Stochastic diffusion in the standard map

Physica 29D (1987) 247-255 North-Holland, Amsterdam

STOCHASTIC DIFFUSION IN THE STANDARD MAP Yoshi H. ICHIKAWA, T. KAMIMURA and T. HATORI Institute of Plasma Physics, Nagoya University, Nagoya, Japan Received 12 November 1986 Revised manuscript received 21 May 1987

A numerical observation of stochastic diffusion in the standard map has been carried out in the domain of small stochastic parameter Ac < A < 1.0, where Ac is given as (2~r) -a × 0.9716 = 0.1546. Multiple periodic accelerator modes manifest their pronounced sharp resonant effect in the stochastic diffusion process even in the region of small stochastic parameter, where the fundamental accelerator modes are not allowed to exist.

1. Introduction

The analysis of nonlinear dynamical maps provides a powerful approach for the investigation of the long-time behavior of nonlinear dynamical systems. In an illuminating monograph, Chirikov [1] had summarized extensive studies on the subjects up to the early 1979. Then, inspired by the advancement of experiments on the improved confinement of nuclear fusion devices, active interests on the stochasticity have evolved during the past several years. Especially, the standard map attracts strong interests to characterize its statistical properties [2]. Applying a nonlinear tandem-mirror map, we have carried out our direct numerical observation of the radial diffusion of collisionless guiding centers in the tandem-mirror devices [3]. It has been shown that the overall behavior of the diffusion process appears to be compatible with the prediction of the standard map reduced by local linearization of the exact nonlinear tandemmirror map. Since the standard map

X.+l = S

P.+t

=

P.

(1)

Pn + Asin(2crX.)

exhibits a peculiar resonant enhancement of the

momentum diffusion in the specific range of the stochastic parameter A, Karney et al. [4] have analyzed the statistical properties of the standard map under the action of external noise. They have shown that the accelerator modes exhibit a universal behavior at large value of the stochastic parameter. Carrying out a detailed analysis of accelerator orbits, we [5] identified that the resonant enhancement of the diffusion is due to the intermittent transition of particles from chaotic orbits into unstable accelerator orbits. Since the fundamental accelerator orbits can exist only in the domain A > 1, such resonant enhancement of the diffusion is presumed to occur in the same domain of the stochastic parameter. With regard to the higher order accelerator modes, Karney et al. have observed that the anomalous behavior found near A = 3/(27r) is due to the existence of a 4th order accelerator mode with a mean acceleration of a = ¼ and that there exist the 2nd order accelerator modes with a mean acceleration of a = ½ in the range 0.6404 < A < 0.6513, yet without establishing their explicit contribution to the diffusion coefficient in the above range. Therefore, we have undertaken an extensive survey of resonant enhancement of the diffusion in the domain A < 1. Our detailed analysis of the observed resonant peaks confirms that

0167-2789/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

248

Y.H. lchikawa et aL / Stochastic diffusion in the standard map

they are indeed due to the contribution of the intermittent transition between chaotic orbits and unstable higher order accelerator orbits. We present a list of the identified higher order accelerator modes in the domain A < 1. o__ 13,., 2

2. Enhanced diffusion in the standard map

The standard map eq. (1) describes discrete temporal evolution of a dynamical system with the Hamiltonian 0

, 1

0

p2 A H = T + ~

cos (2~r ( X -

nt)).

, 2

t 3

4-

(2)

n = --O0

Chirikov [1], Rechester et al. [2] and many others have examined statistical properties of this dynamical system by studying the momentum diffusion coefficient

where the average is taken over the ensemble of particles i. Applying the Fourier-path integral method, Meiss et al. [6] have obtained an analytic expression for the diffusion coefficient (3) as

Fig. 1. The normalized diffusion rate D/DQfor various values of A in the standard map. The thin curve represents the theoretical diffusion rate of eq. (4). The pairs of vertical lines at A = 1, A = 2 and A = 3 indicate the region of stable fundamental accelerator modes.

We indicate also the stable regions of the fundamental accelerator modes, which are determined from the constant phase condition A sin(2rrX (a)) = I,

1= integer,

(5)

and the stability condition D

%

1 - 2j2(2qrA) - y2(2qrA) - 2J2(2~rA)

(a + 4(2

A)) 2

- 4 < 2~rA cos (2~rX (a)) < 0.

'

(4) where J,,(x) is the nth order Bessel function and DQ = A2/4. Although they have improved the previous result of Rechester et al. [2] by taking account of a part of higher order correlation effects, they ignore the contribution of the accelerator modes. In fig. 1, we illustrate the observed values of the diffusion coefficient for numerical experiments together with the theoretical prediction of eq. (4). The initial ensemble of particles is a uniform distribution of 105 particles in the range 0 < Xo(i ) < 1 at Po = 0. The diffusion coefficient D is determined from eq. (3) after time steps of T = 300.

(6)

Eq. (5) determines the coordinate of the accelerator orbits as x(a)~

2~sin-l(/),

(7)

and thus after T-steps, the momentum P~) takes the value

P(r~) = lT + Po(a), while eq. (6) determines the stable domain as Ill
((5)C 1+

(8)

Y.H. Ichikawa et al./ Stochastic diffusion in the standard map

We have, for l = 1, 1 < A < 1.185 . . . . and for 1 = 2, 2 < A < 2.098 . . . . and so on. At A > 1.185 . . . . the fundamental accelerator mode bifurcates into the period-2 accelerator mode. We have shown in the previous report that, in the region A - 1.1, the period-3 accelerator mode causes intermittent transitions of particles between chaotic orbits and the accelerator orbit, and gives rise to the resonant enhancement of the diffusion. Another way of identifying the enhanced diffusion is provided by observation of the spread of the distribution function in the momentum space. Assuming that the process is Gaussian, we can write the diffusion equation as

249

10 °

T

101

~100

I'OLL

1 00

~,.a".162101 , ~~.z,..~,., .T"200

~)2

~ F ( P, t) = D - ~ F ( P, t).

(10)

1 ()q'

I

0

~0

80

120

160

200

P

For the initial condition F(P,O)

= NoS(P - Po),

(11)

solving eq. (10), we obtain

F(P,t)=

N°

exp

[

1 ( P Po)2] 4D t .

(12)

Now, fig. 2 illustrates temporal evolution of the distribution of 105 particles, which are distributed uniformly over the interval 0 < X < 1 with Po = 0. The curved line indicates the analytic expression (12) with an arbitrarily chosen value of D = 1.9 for A = 1.1. The Gaussian curve for eq. (12) represents fairly well the evolution of the low momentum particles, yet there appears an appreciable high momentum tail with a profound peak, which is moving out in accordance with eq. (8) with l - - 1 . This confirms that the leading peak represents the contribution of the fundamental accelerator mode. We can eliminate the contribution of these accelerator modes by shifting the initial value of the momentum P0. Fig. 3 shows evolution of a similar particle distribution starting at P0 = 0.25. Here, as expected, we do not have the leading peaks. Yet, we observe clearly the formation of

Fig. 2. Temporal evolutionof the distribution of 105 particles at A = 1.10 after T~ 100 and T~ 200, for the initial distribution F(P,O)= 105 × 8(P). the high momentum tail. This high momentum tail formation is manifesting its contribution as the enhancement of diffusion process. These observations confirm that the intermittent transitions between chaotic orbits and unstable accelerator orbits are giving rise to the enhancement of diffusion. In general, we can determine the multi-periodic accelerator modes as follows. The p t h iteration of the standard map is defined as

--

P~+p

.

The constant phase condition for

P,+p -

(13)

P~

P~ = 1,

l = integer,

P,+p, (14)

requires that p

A E sin (2rrX~) = l, i~l

(15)

250

Y.H. Ichikawa et a l . / Stochastic diffusion in the standard map

stability Condition

100 .-,

16 ~

T

=I00

- 4 < 3 ( a 1 + a 2 + a3) + 2(axa 2 + e2a 3 + ot3o/1) (16)

-I- alOl20t 3 < O,

I~ ~

ii

a i = 2~rAcos(21rX/),

V~3

i = 1,2,3,

(17)

where X2 = )(1 + e l + a sin (2~rXx),

I 0°

(18a)

)(3 = Xz + P2 + A sin (2~rX1) + A sin (2trX2)

(18b) % 1~~

determine three sets of the accelerator coordinates:

13_

16z 0

gO

80

120

160

200

P

P~(3; 3): (X0 = 0.3183 .... Po = 0.0; X 1 = 1.3183 .... P1 -- 1.0;

Fig. 3. Temporal evolution of the distribution of 105 particles

X 2 = 3.3183 .... /'2 = 2.0),

at A = 1.10 a f t e r T = 100 a n d T ~ 200, for the initial distribution F ( P , 0 ) = 105 x 8 ( P - 0.25).

Qs(3; 3):(X0 = 0.2865 .... P0 = 0.0; X~ --- 1.3576 .... P~ = 1.0711... ;

l determines the size of acceleration after p steps. Thus, the multiperiodic accelerator mode is characterized b y the periodicity p and the step size L The accelerator mode observed at A ~ 1.1 is the period-3 step-3 accelerator mode, while the fundamental mode in the domain 1 < A < 1.815... is the period-1 step-1 accelerator mode and the one in the domain 2 < A < 2.098... is the period-1 step-2 accelerator mode. Here, it would be worthwhile to remark on the difference between the period-1 step-1 accelerator mode (currently called the first order fundamental accelerator mode) and the period-3 step-3 accelerator mode at A = 1.1. For the specified value of A = 1.1, eqs. (5) and (6) determine the coordinates of the stable accelerator mode as P,(1;1): (X=0.31838826...; P=0.0).

Now, as for the period-3 accelerator step-3 accelerator mode, eq. (15) with p = 3 and 1 = 3 and the

X 2 = 3.2865 .... P2 = 1.9288... ), Qu(3; 3 ) : ( x 0 = 0.3094 . . . . P0 = 0.0; x 1 = 1.3336 . . . . Pt = 1.0242... ; x 2 = 3.3094 .... P2 = 1.9757... ), where Ps(3; 3) is nothing but the 3-step iteration of the fundamental accelerator point Ps(1; 1), while the pair of stable and unstable accelerator points Q~(3; 3) and Qu(3; 3) born when the stochastic parameter A becomes larger than 1.097. Increasing the stochastic parameter to the value A = 1.11, we observe that Q~(3; 3) turns out to be also unstable, although P~(3; 3) remains stable. This illustrates that the nonlinear dynamical systems exhibit not only sensitive dependence on the initial conditions, but also depend sensitively on the stochastic parameter A, which characterizes the given systems. Since the unstable accelerator orbits are responsible for the intermittent transitions between the chaotic orbits and the accelerator orbits, it is important to quantify the stickiness

Y.H. Ichikawu et al./Stochastic

0.0

0.2

251

diffusion in the standard map

0.4

0.6

0.8

1.0

A Fig. 4.

Thenormalized diffusion rate D/Da in the region A,
around the islands by analyzing the phase space structure of these coordinates.

3. Stochastic diffusion in the domain A -C 1 Now, it is fully recognized that regions near the last KAM curve surrounding regular regions have finite probability of trapping particles for certain period of time from the chaotic regions, and thus the stickiness of islands gives rise to dominant long-time correlation effect in the stochastic process described by the nonlinear dynamical maps. For the standard map, McKay et al. [7] have shown that the diffusion coefficient near the onset of stochasticity obeys the scaling law of Da(A-A,)‘,

17=3.01,

09)

where the critical stochastic parameter A, is given as A,=(2a)-‘~0.971 =0.1546... . Dana and

X at PO= 0.0.

Fishman [8] claim that the scaling law eq. (19) is valid up to a larger value A = 0.38. Referring to these studies, we have extended the numerical measurement of the momentum diffusion coefficient down to the region A ,< 1.0. The result is shown in fig. 4. Qualitative agreement with the theoretical values given by eq. (4) is maintained fairly well even for the smaller values of A. Furthermore, we observe several sharp resonance peaks and broad peaks in the diffusion coefficient. Suspecting that these resonance peaks are the contribution of accelerator modes, we followed the temporal evolution of the momentum distribution function as discussed in the previous section. Figs. 5 and 6 illustrate the results for A = 0.88 and A = 0.93, where the distribution function is plotted against P2. Typical orbits, which are responsible for these accelerator peaks, are shown in figs. 7 and 8. We can identify that the accelerator mode at A = 0.88 is the period-3 step-2 accelerator mode,

Y.H. Ichikawa et al./ Stochastic diffusion in the standard map

252 1

o°

1 o0 T

=300

T :300

d ,~2

oS ,a2

u_

U_

[

,e

lhllll

,o°{

10 °

•-, i~ I

T

-500

•.~

I 0 llt

0._

I~)2

T

-500

I-

1.0.1__~ 1~ 2 ~It l[~q-

LL

~IIP],IX II I .0

i

i

2.0

3.0

p2

A

i

i

ll..O

I

1,0

5.0

2.0

• 10 ~"

Fig. 5. Temporal evolution of distribution of 105 particles at A = 0.83 after T = 300 and T= 500 for the initial distribution F(P,O) = 105 × 8(P).

while the one at A = 0.93 is the period-7 step-2 accelerator mode. Fig. 9 illustrates an-other accelerator mode, which is the period-5 step-1 unstable accelerator mode. After 35 time steps, the orbit becomes chaotic and then returns into the period-5 step-1 m o d e around 80 time steps. This figure illustrates the intermittent transition between the accelerator orbit and chaotic orbit. We have carried out a detailed survey to identify various accelerator orbits in the domain A c < A < 1.0. Table I lists all of the accelerator orbits having the initial m o m e n t u m P0 = 0. In the last column, we remark how they contribute to the observed diffusion process. We can observe in fig. 4 seven line-like sharp enhancement peaks, and also find three b r o a d enhancement peaks in the ranges A = 0.4750 + 0.0150, A = 0.7525 + 0.0150 and A = 0.8850 + 0.0150. Since the stochastic diffusion is due to the intrinsic orbital instability, which implies the sensitive dependence on the initial condition at the same time, we have carried out the numerical observation for the 105 particles dis-

L

• .0

3.0

.0

p2 Fig. 6. T e m p o r a l e v o l u t i o n of d i s t r i b u t i o n of 105 particles at A = 0.93 after T = 300 a n d T = 500 for the initial d i s t r i b u t i o n F(P,O) = 105 × 8 ( P ) .

10 A

-

0.88

gl O

O

2 O e

I

I

I

I

2

Lt

6

8

10

Fig, 7. O r b i t s of particle starting at X o = 0.33485, Po = 0 for A = 0.88.

tributed uniformly in the domain 0 < X < 1.0 with Po = 0.40. We give the result in fig. 10. Comparing fig. 10 with fig. 4, we observe clearly that the b r o a d peak at A = 0.475 of fig. 5 grows into the line-like sharp peak in fig. 10, while other sharp

253

Y.H. Ichikawa et aL/Stochastic di@ion in the standard map

A

Table I List of period-p step-/ accelerator modes. The last column describes the apparent effect on the diffusion process. “Sharp” resonance in the diffusion process stands for peaks with A A c: 0.0005, while “broad” stands for peaks with 0.01 5 A A. The modes marked by * do not show clear evidence for enhancement of the diffusion process. } indicates that the identified ( p, I) mode remains stable over the indicated interval of the stochastic parameter A.

= 0.93

Effect on D, and other remarks

A

P

I

0.3475 0.3780 0.4060 0.4120 0.4200 0.4750

11 7 11 5 5 4

1 1 2 1 1 1

sharp sharp sharp, most pronounced for PO= 0.4

0.5325 0.5425 >

11

1

*

0.5575 0.5750 0.6020 0.6060

7 17 5 5

1 2 1 2

sharp * * *

0.6420 0.6540 >

4

2

* , pronounced for Pc = 0.2

0.600

3

1

sharp

0.7425 0.7625 >

3

1

broad, much sharpened for PO= 0.4

4 2 2 3 4

sharp sharp * *

* sharp *

Fig. 8. Orbits of particle starting at X0 = 0.32871, Pa = 0 for A = 0.93.

10 A = 0.415 B-

6-

P 4-

0

L’ cl

I

20

1

40

1

I

I

60

80

100

T Fig. 9. Orbit of particle starting at Xc = 0.10218, PO= 0 for A = 0.415.

peaks in fig. 4 are smeared out in fig. 10. Surveying the orbits associated with this sharp peak, we identified it as the period-4 step-l accelerator mode. At the same time, we see that the broad peak of the period-3 step-l accelerator mode in the region 0.7425 -CA -c 0.7625 in fig. 4 is much enhanced and sharpened in fig. 10. Recent studies on the effect of stickiness of the islands in the nonlinear map explore new aspects of transport process of the Hamiltonian systems [9]. Our observations described in this section provide the explicit evidences of interplay of the

0.8000 0.8150 0.8475 0.8625 0.8675

*

0.8780 0.8980 i

3

2

broad

0.9225 0.9400 >

7

2

broad

0.9825

6

4

*

Y.H. Ichikawa et al./ Stochastic diffusion in the standard map

254

~.0

D

3.0

DO 2.0

1.0 \

0,0

~

0.0

'

0.2

0.~

0.8

0.8

1.0

A Fig. 10. The normalized diffusion rate D/DQ in the region Ac < A < 1 for 105 particles uniformly distributed over X at Po = 0,4.

integrable orbits and the chaotic orbits. It is worthwhile that the stickiness of the fixed islands acts to suppress the spreading of particles, while the stickiness around the accelerator islands enhances the effective diffusion.

4. Concluding remarks Owing to the double periodicities inherent in the standard map, we have found that the standard m a p possesses full varieties of periodic accelerator modes, which give rise to resonance enhancement of the stochastic diffusion process. Detailed survey of various orbits starting at ( X 0 0, P0 = 0) has been carried out by changing the stochastic parameter A with an amount AA = 0.0005 in the range A c < A < 1. As summarized in table I, there are two kinds of accelerator modes, the one is stable only in the very small range of AA = 0.001 and the other persists for relatively large range of 0.01 < AA < 0.03. Although the accelerator modes give rise to enhancement of

stochastic diffusion process, not all of them manifest the observable effect in fig. 4. The modes marked by * in table I do not show clear effect on the diffusion process. Because of the sensitive dependence on initial conditions, they may give rise to clear enhancement of the diffusion for another choice of the initial m o m e n t u m Po. Indeed we observed that a broad peak in the range of A = 0.4750 + 0.0150 for P0 = 0 becomes a pronounced sharp peak for P = 0.4 as illustrated in fig. 4. In order to account quantitatively the effect of stickiness of the islands or the intermittent transitions between chaotic orbits and periodic accelerator orbits, it will be crucial to understand overall structure of the various accelerator modes. In this connection, we speculate that the sharp resonance peaks of the period-3 step-1 mode at A = 0.6600 and the period-3 step-2 mode at A = 0.8150 and the b r o a d peaks of the period-3 step-1 mode in the region 0.7425 < A < 0.7625 and the period-3 step-2 m o d e in the region 0.8750 < A < 0.8950 are forming a kind of sequence with the period-3 step-3

Y.H. Ichikawa et al./ Stochastic diffusion in the standard map

accelerator m o d e at A = 1.1. To provide another evidence for such speculations we turn to the period-5 accelerator modes. In the previous analysis, we observed a sharp resonance of the period-5 step-5 m o d e at A = 1.03. Here, we have shown that the period-5 step-1 mode at A = 0,4120 and the period-5 step-2 mode at A = 0.6060 give rise to sharp enhancement of the diffusion. Referring to table I, we notice that the period-5 step-3 mode appears at A = 0.8625, though its contribution to the diffusion is not so profound as of other modes. Although we failed to find the period-5 step-4 m o d e so far, we are tempted to speculate that the sequence of the period-5 accelerator modes is converging to period-5 step-5 mode at A = 1.03 as the sequence of the period-3 modes is converging to the period-3 step-3 mode at A = 1.1. In conclusion, since the intermittent transitions between chaotic orbits and periodic accelerator modes give rise to appreciable contribution even in the region A < 1.0, we emphasize that careful consideration is necessary to apply the standard m a p in analyzing the statistical behavior of a Hamiltonian system.

Acknowledgements The authors wish to express their thanks to valuable discussions of Dr. J.D. Meiss and other

255

participants of the series of workshops on statistical physics in fusion plasmas under the U S - J a p a n Fusion Cooperation Program. They are also obliged to Professor E.A. Jackson, the 1985 visiting professor of the Joint Institute for Fusion Theory under the U S - J a p a n Fusion Cooperation Program, for his enlightening suggestions. This work was supported in part by a Grant-in-Aid for Science Research of the Ministry of Education, Science and Culture under contract number 58380001.

References [1] B.V. Chirikov, Phys. Rep. 52 (1979) 263. [2] A.B. Rechester, M.N. Rosenbluth and R.B. White, Phys. Rev. A 23 (1981) 2664. [3] Y.H. Ichikawa, T. Kamimura and C.F.F. Karney, Physica 6D (1983) 233. [4] C.F.F. Karney, A.B. Rechester and R. White, Physica 4D (1982) 425. [5] Y.H. Ichikawa, T. Kamimura and T. Hatori, Statistical Physics and Chaos in Fusion Plasmas, C.W. Horton and L.E. Reichl, eds. (John Wiley, New York, 1984), p. 21. [6] J.D. Meiss, J.R. Cary, C. Grebogi, J.D. Crawford, A.N. Kaufman and H.D.I. Abarbanel, Physica 6D (1983) 375. [7] J.D. Meiss, R.S. MacKay and I.C. Percival, Phys. Rev. Lett. 52 (1984) 697; Physica 13D (1984) 55. [8] I. Dana and S. Fishman. Physica 17D (1985) 63. [9] J.D. Hanson, J.R. Cary and J.D. Meiss, J. Stat. Phys. 39 (1985) 327.