localization). The position of the localized region of Lyapunov vectors moves as a function of time [17]. A variety of systems with many degrees of freedom show Lyapunov localization, for Corresponding author. Tel.: +61 2 93855667; fax: +61 2 93856060.
E-mail address: [email protected] (T. Taniguchi). 0378-4371/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2006.09.017
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example, the Kuramoto–Sivashinsky model [18], a random matrix model [19], map systems [20–23], coupled non-linear oscillators [24], and many-disk systems [25,26]. The magnitude of Lyapunov localization can be measured quantitatively by the Lyapunov localization spectrum, which is defined as a set of exponential functions of entropy-like quantities for the normalized amplitudes of the Lyapunov vector particle components [26]. It is shown in Ref. [26] that at low density the Lyapunov localization spectra show a critical bending point, which limits the number of strongly localized Lyapunov vectors. The present study addresses a question that was explored but unresolved in a previous paper [26], that is, how many strongly localized Lyapunov vectors are present in the low density limit, in other words, what kind of physical characteristics of systems determine the critical bending point of Lyapunov localization spectra? While it is clear from the study of the clock model [17,27,28] that the Lyapunov vector associated with the largest Lyapunov exponent is strongly localized, and that this arises from the cumulative effect of collisions, the number of such localized Lyapunov vectors was unknown. Here we give the answer to that question for a quasi-one-dimensional system, using a simple model distributing bricks randomly in a one-dimensional region. While the mechanism of cumulative collisions remains in general, constructing an equivalent brick model for more general situations is technically difficult. To calculate Lyapunov vectors shown in this paper we use the numerical algorithm developed by Benettin et al. [30,31] and Shimada and Nagashima [32], as characterized by intermittent reorthogonalization and renormalization of Lyapunov vectors. In this algorithm, Lyapunov vectors are calculated as follows. First, the rate of growth of the difference between two nearby phase space trajectories is calculated to give the largest Lyapunov exponent and the corresponding (normalized) Lyapunov vector. Second, eliminating the direction defined by the Lyapunov vector for the largest Lyapunov exponent, we reduce the dimension of the phase space by one. In that reduced phase space we can again define a direction that corresponds to the next largest Lyapunov exponent and its associated Lyapunov vector. Continuing this procedure we define the same number of Lyapunov exponents and the corresponding Lyapunov vectors as there are independent directions in phase space. While the Lyapunov vectors associated with the largest exponents (in absolute value) are strongly localized, the vectors associated with the smallest exponents (in absolute value) have delocalized Lyapunov vectors, referred to as Lyapunov modes. The Lyapunov vectors, as defined here, have already been investigated in many chaotic systems with many degrees of freedom [5,6,9,12–16,26]. 2. Lyapunov localization spectra at low density The system, which we consider in this paper, is quasi-one-dimensional in which the radius of the particles R ¼ 1, the mass of each particle m ¼ 1, the total energy of system E ¼ N with the number N of disks, and the system width Ly ¼ 2Rð1 þ 106 Þ and length Lx ¼ NLy ð1 þ dÞ with parameter d, related to the density by r ¼ pR2 =½ð1 þ dÞL2y . Note that in this system the particles remain ordered because the vertical size Ly prohibits the exchange of particles [9,13,26], justifying the name ‘‘quasi-one-dimensional system’’. A schematic illustration of quasi-one-dimensional system is in Fig. 1. We use periodic boundary conditions in both the longitudinal and the transverse direction. (Quasi-one-dimensional systems with different boundary conditions are considered in Refs. [9,26].) We introduce the normalized Lyapunov vector for the nth Lyapunov exponent lðnÞ as dCðnÞ ðnÞ ðnÞ ðnÞ ðdCðnÞ is the contribution of the jth particle, and we construct the normalized 1 ; dC2 ; . . . ; dCN Þ where dCj ðnÞ 2 ðnÞ Lyapunov vector component amplitude for each particle j as gðnÞ j jdCj j . The normalization of dC PN ðnÞ ðnÞ imposes the condition j¼1 gj ¼ 1 for gj . Recently, a quantity to measure the strength of Lyapunov
1
2
N
Fig. 1. A quasi-one-dimensional hard-disk system with the disk radius R, the length Lx and the width Ly satisfying the inequality 2RoLy o4R. In this system, the disks remains in the same order and they can be numbered 1; 2; . . . ; N from the left to right.
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localization for each Lyapunov exponent was proposed [26]. The localization of the nth Lyapunov vector is " # N X ðnÞ ðnÞ ðnÞ hgj ln gj i . (1) W exp j¼1
P ðnÞ ðnÞ ðnÞ The bracket h i in Eq. (1) indicates the time-average. The quantity N j¼1 hgj ln gj i in the definition of W ðnÞ can be regarded as an entropy-like quantity [21,22,29] as gj is a distribution function over the particle index j. The quantity WðnÞ satisfies the inequality 1pWðnÞ pN, and can be interpreted as the effective number of particles contributing strongly to the normalized Lyapunov vector. A Hamiltonian system satisfies the conjugate relation WðDjþ1Þ ¼ WðjÞ for any j with the phase space dimension D [26], because of the symplectic structure. The set of quantities fWðnÞ gn which we call the Lyapunov localization spectrum, has been calculated previously in many-particle systems with hard-core interactions [26] and soft-core interactions [14]. In these systems, the value of WðnÞ usually increases with the Lyapunov index n, and this implies that Lyapunov vectors for the largest exponents are the most localized. The quantity WðnÞ has a minimum value of 2, when only two particles contribute strongly to the value of WðnÞ , and this value for WðnÞ is observed numerically for a number of the largest Lyapunov exponents in the limit as the density approaches zero [26]. This suggests that there is a maximum number of Lyapunov vectors of the type shown in Fig. 2 for which only two particles contribute strongly. Fig. 2 is a graph of gð1Þ j as a function of the collision number nt and the particle index j. As shown in Fig. 2, the localized region of Lyapunov vector moves with time [17], but the localized region is almost always concentrated on two nearest-neighbor disks. It was also shown that WðnÞ can detect not only the localized behavior of Lyapunov vectors, but also the de-localized behavior observed in the Lyapunov modes [26]. One of the important characteristics of the Lyapunov localization spectrum is the bending behavior at low density [26]. In Fig. 3 we show an example of such bending behavior [33] in a quasi-one-dimensional system of 50 hard disks. In Fig. 3, five different Lyapunov localization spectra are represented at different densities. In the low density limit, this figure shows a bending point in the Lyapunov localization spectra at nc . At this bending point n nc the Lyapunov localization spectra WðnÞ changes from a linear dependence upon the Lyapunov index (n) to an exponential dependence. This bending becomes sharper at lower density, and the numerical results suggest that in the low density limit the Lyapunov localization spectrum converges to the solid line given by ( 2 in nonc ; ðnÞ W ¼ (2) g aN expfbn=ð2NÞg in nXnc ;
0.5 0.4
γ j(1)
0.3 0.2 0.1 0 449100 449150 449200
nt
449250
0
45 50 35 40 30 25 15 20 5 10 j
Fig. 2. The normalized amplitude gð1Þ j of the Lyapunov vector particle component corresponding to the largest Lyapunov exponent as a function the collision number nt and the particle index j. The system is a quasi-one-dimensional hard-disk system with d ¼ 105 and N ¼ 50, and particle indices are 1; 2; . . . ; N labeled from left to right.
ARTICLE IN PRESS T. Taniguchi, G.P. Morriss / Physica A 375 (2007) 563–570
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0.6
0.5
0.3
(
(n)-
2) / N
0.4
0.2
d= 1 d= 10 d = 100 d = 500 d = 10000
0.1 nc/(2N) 0 0
0.1
0.2
0.3
0.4
0.5
0.6
n / (2N) ðnÞ
Fig. 3. The normalized Lyapunov localization spectra fðW 2Þ=Ngn as functions of the normalized Lyapunov index n=ð2NÞ for a quasione-dimensional 50 hard-disk system at different densities given by d ¼ 1 (diamonds), d ¼ 10 (circles), d ¼ 100 (triangles), d ¼ 500 (inverted triangles), and d ¼ 10000 (squares). The parameter d is inversely proportional to the density as r ¼ pR2 =½ð1 þ dÞL2y . The smallest Lyapunov index corresponds to the largest Lyapunov exponent. The solid line is the asymptotic form of the Lyapunov localization spectrum as the density approaches zero, and n nc is the critical bending point.
with constants a, b, and g 2 þ aN expfbnc =ð2NÞg [so that WðnÞ jn¼nc ¼ 2 in Eq. (2)]. In other words, we can estimate the critical value n nc defined as the bending point of the Lyapunov localization spectrum by the value of the fitting parameter nc in the fit of the Lyapunov localization spectrum to the function (2). The bending behavior of Lyapunov localization spectra, shown in Fig. 3 is associated with a similar bending point in the Lyapunov exponent spectra. Moreover, the existence of a linear dependence of WðnÞ on n in the region npnc , at low density, is connected with some other known kinetic properties, for example, that the mean free time is inversely proportional to density, and the Krylov relation that the largest Lyapunov exponent lð1Þ depends on the density r like lð1Þ r ln r [26]. These results suggest that the existence of the linear dependence of WðnÞ is connected to the density range where kinetic theory provides an accurate description. These points were investigated in detail in quasi-one-dimensional systems, although the bending behavior of the Lyapunov localization spectra is also observed in fully two-dimensional square systems [26]. However, no mechanism was proposed to explain this bending behavior. In this paper, we construct a mechanism that leads to the bending behavior observed in the Lyapunov localization spectra, and predicts the critical value nc for quasi-one-dimensional systems.
3. Randomly distributed brick model To explain the maximum number of the most localized Lyapunov vectors nc , and thus the bending behavior of the Lyapunov localization spectrum at low density, we begin with some properties of the Lyapunov vectors of many-particle systems. The first property expresses the fact that different Lyapunov vectors sample independent directions in phase space. [I] Lyapunov vectors with different Lyapunov indexes are orthogonal [31]. The second property of Lyapunov vectors is based on the numerical observation that particle interactions occur between a minimum of two different particles, so for quasi-one-dimensional systems Wð1Þ ! 2 as r ! 0 as shown in Fig. 3 and Ref. [26]. [IIA] In the low density limit, for hard particles, all the normalized Lyapunov vectors in the linear region 1pnonc have strong contributions from only two particles.
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The third property is justified only for quasi-one-dimensional systems, and restricts the property [IIA] further to [IIB] The two particles, whose normalized Lyapunov vector components are non-zero values in the low density limit, are nearest-neighbors. In Ref. [26] it is observed that in the largest Lyapunov exponent region, at low density, the two strong contributions to the normalized Lyapunov vector appear due to a collision. Moreover, in quasi-onedimensional systems, particle collisions occur only between nearest neighbor particles. Therefore, property [IIB] is justified. Using the properties [IIA] and [IIB], in the low density limit, all the normalized Lyapunov vectors dCðnÞ in the largest Lyapunov exponent region 1pnonc can be well approximated by ðnÞ dCðnÞ ð0; 0; . . . ; 0; dCðnÞ mn ; dCmn þ1 ; 0; . . . ; 0Þ,
(3)
where 0 is the null vector. In general, the particle number mn depends on the Lyapunov index n and on the time. Here, the particles in the quasi-one-dimensional system are numbered from left to right, and mn and mn þ 1 are nearest neighbor particles. (For periodic boundary conditions in the longitudinal direction particles N and 1 are nearest neighbors.) Now we consider the restriction imposed by condition [I] on normalized Lyapunov vectors of the form (3). To ensure that condition [I] is satisfied, it is sufficient that the particle numbers mn and mn þ 1 in one normalized Lyapunov vector are distinct from those of the other Lyapunov vector mn0 and mn0 þ 1. This ensures that the two Lyapunov vectors are orthogonal regardless of the details of the various components for particles mn and mn þ 1 in the first vector, and mn0 and mn0 þ 1 in the second. This further restricts the maximum number of independent normalized Lyapunov vectors of the form (3), and puts an upper limit on the critical value nc . This situation is explained using a simple model whose schematic illustration is given in Fig. 4. In this randomly distributed brick model, N boxes corresponding to each of the particles are arranged on a line. Each normalized Lyapunov vector represented by Eq. (3) must have strong contributions for only two particles ðnÞ dCðnÞ mn and dCmn þ1 . These are shown as a gray-filled rectangular brick filling boxes mn and ðmn þ 1Þ in Fig. 4. To satisfy condition [I], these bricks must not overlap. The critical value nc is then the average number of bricks which can be put without overlaps on the N different boxes. Notice that there is no accumulation in this model. Bricks are dropped until no further adjacent vacancies remain. There is one more important point to obtain an explicit value of nc from the above mechanism: [III] The particle number mn in Eq. (3) is chosen randomly with respect to the Lyapunov index nðonc Þ, so that there is no overlap among the non-zero Lyapunov vector components for normalized Lyapunov vectors with different Lyapunov indices. This means that the bricks shown in Fig. 4 must be randomly distributed without overlaps. Therefore, randomly constructed configurations with any number of single empty boxes at non-neighboring positions are possible. Taking an ensemble average of the values of n~ c for each possible configuration we obtain the critical value nc . Fig. 5 shows the normalized critical value nc =ð2NÞ from such a numerical simulation for different numbers of particles N. The data in this figure shows that nc =ð2NÞ is independent of N, giving an estimate of the normalized critical value as nc 0:216. 2N
(4) 1
2
3
4
5
N-5 N-4 N-3 N-2 N-1
6
N
... μn μn +1
μn' μ n' +1
Fig. 4. A schematic illustration of the randomly distributed brick model, used to explain the maximum number of independent normalized Lyapunov vectors with only two dominant particle contributions. The boxes numbered j correspond to the jth particle component of the Lyapunov vector. Gray-filled rectangular bricks occupying boxes mn and mn þ 1 represent the dominant contributions of possible normalized Lyapunov vectors for which nonc . The value nc is the average number of bricks that can be randomly arranged on the set of boxes such that no pair of adjoining spaces remain.
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0.3
0.25
nc / (2N)
0.2
0.15
0.1
0.05
0 40
50
60
70
80
90
100
N Fig. 5. The normalized critical value nc =ð2NÞ given by the randomly distributed brick model as a function of the number of particles N. The solid line is a fit of the numerical data to a constant function nc =ð2NÞ ¼ x where the parameter x 0:216.
Table 1 The values of the fitting parameters a and b given in Eq. (2) for Fig. 6 N
a
b
40 50 60 70
2:83 0:04 2:51 0:02 2:28 0:01 2:10 0:02
7:83 0:02 7:24 0:01 6:08 0:00 6:39 0:01
The accuracy of this value can be used as a check of the proposed mechanism for the bending behavior of Lyapunov localization spectra at low density. To draw the solid line shown in Fig. 3 we have already used the parameters a and b in Eq. (2) given in Table 1 for N ¼ 50. The result shown in Fig. 3, using the critical value (4) gives a very satisfactory fit. In order to check that the result (4) is satisfied for any number of particles N, we present Fig. 6. It is the ½n=ð2NÞ-dependence of ½WðnÞ 2=N in quasi-one-dimensional systems of different sizes at the same density. Notice that values of ½WðnÞ 2=N themselves decrease slightly as the number of particles increases, meaning that the fitting parameters a and b in Eq. (2) depend only slightly on N, as shown in Table 1. However, all the data for different numbers of particles in Fig. 6 are nicely fitted using the same critical value nc =ð2NÞ, given by Eq. (4). 4. Conclusion and remarks To conclude, we have shown that a model of randomly distributed bricks on a line (Fig. 4) can predict the maximum number of Lyapunov vectors which have a Lyapunov localization WðnÞ ¼ 2, in the low density limit. These are the Lyapunov vectors for which nonc in the linear region that lead to the bending behavior of Lyapunov localization spectra. We showed that this behavior comes from a restriction on the maximum number of the most localized Lyapunov vectors with dominant components for only two particles. The randomly distributed brick model was applied to quasi-one-dimensional systems, giving the specific value nc =ð2NÞ 0:216 for the critical value nc of the Lyapunov localization spectra for any number of particles N. Numerical simulations give a value of nc =ð2NÞ 0:219 0:005. Our explanation for the bending behavior is
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0.5 0.45 0.4
2) / N
0.3
(n)-
0.25
(
0.35
0.2 0.15
N = 40 N = 50 N = 60 N = 70
0.1 0.05
nc /(2N)
0 0
0.1
0.2
0.3
0.4
0.5
0.6
n / (2N) ðnÞ
Fig. 6. The normalized Lyapunov localization spectra fðW 2Þ=Ngn as functions of the normalized Lyapunov index n=ð2NÞ in quasione-dimensional systems of different size; N ¼ 40 (diamonds), N ¼ 50 (triangles), N ¼ 60 (inverted triangles), and N ¼ 70 (squares) for a density given by d ¼ 10; 000. The lines are the asymptotic forms of Lyapunov localization spectra in the low density limit for these different values of N.
independent of the system width Ly , as long as the particle order is invariant and the system remains quasione-dimensional. We have checked this using Ly ¼ 2Rð2 106 Þ and the critical value is unchanged. The critical value of the Lyapunov localization spectra depends upon the shape of the system. Numerical results show that the critical value of the Lyapunov localization spectra for a square system is smaller than that for the quasi-one-dimensional system [26]. The randomly distributed brick model is specific to the quasione-dimensional system and would need to be generalized for square systems, and property [IIB] can no longer be assumed. The calculation of the critical value nc for a general two (or three) dimensional systems remains as a future problem. Acknowledgments The authors appreciate the financial support by the Japan Society for the Promotion Science. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
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