Power spectra and random walks in intermittent chaotic systems

Physica D 69 (1993) 436-446 North-Holland SDI: 0167-2789(93)E0254-9

Power spectra and random walks in intermittent chaotic systems G . Z u m o f e n a a n d J. K l a f t e r b "Laboratorium fiir Physikalische Chemie, ETH-Zentrum, CH-8092 Ziirich, Switzerland bDepartment of Chemistry, Massachusetts Institute of Technology, Cambridge, MA 02139, USA and School of Chemistry, TeI-Aviv University, Tel Aviv, 69978, Israel ~

Received 4 May 1993 Accepted 21 June 1993 Communicated by B.V. Chirikov

We investigate anomalous diffusion and the corresponding power spectra generated by iterated maps and analyze the motion in terms of the probabilistic continuous-time random walk approach. Both stationary and non-stationary conditions are considered demonstrating the dependence of the mean-squared displacement and of the power-spectra on the initial conditions. The theoretical results are corroborated by numerical calculations and excellent agreement is obtained.

1. Introduction A n o m a l o u s diffusion has been a topic of growing interest in a variety of disciplines such as physics, chemistry and biology mainly because it offers some insight into basic dynamic processes in complex system [1-4]. A n o m a l o u s diffusion is characterized by a time evolution of the meansquared displacement (rZ(t)) which deviates f r o m the linear behavior known to hold for simple Brownian motion, namely (r2(t))~t ~,

a¢l.

(1)

A sublinear, dispersive (a < 1) behavior is observed when the motion is inhibited by geometrical or temporal restrictions on all scales [1-3]. For the superlinear, enhanced diffusional behavior, on which we focus here, accelerating mechanisms have to be introduced [4,5], or

Permanent address.

starting from a ballistic motion, restrictions can lead to an enhanced diffusion, intermediate between regular and ballistic motion. E n h a n c e d diffusion has been observed, for instance, in turbulent diffusion [7,8], in Josephson junctions [9], in problems related to self-avoiding r a n d o m walks [5,10], to directed polymers [11-13], and to motion in r a n d o m velocity fields [5,6]. Interesting example of enhanced diffusion are provided by the dynamics of non-integrable hamiltonian systems where the anomalous behavior depends on the potential structure and on the particles' energy and is related to a complex behavior in the phase space which contains both domains of stability and of chaotic motion [1418]. It has been shown that iterated maps are an instructive model for the description of the intermittent chaotic motion [19-31]. It has been established that continuous-time random walks ( C T R W ) offer a m e t h o d for the description of transport resulting from iterated non-linear maps with periodic symmetry [23,29,32]. For the appropriate treatment of this problem the C T R W approach has been extended

0167-2789/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

G. Zumofen, J. Klafter / Intermittent chaotic systems

to motion characterized by a constant velocity between points of halt [29]. Furthermore, spacetime coupled and decoupled memories were considered. In the case of the dispersive motion a reasonable description of the motion can be based on decoupled memories, while for enhanced diffusion coupled memories are indispensable. In the latter case the L6vy-walk formalism is applied [29,32-35]. As a matter of fact in many instances the time dependent mean-squared displacement is sufficient for the description of the diffusional process. For a more profound analysis of the motion, however, the propagator, the spatial distribution of the time-dependent probability of the particles' location has to be considered. For simple Brownian motion this probability distribution follows a Gaussian whereas for anomalous diffusion distributions are observed which deviate from the normal distribution. While for the dispersive motion a modified Gaussian decay in the wing is the rule, scale invariance can be obtained for enhanced diffusion in the framework of L6vy walks. For this case basic expressions have been derived for the asymptotic behavior of the propagator and of the meansquared displacement [29-31]; in this paper we will consider this formalism. A problem which arises naturally for transport in iterated maps [31] as well as in other nonlinear systems is the dependence of the dynamics on the initial conditions. This dependence is equivalent to the problem of stationarity in the theory of the CTRWs [3,29,31]. Usually, within the C T R W approach, it is assumed that the time-origin of the a dynamical process coincides with that of the observation process, to which we refer to as the non-stationary case. For stationary conditions the time origin of the diffusion process is set to - ~ , which means that any observation probes an ongoing process. Interestingly, the stationary and non-stationary situations arise in iterated maps depending on the way averages are realized; averages can be taken over space or time. In the case of broad dis-

437

tributions the two ways of taking averages can lead to qualitatively different results. Consequently, in the analysis of the meansquared displacement and of the propagator one has to distinguish between the stationary and non-stationary situation. It turns out that differences between the two cases are more pronounced for the propagator than for the meansquared displacement. The situation is different for power spectra which in principle display stationary state properties. This is evident from the definition of the power-spectrum which for the case treated here is T

S(o~)__lrim(2T)_llfdte_i~tv(t)

2,

(2)

0

where v(t) denotes a time-dependent stochastic velocity (here and in what follows we do not consider the factor ~ for the normalization which is present in the original definition of S(to)). An equivalent description is given by the W i e n e r Khinchin theorem which is the cosine transform

C(t)

(3)

of the velocity-velocity correlation function which is defined as

C(t)

S(to) = i dt cos(wt) 0

C(t) = (v(t) v(O)) T-t l,

= lim T 1 / T~

d~- v(T)

v(t + "r).

(4)

J 0

As a consequence of taking T--->~ it follows that stationary conditions hold. However, even when a cutoff is introduced and T is taken to be finite, for instance due to a finite system time T = t . . . . S(to) mimics stationary-state behavior for to T -1 because the way of averaging over time in eq. (4) implicitly assumes that there is no particular time origin for the observation. From a random walk point of view stationarity implies that renewal properties apply to the motion [3,36]. Furthermore, the renewal theory

G. Zumofen, J. Klafter / Intermittent chaotic systems

438

determines how the velocity-velocity correlation function is related to the probability distribution of the single motion events. For non-stationary conditions, on the other hand, the first motion event is assumed to be initiated at the time origin of the observation. The mean-squared displacement and the velocity-velocity correlation function are related by the expression t

(r2(t)) = 2 / d~" (t - z)C(z).

(5)

0

Consequently, the power-spectra are also related to the mean-squared displacement. Thus in the analysis of power spectra the expressions obtained from the propagator for stationary conditions must be identical to those derived from the renewal theory. The purpose of this paper is to verify this fact in the case of enhanced diffusion obtained from non-linear iterated maps.

2. Random walk description of map-generated enhanced diffusion In this section we study iterated maps that have been shown to produce enhanced diffusion by intermittent chaotic motion [23,24,29-31]. We denote the map by g(x) and write for the nth iteration (6)

x , +1 = g ( x , ) .

Assuming periodic and inversion symmetries we have g(x + N ) = g(x) + N ,

g(-x) = -g(x),

(7)

where N is an integer and denotes the box number. With the help of these two rules the definition of the map is required only for the reduced range 0 < x < ½. Following the formulation in refs. [21,22], one may decompose the coordinate x of the trajectory into the box n u m b e r N and the position ~ within a box

x. = N. +

(8)

Thus the box number and the reduced coordinate £ have to be iterated individually £,+1 = ~(£n),

0<£<1,

(9a) (9b)

Nn+ 1 = ~,(£,) + N , ,

where g(x-) is the reduced map for the reduced coordinate £ and ~ ( ~ is used to increment or decrement the box number N. For the mean-squared displacement after t iterations we have (r2(t))

:

(10)

((Xn+ t - - X n ) 2 ) ,

with the initial coordinate x 0 chosen arbitrarily. The problem of how to take the average will be discussed below. The propagator P(r, t) is calculated accordingly P(r, t) = ( a ( r - x , + t + x , ) ) .

(11)

We follow a map studied by Geisel, Nierwetberg, and Zacherl [24] which generates enhanced diffusion. For the reduced range the definition is g(x)=(l+e)x+axZ-1,

O<-x<-½,

(12)

where a is set to a = 2Z(l - e/2). The quantity e in eq. (12) is supposed to be small and serves as a cutoff to enable the normalization of the invariant measure for the reduced map for z > 2. The map in eq. (12) and the corresponding reduced map are shown in figs. 1 and 2. The periodic symmetry and the discontinuities at the cell boundaries are evident from fig. 1. The reduced map function, as shown in fig. 2, exhibits marginally stable fixed points at the cell boundaries for e--~0. Thus these fixed points give rise to a slow change of the reduced coordinate £ in their neighborhood. From this an intermittent behavior emerges, i.e. the reduced coordinate is almost constant for many iterations and then varies rapidly for a few iterations which are followed again by a period of small changes. For the coordinate x this means that during the laminar phases iterations are associated with an

G. Zumofen, J. Klafter / Intermittent chaotic systems

439

ft)

3

2 .

.

.

.

.

.

.

.

1

I

-1

I

200

-I

o

1

2

X

Fig. 1. The map g(x) for the enhanced diffusion, eq. (12), for z = ~. Note the discontinuities for x = N. 1

(xJ z %/

/

=

/, 0

0.5 S

Fig. 2. The reduced map if(x) for the enhanced diffusion and for z -_ 7' 5 The branches on the left and on the right are associated with a decrement and an increment of the cell number N, respectively. Note the marginally stable fixed points at £ = 0 and £ = 1.

increment, or decrement, of the box number N; this can b e i n t e r p r e t e d as a m o t i o n at a c o n s t a n t v e l o c i t y . T h e l a m i n a r p h a s e s a r e i n t e r r u p t e d by

400

t

Fig. 3. The time evolution of the velocity u(t) as a function of the number of iterations t. In the upper part the result of a typical realization of iterated maps is shown for z = ~ (3' = 3). In the lower part of the figure the idealized signal is shown which is considered for the probabilistic description of the motion. i t e r a t i o n s which a r e a c c o m p a n i e d b y c h a n g e s o f d i r e c t i o n s a n d b y s t e p s o f s h o r t length. This b e h a v i o r is e v i d e n t f r o m fig. 3 w h e r e an e x a m p l e o f t h e e v o l u t i o n o f t h e v e l o c i t y with t h e n u m b e r o f i t e r a t i o n s is d i s p l a y e d . T h e v e l o c i t i e s v(t) a r e d e f i n e d b y v ( t ) = x t + 1 - x , . T h e figure d e m o n s t r a t e s t h a t t h e e v o l u t i o n follows l a m i n a r p h a s e s with velocities at an a l m o s t c o n s t a n t v a l u e i n t e r r u p t e d by s h o r t c h a o t i c m o t i o n s . In o r d e r to m o d e l this b e h a v i o r w e c o n s i d e r t h e following a s s u m p t i o n s : v(t) d e n o t e s a stochastic t i m e d e p e n d e n t f u n c t i o n . T h e m o t i o n t a k e s p l a c e at a c o n s t a n t v e l o c i t y for s o m e t i m e (number of iterations) after which the direction a n d t h e l e n g t h o f t h e n e x t m o t i o n e v e n t is c h o s e n randomly. The motion events are uncorrelated and the lengths of the motion events follow i n d e p e n d e n t l y a p r o b a b i l i t y d i s t r i b u t i o n @(t). This i d e a l i z e d p i c t u r e o f t h e m o t i o n , as illust r a t e d in t h e l o w e r p a r t o f fig. 3, defines t h e p r o b a b i l i s t i c d e s c r i p t i o n o f t h e m o t i o n which was i n t r o d u c e d for t h e v e l o c i t y m o d e l in t h e C T R W a p p r o a c h [29,30]. T h e a p p r o a c h is b a s e d on t h e

G. Zumofen, J. Klafter / Intermittent chaotic systems

440

m e m o r y function tO(r, t), the probability to move a distance r in time t in single motion event. For the velocity model a space-time coupled memory was applied, defined as

tO(r, t) = ½a(lrl - t) tO(t),

(13)

where the delta-function accounts for the motion at a constant velocity and where length and time are given in dimensionless units. CTRWs with coupled memories tO(r, t) can be referred to as L6vy walks [32-35]. Actually, a variant of eq. (13) was considered in refs. [29,30] to account for the spatial scale invariant set of turning points. H e r e we make use of eq. (13) which may be more directly related to the temporal intermittency picture. The two approaches, however, are equivalent and lead to identical results in the analysis of the asymptotic behavior. In a previous work we introduced the velocity model [29] and we analyzed the mean-squared displacement and the propagator for various regimes of enhanced diffusion. Here we first present expressions for the propagator from which we obtain asymptotic forms for the meansquared displacement; these forms are then used as a basis for the power-spectra. In ref. [24] the renewal theory was applied to determine expressions for the mean-squared displacement. We shall confront the two approaches. We begin with tO(t), the probability distribution to stay in laminar phase for t iterations and adopt the expression derived in ref. [24],

where 3' = 1 / ( z - 1) and b =2z-l/[a(z - 1)]. The constant b simplifies when we use the expression 1 for the constant a for e--+ 0; then one has b -- - ~3'. Although b is specified through z and e we consider this constant to be an adjustable parameter because we noticed from numerical experiments that the asymptotic scaling behavior of tO(t) can be verified whereas the prefactor follows only tentatively the predicted z-dependence. In this work we consider 3' as the relevant exponent for the description of the motion events. In a previous work [29] the motion events were based on the spatial distribution tO(r)--r-"; the relationship between the two exponents is simply /*=3'+1. Depending on the exponent y it has been found that the motion shows a range of behaviors, namely regular diffusion for 3' > 2, intermediate enhanced diffusion for 1 < 3' < 2 and ballistic type of motion for 0 < 3' < 1. The ballistic type regime is characterized by a diverging mean sojourn time i where the mean sojourn time is the first moment of the tO(t) distribution

i = J dt ttO(t).

(17)

0

Expressions (14) for l ~ t ~ e -1 and (15) for 1 < t show a scale-invariant power-law dependence tO(t)- t -z/(z-l~. In our analysis we use

A diverging t-is associated with an invariant density for the reduced map that is not normalizable and consequently a cutoff has to be introduced [31]. This peculiarity is related to the problem of stationarity. Usually, for stationary conditions the case of a diverging i is of no importance, because the particle is trapped in an ongoing process. Consequently, the particle remains in a moving mode so that a purely ballistic behavior results. Empirically, we found that iterated maps represent a showcase to demonstrate the significance of the stationary and nonstationary conditions. As mentioned above the method of taking averages in eqs. (10) and (11) plays a key role: averages can be taken over space or time. We thus write

tO(t)

(r2eq(t))

tO(t)

=

2(2z-1e + a) e ~(z-~)' [(2z_ 1 + a/e) e "~-I)' -- a/ey/~-~) '

(14)

which for l i m e ~ 0 is tO(t) = 2a/[2 z-I + a(z - 1)t] z/(z '~.

= y b V / [ b + t] v+' ,

(15)

(16)

=

((Xn+ t -- X n ) 2 ) n

(18)

G. Zumofen, J. Klafter / Intermittent chaotic systems

for stationary conditions with the average taken over the steps and =

((xt - x 0 ) 2)xo

441

qt(r, t) denotes the probability density to move a distance r in time t and to continue the motion in a single motion event

(19)

for non-stationary conditions with the average taken over the uniformly distributed coordinate x 0. The two ways of averaging lead to different results for the asymptotic behavior for y < 2. This provokes the question whether self-averaging takes place in this systems. For 3' < 1 and for a finite number of iterations the result of eq. (18) depends on x 0, that is, the result differs from sample to sample. Only in the limit of an infinite n u m b e r of iterations does the result coincide with the theoretical asymptotic behavior. In our previous papers we considered a slightly different definition for the two methods of taking averages. In order to optimize the numerical implementation we did not take averages over x 0 explicitly to obtain non-stationary state results but we regarded it as adequate to take the average over sequences of iterations for which the first iteration was a chaotic step or an injection step into a laminar phase. More precisely, one single walk was computed and averages were taken according to eq. (18) but with n chosen for steps which met the condition Ix, + 1 -x,I < 1 . In the probabilistic approach such a condition fixes the instants at which new motion events are initiated. Depending on the way of averaging two different propagators can be determined. We have derived expressions for both cases [31]; we do not repeat the derivation here but give the equations necessary for the subsequent analysis. For stationary conditions we have [31]

qt(r, t) =

18([rl -

t) f dr ~b(~-).

~0(r, t) and ~ ( r , t) are relevant quantities for the average motion events. For the exceptional first motion event overlapping with the observation time analogous distributions have to be defined. We follow Haus and Kehr [3] and let h(r, t) denote the probability distribution to move a distance r in time t and to stop for choosing a new direction and a new sojourn time at random for the first motion event

h(r, t) = t-aqP(r, t) .

(22)

Furthermore, H(r, t) is the probability distribution to move a distance r in time t and to continue the motion in the first motion event

H(r, t) =

½?-18(Ir I -

t) J d~- 0 - t)~b(~-).

U) = ltt(k, u) h(k,

+ H(k, u),

u)/[1

(23)

t

We now give the F o u r i e r - L a p l a c e transforms of eqs. (13) and (21)-(23). For ~0(k, u) we choose the form ~O(k, u) = ½16(q) + 6 ( 4 ) 1 ,

(24)

with q = u + ik and ~ = u - ik and with

6(q) = f dt e-q'qJ(t).

(25)

We also have q'(k, u) = ½{q-~[1 - 6(q)]} + C C ,

peq(k,

(21)

t

(26)

- i#(k, u)]

(20)

where the quantities ~, h, H are related to qJ [3,31]. We recall that qJ(r, t) is the probability density to move a distance r in time t in a single motion event and to stop for choosing a new direction and a new sojourn time at random.

where CC denotes the conjugated complex form of the first part on the rhs of the equation. From eq. (22) it follows that h ( k , u ) = ~ ( k , u ) / L Furthermore, we derive

H(k, u) = ½q-~{1 - (i-q)-1[1 - th(q)]} + C C . (27)

442

G. Z u m o f e n , J. Klafter / Intermittent chaotic systems

In order to recover the propagator for nonstationary state conditions we simply have to replace the distributions h and H by the corresponding distributions of the average steps: h(r, t) = O(r, t) and H(r, t) = gt(r, t) which leads to [29,30]

P"e(k, u) = qt(k, u)/[1 - O(k, u)].

(28)

We now calculate the mean-squared displacement making use of the expressions for the propagator. In Laplace space one has (using eq. (20))

2 2 (r.~(U))=--OkP

ne

(k, H )lk=O

2

= -a~{~v(k, ~)/[1 - 4,(k, u)]}l~=o

= 2u

3

-I- {2/U2[1 -- ~b(u)]} Ouqb(u ) , (33)

which clearly differs from eq. (29) and is not related in an obvious way to C(t). According to eq. (3) the power-spectrum is the Fourier transform of the velocity-velocity correlation function. Using the Laplace transform of this function one has S(to) =lim [C(u = r / + ito) + C(u = r / - iw)]. */---~0

(k, U)[k= o

(34)

= - ( 2 / { u ) 02qt(k, u)

We proceed by calculating the asymptotic forms for the mean-squared displacement and for S(w). We first note that based on eqs. (16) and (25) ~b(u) can be given analytically as [38]

2

eq

(r2eq(U)) = - O k P

- ~-- I U - 2 o 2~q,(k, u)

- O2H(k, U)lk=O = 2u-3

_

(2/{u4)[1

_

6(U)] .

(29)

This is one of the main results of this paper. Starting with the propagator we reproduce the expression which was obtained from renewal theory in refs. [24,37]. For comparison we present also the renewal approach for which the velocity-velocity correlation function is given as

¢(u) = yb~u ~ eb"F(--y, bu),

(35)

where F(T, x) denotes the incomplete G a m m a function. For non-integer Y values and for small x we apply the series expansion [38] x J 2 7_

F ( - T , x) = F ( - T ) (1 - e -x

s:0

r(j¥1-3`)/ (36)

C(t) =

{-1

i d-r (z - t)~b(-r) ,

(30)

For integer 3` values eq. (35) can be written as

l

6(u) = 3' eb"E~+,(bu),

which in Laplace space is C(u)

= u -1 -

U'u

211 - 4 , ( u ) ] .

(31)

The mean-squared displacement is related to the velocity-velocity correlation function according to eq. (4) which in Laplace space reads (r~q(U)) =2U-ZC(u).

(32)

Inserting C(u) of eq. (31) into eq. (32) we obtain eq. (29). For non-stationary conditions we make use of P"e(r, t), eq. (28), and obtain

3` = 1, 2 . . . . .

(37)

where E,(x) is the exponential integral [38]. For further derivations we consider the recursion relation

En+ l (x )

=

n - i [ e -x - xE,(x) ]

(38)

and the series expansion

E,(x) = - ' ~ - l n x - ~ 1)Jx------~J (j=, JJ! '

(39)

where ~ is the Euler constant ~ = 0.5772 . . . . For 3` -< 1 the mean-waiting time diverges [ = ~, while for 3` > 1 it is finite [= b/(3` - 1). For the

G. Zumofen, J. Klafter / Intermittent chaotic systems

non-stationary case, eq. (33), the asymptotic forms for the mean-squared displacement are f(1 --y)t 2 ,

O
t2

y=l,

'

(r2ne(t)) -- ~

2(y - 1)b ~-1

|(3-y)(2-y)

t 3-y ,

]2bt In t ,

1<7<2, y=2,

k2b(y - 2)-1/,

2< 7 . (4O)

We now give the analogous expressions for the stationary case. As pointed out above the average sojourn time t- diverges for 0 < y -< 1. Consequently, there are no turning points and thus the motion is purely ballistic, (r2(t))/t 2= 1. However, in order to extend the random-walk aspect to the 0 < y-< 1 regime one has to introduce cutoffs .which is achieved by either setting e # 0 in eq. (12) or by assuming a finite system time tmax < o0 such that t-is finite; both methods lead to equivalent results. We consider a finite system t i m e /max SO that

the prefactor for non-stationary conditions is (1-y) 2 regime (r~q(t)) = (r2e(t)) to leading order including the prefactors, therefore this regime was not included in eq. (42). Using eqs. (31) and (34) we find for the power-spectrum S(,o) f2F(2 - y) cos(y~r/2)t ~ o ~

/

v-2 ,

Ir/o~ In tm~x ,

)-2F(2

- y ) c o s ( y ~ r / 2 ) b~-~oJ ~ 2

/2blln(bco)l, kconst.,

0
< 1,oJ >>tm] x ,

y = 1, oJ >>tmlax, l
(43)

/max

i-= J dt ~b(t) 0

_

443

y

-1 1-y

b(1-y)

t....

0
[blnt ....

(41)

For y > 1 the mean waiting time is still given by {= b/(y - 1). Making use of eq. (29) we obtain for y < 2 (r~, (t)) t2 1

(3

1-~,

--

y )(2 - y )tma x

t2(1_

lnt ] lntmax / ,

tl-v '

0 < y < 1 t "~/max ' ' y= 1 ,t~t

....

2b ~- t

.(3- y ) ( 2 - y )

t3 ~,

1
(42) We notice that for the ballistic regime y -< 1 the correction terms vanish for tmax---*oo. For y < 1

Here we have ignored the 6(to) contribution for 31 ~ 1. We notice that for 0 < y < 2 the power spectrum characterized by the power law S to *-2 which does not show a transition between the two regimes of the intermediate enhanced and ballistic motion. Only the prefactor indicates that y = l is marginal and that for y < l the power spectrum vanishs for tmax--"~oo. Eqs. (43) are in agreement with the results obtained in ref. [24]. We now present numerical results obtained from iterated maps. We first concentration on the mean-squared displacement in the ballistic type regime. Some numerical results were presented already in ref. [24]. Furthermore for non-stationary states and z = 2 results were given in ref. [31]. In fig. 4 we show the mean-squared displacement for non-stationary states and for z > 2. The results obtained from iterated maps are plotted a s (r2q(t))/t 2 for z = 7 , 3 and 5 corre-

G. Zumofen, J. Klafter / Intermittent chaotic systems

444



1.0

10 1

te 0.8

z = 3

z =5

~z

tm.x

10 -~

10 -3

0.6 z=3

10 '*

0.4

10 5

0.2

i 0

.........

k 10

Fig.

........

L 10 z

..i 10 a

10 4

10 6 . . . . . .

. .

1

........ 10

,

..................

10 2

L _ ~

10 3

10 5

10 4

10 5

4 . The mean-squared displacement obtained from iter-

ated maps for the ballistic type of motion for non-stationary conditions. Plotted is (r2,¢(t))/t 2 vs. t for z = ~ , 3 and 5 as indicated. The numerical results were obtained from 104 realizations of the walk. The predicted asymptotic behavior, according to eq. (40), is given by the dashed lines at ordinate i 7' a n d ~3. values equal to l - 3' -_ ~,

sponding to 3" values of 3, ½ and ¼, respectively. A v e r a g e s over 104 realizations were taken according to eq. (19). With increasing time the curves approach asymptotic values which are indicated by dashed lines at ordinate values of (1-3,). The results indicate that the meansquared displacement behaves ballistically and that the prefactor is really given by eq. (40). H o w e v e r , from the numerical results were not able to determine correction terms to the leading term of eq. (40) which followed a systematic d e p e n d e n c e on z. Rather we observed that the behavior at small and intermediate times t depended on the details of the numerical realization of the maps. In fig. 5 results are s h o w n for stationary state conditions with averages taken according to eq. (18). T o demonstrate the correction to the ballistic behavior we plotted 1 - (r2e(t))/t 2. For the interpretation according to eq. (42) we set the system time tmax equal to the number of iterations. The simulation results obtained for z = 3 (3" = ½) follow reasonably the power-laws and the d e p e n d e n c e on t . . . . as presented in eq. (42). Numerical experiments, not shown here, indicate a dependence on x 0 in particular for smaller tmaxWe continue by discussing the numerical re-

t

F i g . 5. The mean-squared displacement for the ballistic type

of motion for stationary conditions. Plotted is 1 - ( r ~ q ( t ) ) / t 2 vs. t as full lines for z = 3 ( Y = ½ ) and for various t . . . . as indicated. The dashed lines give the theoretical behavior according to the correction term in eq. ( 4 2 ) .

suits calculated for the power spectra S(co). Walks were generated from iterated maps for the number of iterations equal to the system time /max ~ 106" Then the power spectra were calculated according to eq. (2) and averages were taken over 104 realizations of the walks with random initial coordinates x 0. In fig. 6 we monitor the power-law behavior of the power spectrum S(~o) -- to v-e, 0 < 3, < 2. Plotted is S(o~) vs. ~o on log scales for z = ~-, 2 and 3, corresponding to 3, = 3 , 1 and ~, 1 respectively. The numerical results follow reasonably 104 los

"". ""-.

101021

"""

z = 2

""'" /

10-1

)

~

z = 10 e

10-3

, 10 . 4

........

L 10 - 3

........

, 10 . 2

........

L

......

T:

10 1 5)

S(o~) maps. S(~o) is given by full lines indicated. The dashed lines give the z = ~3 the prefactor was corrected by Fig.

6. The power spectrum

obtained from iterated for z = ~ , 2 a n d 3, as predicted behavior. For a factor of ½ for clarity.

G. Zumofen, J. Klafter I Intelmittent chaotic systems

the theoretical power-laws for small to values. Also quantitatively the predictions are reasonably corroborated for z > 2, (3, < 2) while for z = 5 (y = 3) the numerical results deviate from the predictions by approximately a factor ½. We explain this by the constant b introduced in eq. (16) with a z-dependence which could only tentatively be confirmed in our numerical calculations. In fig. 7 we show the dependence of S(to) on the system time tma x. Plotted is S(to) for z = 3 (y = ½) for various system times tma x. The numerical results are in good agreement with the predicted power-law behavior and with the dependence o n /'max according to eq. (43). The saturation (leveling off) at small to values is due to the fact that the frequency range is extended beyond the corresponding time range bounded

445

1 ~/'z/ax

lO-e

=3

' ~/

= I0z 10 -4

10-8

z = 4 / ' % ~ , 104 \ ' \

10-a

, ~\~, 0 -.~. e

10 -2

1

10 2

10 4

10B

~) ~raax

Fig. 8. The power spectrum S(to) obtained from iterated maps in the scaling representation. Plotted is S(to)/tm,x vs. tOtmaX for z = 3 and z = 4 as full lines for various tmax as indicated, The dashed lines give the predicted behavior according to eq. (43).

b y tma x-

In fig. 8 we show further results on the power spectrum for the z > 2 (y < 1 ) regime. In a scaling representation S(to)/t~ v s . totma x is plotted for z = 3 ( y = ½ ) and z = 4 ( y = l ) for various system times /max" For both z values we observe a remarkable data collapse onto a single curve which supports the predicted to and tma xdependence given in eq. (43). In fig. 9 we pay special attention to the marginal case z = 2 (T = 1). Plotted, again in the scaling representation, is S(to) lnlmax/tma x VS.

',

1T

tO -z tin,,., = i 0 z \ ~ ,

lOa"~kk " ~" 10-4 ~"

10-B

1°' \ ~ ,

........................, . . . . . . . . . . ~

10 - 2

1

10 2

lOa~',

.......\.,......

10 4

10 B

CO ~max

10 4

Fig. 9. The power spectrum S(to) obtained from iterated maps in the scaling representation for the marginal case z = 2 . Plotted is S(to) lntm~x/tm~X vs. tOtmaX as full lines for various tmax as indicated. The dashed line gives the predicted behavior ¢r/to.

s(~) lO 2

totma x for various system times tma x. A data collapse onto a single curve and a reasonable agreement with the predicted power law "tr/to is observed.

10 2

10-4

i0 -5

10 4

10-3

10-2

10-1

]

10

3. Conclusions Fig. 7. The power spectrum S(to) obtained from iterated maps. Plotted is S(to) for z = 3 from top to bottom for system times /max -- 102, 103, 104, 105 and 106. The dashed lines give the predicted behavior according to eq. (43).

We have studied a probabilistic approach to the enhanced diffusion generated by iterated

446

G. Zumofen, J. Klafter / Intermittent chaotic systems

maps using the CTRW velocity model. The quantitative and qualitative analyses of the timedependent mean-squared displacement and of the power spectra have demonstrated that iterated maps serve as useful examples for the study of enhanced diffusion, of the corresponding power spectra and of the role played by the initial conditions. We have shown that the expressions obtained for the mean-squared displacement from the propagator under stationary conditions and from renewal theory are identical. The CTRW approach appears as a consistent probabilistic description of the enhanced diffusion as obtained from iterated maps.

Acknowledgements We thank Prof. T. Geisel [37] for providing us with his unpublished notes on continuous-time random walks and on power spectra. We are indebted to Prof. K, Dressier, Dr. R. Stoop and G. Poupart for helpful discussions and to F. Weber for technical assistance. J.K. thanks Prof. R. Silbey of the MIT for the support and the hospitality during the time this work was carried out. A grant of computer time from the Rechenzentrum der ETH-Z/irich is gratefully acknowledged.

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