Glassy dynamics in a spatially distributed dynamical system

Physics Letters A 174 (1993) 103-110 North-Holland

PHYSICS LETTERS A

Glassy dynamics in a spatially distributed dynamical system H i r o k a z u Fujisaka, K u n i h i r o E g a m i

Department of Physics, Kyushu University33, Fukuoka 812, Japan and

TomojiY a m a d a Division of Electronic Physics, KyushuInstitute of Technology, Tobata804, Japan Received 8 May 1992; revised manuscript received 11 September 1992; acceptedfor publication 10 December 1992 Communicatedby A.P. Fordy

A numerical verification of a glassystate, dynamicalglass, in a spatially distributed dynamical systemis reported. The present glass state is temporallyperiodic and spatially random, which is caused by the existenceof extremelymany statisticallyequivalent glass attractors.The relaxation processis slow,and the distribution function for relaxationtimes obeysthe scalinglaw for different systemsizes.

Recently, the glassy state and the slow relaxation associated with it received much attention in various contexts, e.g., dielectrics, structural glass, spin glass, protein folding, etc. (see, e.g., ref. [ 1 ] ). One prominent statistical characteristic of the glassy dynamics is the slow relaxation process, which is typically known as the Kohlrausch law of the time correlation function. The origin of this anomalous relaxation is widely believed to be that the system yields extremely many metastable states. Recently Daido [ 2 ] suggested the possibility of the existence of the glass state in the phase model of an oscillator assembly provided that the interactions among the oscillators are random and frustrated. The fundamental purpose of the present Letter is to report on the observation of the glassy state in the spatially one-dimensional extended dynamical system [ 3,4 ]

(or>0, f ¢ ( A z ) d A z = 1 ). This is the coupled oscillator system whose element dynamics is given by the one-dimensional map xn ÷ ~= f ( x n ) . In the second half of the present Letter we will discuss the scaling structure of the distribution function for relaxation times from the unstable initial state to the final glassy state. The coupled oscillator system ( 1 ) was derived with a certain kind of perturbation expansion with respect to spatial non-uniformity by starting with the ordinary reaction-diffusion equation [3]. The important aspect of (1) is that it has spatially translational invariance, i.e. the oscillator dynamics is homogeneous, and contains no external randomness. The stability property of the motion is examinedwith the quantity d ~ = { L - l f [y~(z)]2dz} 1/2 where the perturbation y~ (z) obeys

x,+ t (z) = e x p ( a d 2 / O z 2 ) f ( x , ( z ) )

Yn+l(Z)= ~ O ( Z - - Z ' ) f ' ( x . ( z ' ) ) y . ( z ' ) dz'

= ~ O ( z - z ' ) f ( x ~ ( z ' ) ) dz',

( 1)

where n ( = 0, l, 2 .... ) is the time step, z is the coordinate and 0 is the interaction kernel,

q ) ( z - z ' ) = (4not) -1/2 e x p [ - ( z - z ' ) 2 / 4 0 t ]

(2)

( f ' ( x ) = d f ( x ) / d x ) . One should remark that the translational invariance implies that (2) yields the particular solution y . ( z ) = OzX.(Z) except near the boundaries. This solution is however irrelevant to

0375-9601/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.

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the stability of the spatio-temporal behavior. For a large n, d, ~ e a", where A = lim,~oo n - ~log d, is the largest Lyapunov exponent for the coupled oscillator system ( 1 ). If A < 0 ( > 0), the spatio-temporal evolution {x,(z)} obeying (1) is stable (unstable). Depending on the form o f f ( x ) and the initial condition, ( 1 ) exhibits various spatio-temporal behaviors. A spatially synchronous oscillation for a sufficiently large system size is stable only when the element dynamics x,+ 1= f ( x , ) is periodic [ 3 ]. One can easily deduce that the synchronous chaos as a particular solution, which is possible only when the element dynamics is chaotic, is always unstable for a large system. Generally speaking the spatio-temporal behavior depends on the system size L. A relevant length scale exists only when the element dynamics is chaotic, and is given by Lc = 2 ~ r x / - ~ [ 3 ], 2 being the Lyapunov exponent of the element dynamics. If the system size is smaller than Lc, no unstable region in the wavenumber space exists and the synchronous state is chaotic. On the other hand for L > L~ the synchronous state is linearly unstable. In this Letter numerical integrations were carried out by discretizing the total space L with the mesh size Az, small enough, and with the periodic boundary condition. This corresponds to solving the coupled-map system [ 3 ] N X ,+1 U) =

E ~ _ , f ( x . (t) ),

I=1

(3)

where x ,U) = x , ( z j ) with the position zj ( = j A z + const) of the jth oscillator,

~_t=O( zj- z~),az = (4nOtE) -1/2 e x p [ - (j--l)2/4OtE] and aE=a/ (Az) 2, Zt R_t= 1, N ( - L / A z ) being the number of oscillators. Equation (3) is independent of the mesh size Az as long as it is small enough. Correspondingly (2) is given by Y.+lu)=Z~=~ ~ _ t X f'(x~t))y~ t), and d.=[N-IZ~c=l(y~t))2] 1/2. The coupled-map system similar to (3) with different interactions ~ _ t is studied by many authors (see, e.g., refs. [5,6] ). Hereafter we adopt the logistic parabola f ( x ) = a x ( 1 - x ) with the control parameter a. The parameter a is set to be 3.71, for which the uncoupled system x°+l = f ( x °) is chaotic, i.e. 2 is positive 104

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(2--- 0.363 ). The effective coupling constant o(E is set to be 1. The initial condition for Xo(Z) at a point z is put as Xo(Z) =p+p', where p (p') is a value randomly generated with numbers uniformly distributed in the region 0.1 < p < 0 . 6 (Ip'l <0.1 or 0.01 ). There is no correlation between Xo(Z) and Xo(Z') provided that z ~ z'. Figure 1a shows how the spatial pattern is formed in the course of time for 512 oscillators for a certain initial condition. Figure 2a corresponds to the pattern at the final time 16368 shown in fig. 1a. The final pattern consists of a periodic-like spatial oscillation with two different, globally local maxima (minima) x-~0.85 and 0,92 (0.32 and 0.49). Numerical calculation shows that the spatial periodicity is independent of the initial condition. It seems that there is no apparent rule on the positions of the local maxima (minima), the pattern is spatially irregular. However, this is temporally periodic with period 4, and the same spatially "irregular" pattern repeats in each 4 time steps. Figure lb displays the temporal evolution of the patterns at the same time as fig. la for the initial condition set such that x~/) [b] - x ~ >[a] = 0 f o r j ~ 256 and 0.005 for j=256. An infinitesimal difference of initial conditions thus produces a huge difference in final patterns. One should remark that the difference between figs. 2a and 2b is observed in the peak heights of local maxima and minima. For almost any initial condition spatial patterns in a sufficient time are similar to those in fig. 1. The spatial periodicity is independent of the initial conditions, and the pattern is thus characterized by the positions of local maxima and minima with different heights. The above observation implies that the final pattern is sensitive to the initial condition. One can expect that the patterns in fig. 1 are simply two of many attractors existing in the high-dimensional state space. The significant characteristic of the final pattern is the breakdown of translational invariance, although initial conditions satisfy the statistically spatially translational invariance. This breakdown is purely due to the dynamical origin. Figure 3a displays how the coarse-grained local expansion rate A. defined by A, = ( log (d, +r/d, ) ) / T, where ( ) is the average with different initial conditions and we put T = 100, evolves in time for different initial conditions. For an arbitrary initial condition after a long transient time the phase point

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Fig. 1. Illustration of the temporal evolution of {x, (j) } obeying (3) with f ( x ) = 3.7 Ix( 1 - x ) for each 8 time steps and the oscillator n u m b e r N = 512 for slightly different initial conditions. The difference o f the two initial conditions for (a) and (b) was given only in x~ 25~) . For details see the text.

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Fig. 2. The glassy patterns after a long time (n= 16368) for the initial conditions corresponding to fig. 1. Apparently even for an infinitesimal difference of initial conditions the final states take quite different, random spatial patterns. suddenly relaxes onto a final attractor, which is apparently temporally periodic because the global expansion rate A is negative. We therefore conclude that the abovc system eventually approaches a temporally periodic, spatially irregular motion for the aforementioned initial conditions. Let t be the time-length that a phase point needs to fall down into the final temporally periodic attractor, i.e. thc duration of the rc106

laxation (transient) process, t strongly depends on the initial condition because o f the trajectory instability in the relaxation process (fig. 3a). Let P(t) be the distribution function that the time o f the transient process is in-between t and t + 1. Numerically wc find that P(t) has a peak at t=t* (fig. 3b) and that this depends on the system size L as t*ocL~, where t/_~ 1 (fig. 4). As shown in fig. 5a, the distri-

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Fig. 3. (a) Temporal evolutions of coar~-graincd local expansion rates for the system size N=512 (the number of oscillators) for different initial conditions. The orbit for the initial condition with the symbol • dropped into the glass attractor at the time step 16300. The relaxation times to the glass states with the negative expansion rates stronglydepend on initial conditions. (b) Plot of the distribution function for relaxation times for different system sizes L. The numbers in the figure are the numbers of oscillator N. b u t i o n function P(t) takes the power law form oct 2 for t smaller than t*, a n d

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for t larger than t*. Here fl--- 2, i n d e p e n d e n t l y of the system size L (fig. 5b). F u r t h e r m o r e P(t*) is approximately inversely proportional to L. The state discovered above has characteristics c o m m o n l y observed in glass states [ 1 ] a n d will be 107

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N Fig. 4. The system size dependence of the characteristictime t*. called the dynamical glass, and the final attractors the glass attractors. One should remark that the present glass is quite different from the oscillator glass proposed by Daido [ 2 ] in the sense that the present state is generated neither by randomness nor by frustration of the interactions among the oscillators which he assumed, but a purely dynamical result due to the existence of many statistically equivalent attractors of the "pure" system ( l ) or (3). The number of glass attractors can be roughly estimated as eeL, where c is a certain positive constant. This is evaluated by noting that the glass patterns, e.g. figs. 2a and 2b, are mainly distinguished by the positions of local extrema of the spatial variation of a state variable. Recently Kaneko reported temporally frozen, spatially irregular patterns in his coupled map system [ 6 ]. The patterns are quite different depending on the initial conditions. Probably his frozen pattern and our state are essentially the same. The slow relaxation in the spin glass state is due to the existence of a huge number of local free-energy minima. In the present oscillator glass this corresponds to the existence of a huge number of statistically "equivalent" glass attractors. It is natural to expect that the number of glass attractors increases as the system size is increased. The slow relaxation is observed when the system is "quenched", i.e., when 108

the system relaxes into one of a huge number of attractors. As discussed, e.g., in ref. [ 7 ] for glassy relaxation, one can expect that the present type slow relaxation has two characteristics. One is the hierarchical structure and the self-similarity in the relaxation process from a large scale dynamics to a small scale dynamics in the state space spanned by {x(z)}. After the power law for the distribution function for relaxation times is derived by assuming a simple hierarchical self-similarity, we will discuss the scaling structure of the distribution function over the whole t region. Let ~o be the largest scale of the minimum region f~ covering the whole set of attractors in the Hilbert space spanned by {x(z)). The set of attractors, precisely speaking the set of attracting regions, is assumed to have a hierarchical structure in the following sense. Let the attractor scale ~ in the hierarchy levelj be given by ~j= ~oe-j where the minimum size of attractors is ~min, and 0> 1 ). The relaxation process under consideration is assumed to be composed of three stages, i.e. the first is the process that the phase point relaxes into £~ by starting with a rather arbitrary initial condition given above, the second is the long wandering process from the large-scale motion to small-scale motions, and the third is a sudden drop into one of

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t/t* Fig. 5. Scaling plot of the distribution function P(t) for relaxation times to glassy states. The symbols have the same meanings as in fig, 3, and correspond to different sizes. The solid line is the master function g ( x ) = 2x2/(x4+ 1 ).

the smallest attractors (glass attractors) with the scale ~mi,. The numerical result suggests the a s y m p t o t i c form P ( t ) ... ( t / t o ) 2 for t a p p r o p r i a t e l y small, where to m a y be regarded as the d u r a t i o n in the first relaxation process in the first level. Let Tj be the residence t i m e in the j t h hierarchy level. Namely, after the phase p o i n t started to relax over the scale ~j it takes

a time zj to fall down into the attracting region with the next scale 5+ 1. However, in the course o f the successive relaxation process the phase p o i n t is able to d r o p into a final glassy attractor. The probability that the phase point relaxes into the glass attractor at a t i m e tj is assumed to satisfy the similarity law P ( tj+ t ) / P ( t j ) = e - b for I <
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t j = ~ = o rl ( j ~ 0 ) , a n d b is a certain positive constant. This yields P(tj)oce-bL Assuming that zl=A~i-s ( s > 0 ) , where s a n d A are certain constants, we i m m e d i a t e l y find ( 4 ) with fl=b/s. The asymptotic form ( 4 ) is valid for to<< t<< to(~O/~mi,) s (to~--eSzo). Thus in o u r m o d e l the exponent fl is det e r m i n e d by the e x p o n e n t s, which is relevant to how the residence t i m e depends on the hierarchy level and the p a r a m e t e r b which is related to the p r o b a b i l i t y that the phase p o i n t drops into the final glass attractor. Figure 5 displays the scaling behaviors o f P(t), which suggests the existence o f the m a s t e r function g(x) defined b y P(t)/P(t*)=g(t/t*). We propose the form

g(x)= (l+½fl)x2 x:+p+

(5)

½/~ •

F o r f l = 2 we o b t a i n g(x) =2x2/(x4+ 1 ). This master function explains numerical results quite well (fig. 5). The peak position t* and the d u r a t i o n to in the first relaxation level are connected to each other via to~t*/ 2x~(t*). Since t*~L a n d P(t*)~L -I, one finds to ~ L 3/2. We can s u m m a r i z e the results o f this p a p e r as follows. ( i ) We found a glass state in a coupled-oscillator system ( 1 ) or ( 3 ) . After a long time the phase p o i n t suddenly relaxes into the glass state, which is t e m p o r a l l y p e r i o d i c a n d spatially irregular. (ii) The system m a y have manY statistically equivalent glass attractors, one o f which is selected for a given initial condition. The final glassy pattern is sensitive to the initial condition. ( i i i ) The d i s t r i b u t i o n function for relaxation times exhibits p o w e r law behaviors in two characteristic t i m e regions, a n d t u r n e d out to satisfy the scaling law for different system sizes. It should be n o t e d that the present glass state is observed for a wide p a r a m e t e r region o f a. F u r t h e r m o r e we can naturally expect this glass state can be found in other, spatially d i s t r i b u t e d d y n a m i c a l systems. This is one new p r o m i n e n t characteristic o f spatially distributed d y n a m i c a l systems.

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Finally the following r e m a r k is added: Some years ago Ruelle suggested the possibility o f the existence o f turbulent crystals by extending the analogy between structures in solids a n d oscillatory behaviors in d y n a m i c a l systems [ 8 ]. This r e m i n d s us o f the possibility o f the similarity between the present glass and a turbulent crystal. However, we have no solid basis for proving the similarity. The authors thank referees o f Physics Letters A for their helpful c o m m e n t s and suggestions on the manuscript.

References [ 1] K.L. Ngai, Comm. Solid State Phys. 9 (1979) 127; 9 (1980) 141; Ann. NY Acad. Sci. 484 ( 1986); J.L. van Hemmen and I. Morgenstein, eds., Heidelberg Colloquium on Glassy dynamics (Springer, Berlin, 1987 ); H. Takayama, ed., Springer series in synergetics, Vol. 43. Cooperative dynamics in complex physical systems (Springer, Berlin, 1989), and references therein. [2 ] H. Daido, Prog. Theor. Phys. 77 (1987) 622; Phys. Rev. Lett. 68 (1992) 1073. [3l T. Yamada and H. Fujisaka, Prog. Theor. Phys. 72 (1984) 885; H. Fujisaka and T. Yamada, Prog, Theor. Phys. 75 (1986) 1087. [ 4 ] F. Kaspar and H.G. Schuster, Phys. Lett. A 113 (1986) 451. [ 5 ] T. Yamada and H. Fujisaka, Prog. Theor. Phys. 70 ( 1983 ) 1240; K. Kaneko, Prog. Theor. Phys. 72 (1984) 480; Phys. D 37 ( 1989 ) 60, and references therein; I. Walter and R. Kapral, Phys. Rev. A 30 (1984) 2047; T. Hogg and B.A. Huberman, Phys. Rev. A 29 ( 1984 ) 275; H. Chat6 and P. ManneviUe, Physica D 32 (1988) 409, and references therein; T. Bohr and O.B. Christensen, Phys. Rev. Lett. 63 (1989) 2161; D. Stassinopoulos and P. Alstrom, Phys. Rev. A 45 (1992) 675, and references therein. [6] K. Kaneko, Physica D 34 (1989) 1. [7] R.G. Palmer, D.L. Stein, E. Abrahams and P.W. Anderson, Phys. Rev. Lett. 53 (1984) 958. [ 8 ] D. Ruelle, Physica A 113 (1982) 619; Physica D 7 ( 1983 ) 40.