Diffusion in very chaotic hamiltonian systems

Volume 82A, number 8

PHYSICS LETTERS

20 April 1981

DIFFUSION IN VERY CHAOTIC HAMILTONIAN SYSTEMS~ Henry D.I. ABARBANEL and John David CRAWFORD Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720, USA Received 1 December 1980 Revised manuscript received 29 January 1981 We study nonintegrable hamiltonian dynamics: H(I9)

=

J1~(I)+ kIt

1 (1,9), for large k, that is, far from integrability. An integral representation is given for the conditional probability P(I,9, tiI0,00, t0) that the system is at I, Oat t, given it was at J~,O~at t0. By discretizing time into steps of size e, we show how to evaluate physical observables for large k, fixed e. An explicit calculation of a diffusion coefficient in a two degrees of freedom problem is reported. Passage to e 0, the original hamiltonian flow, is discussed.

In this note we study hamiltonian systems which are very far from integrable. For action (I), angle (0) canonical co-ordinates we are interested in hamiltonians of the form H(!,O) = I1~(i)+ ku1 (1,0),

(1)

when k is large. The small-k behavior of the motion generated by (1) has been extensively studied in perturbation theory. We are interestedhere in the opposite limit where one encounters complicated, irregular motion [1]. Our central tool is the conditional probability P(z, tIw, t0) that the system is at z = (1,0) at t given it resided at W = (‘o~°o)at t0. The motion arising from (1) is, of course, deterministic, but we expect the irregular behavior at large k to be describable by probabilistic ideas such as diffusion. The use of the conditional probability to discuss this intrinsic stochasticity is quite appropriate. Intrinsic stochasticity [1] refers to the apparent random behavior of a deterministic system. The onset of this behavior is known from numerical studies to be quite abrupt as parameters of a physical system are increased beyond some critical value. The intrinsic stochastic behavior is characterized by a non-zero Kolmogorov—Sinai entropy and by exponentially fast separation of neat orbits in phase space [1] When a deterministic system exhibits this irregular, apparently random behavior, we expect diffusion to be a characteristic phenomenon which can describe the behavior of an orbit or groups of orbits in phase space. Our goal in this note is to calculate a diffusion coefficient, defmed below, in the regime far beyond the critical parameter values, namely, the regime where the deterministic system appears very stochastic. The conditional probability satisfies Liouville’s equation [2] -

[a

~

Iau ~

aH

a~1

~

(2)

with the initial condition P(z, t01 w, t0) = ~(z

—

w).

The general solution to this is given in terms of the solution to Hamilton’s equations of motion, X(w, t), which at t = 0 satisfies X(w, 0) = w. We have *

Work supported in part by US Department of Energy, under contract W-7405-ENG-48, and by ARPA Grant No. 4805-2.

378

(3)

Volume

82A, number

PHYSICS

8

20 April 1981

LETTERS

P(z, tl w, to) = 6(z - X(w, t - to)).

(4)

If we split the interval to to t into N segments of length E, t - to = NE, we may repeatedly of a conditional probability

use the property

t/z!, t’)P(z’, t’l w, to)

P(z, tl w, to) =sdz’P(z,

(5)

to write Pb,fl~,fo)=~~dr(l)6(x(0)-w)S(x(,\?-z)~P(x(s),t~=to+srlx(s-l),f~_,=to+(s-1)~). s=l I=0 We require, then, the elementary ~(x(s),tJx(s

- l)J_,)

conditional

probability -

= 8(x(s) - X(x(s - l),E)).

Now we must discretize the equations

(6)

(7)

of motion:

(8)

~-VW, Oldt = Wtw, t)), which we do in the form e-l [X(w, fS) -

Jxw_J

=ww,ts_~))’

(9)

leading to N P(Z,dW,

to) =J,fj

dr(l)6(x(O)

- w>S(x(ql

-zyj

6(x(s)

-x(s

-

1) - EB(X(S -

In the case of hamiltonian dynamics coming from (l), we take for the discretized equations phase space volume preserving mapping Z(n) = Z(n - 1) + &Zz(z(~),e(n

(10)

1))). of motion the

- 1))

(11)

e(n) = qn - 1) + eo(Z(n)) + &Zl’(Z(n),B(n - l)), where I(&), e(t,) are written as Z(n) and 0(n), o = SZo/aZ, h = -XZ1 /a& h’ = aZZ, /aZ. The conditional ty P(Z, 0, tlZo, 0,) to) follows directly from (10). In the limit E + 0, we return to the actual hamiltonian (10) becomes ~(Z,~,tlZo,~oPto)=J x Mr)

(12) probabiliflow, and

I-I ~(r)d~(r)~(Z(tO)-ZO)~(Z(O-I)~(~(tO)-~O)~(~(t)-e) 7

- kZ@(r), x(r))G(r)

-0(5(T))

- kh’(J(T),X(T))),

(13)

up to an overall normalization factor. This functional integral representation has been given earlier [3]. Our goal is to utilize (10) to study physical properties of our hamiltonian dynamics at fured E for large k, and then we use this information to extrapolate back to E = 0. We proceed by representing each 6(x(s) - x(s - 1) eB(x(s - 1))) in (10) by a Fourier series. For the delta functions representing evolution in the angle variables, this is quite natural since each O(t,) = (f31(t,), f3,(t,), ...) lies in 0 < Cl,(t,) < 1. For the actions to remain in a finite interval, we require that the energy surface ZZ(Z,0) = constant, on which the orbits lie, be bounded. Then if any la appears in H with maximum power p, the values of Z,(t,) lie more or less in the range 0 < IZ,(t,)l < I_Elllp. Thus, on the energy shell, each Z,(t,) takes values in some finite interval. Denoting the length of this interval by L,, we write (13) as

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Volume 82A, number 8

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P(1,0,t!Io,0o,to)=ffldJ(~dx((1(0)—I~(x(0)_00)6(J(AO_I)o(x(AO_O)

x

~ s1

exp2xil3- [x(s) —~(s 1)— ew(J(s)) —

1 ~

X

—

ekh’(J(s),X(s

1)

—

—

1))]

l~~oo

exp 2riL;~mas•[~(s)

U L~

—

~(s

—

kha(1(S)~(s

—

1))]),

(14)

where l~is a vector of integers. To evaluate P(I, 0, tI! 0o~ t 0 ~ 0) for large k, fixed e, we choose that combination of l~and mas values which result in the smallest number of oscillating integrands of1a5the form exp {ik[functions ofI(s) and x(s)] }. This = mas = 0. Then we can arrange to have the next smallest number number will be zero, resulting from the choicechoice ofI~and m smallest of oscifiating integrands by judicious 5. Iffrom this number is n1 then 1 ~/2 which we derive the stationary phasecorrections approxito the leading termwe willgobeon(oscifiating function of k)/k’ mation. After that to n 2 > n1 oscifiating integrands, then n3 > n2, etc., so deriving an asymptotic series for P(z, tj w, t0). We illustrate these general remarks by the hamiltonian with two degrees of freedom ,

H(I,0)=~(I~ +122 +1112)+(k/2ir) (cos2ire1 +pcos2irO2),

(15)

which has a bounded energy surface and for small k always has overlapping resonances [4] in the physical region. By the surface of section method [1] we have numerically concluded that for p ~ 0, the motion generated by (15) is non-integrable. In fig. 1 we show the 11,01 plane at 02 = 0 for p = 1.0, k/2ir = 110. Using the technique outlined above we evaluated the diffusion coefficient defined by (t = t0 + Ne) D~I0,k)nlim ~~(t~I~N)_I0)2)0

(16) 2

lim ~fd1dodO0P~I,

N—i =

urn N—~oo

X

2

II ~ s1

11

exp2iril

5- [z(s)—X(s

1)— ew(Io

—

+

ek~ h(0(r)))].

e,p fixed

2

÷J0(Kp/2)2[J2(K)

(K

(18)

r0

l~~_~oo

The leading terms in D are D(1 2k2{~(l 0,k) klarge ~

where J

2

d2x(Oe2k2(~ h(0(r))) r~’O

~o

0

(17)

,tII0,O0,t0)~I—10)

2k) 2ire +~2)_[J =

2J 2(K)J0(Kp12)+p 2(Kp)J0(K12)1

2] +p2J

2 —J

+J 3(K)

1(K)

2 +J

2[J 0(K/2)

2(Kp)

2 —J 3(Kp)

2}

+ ~.},

(19)

1(gp)

1(z) is the ordinary Bessel function. The leading term in (19) arises from (18) when all i~= 0, as indicated before. The next term comes from = 0,r 1,2, ...,N;125 = 0, sarise = 1,2, ...,N; 11s =the 0,scombinations111 = 1,2, N. The~terms quadratic in Bessel functions from theand choices 12, + 121+1 = 0, r = 1,2,..., N; ‘l,r +ll r+2 = o~ ~ = 0 O1l~ +ll r+2 = ~ for r = 1, ...,N;1 2,5 = 0,s = 1, ...,N; and then interchange I and 2. The general pattern is to take longer and longer steps in the Fourier coefficient (1~)space. The longer the .,

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Volume 82A, number 8

PHYSICS LETTERS

I

I

I

20 April 1981

I

I

I

I

I

I

I

I I

I

I

I

0130-

-

~:::

‘:

0.115

I

0

I

I

I

I

0.2

04

ti

i

9I

I

I

0.6

08

1.0

Fig. 1. Surface of section plot for an orbit generated by our hamiltonian, eq. (15). We show the I~,e~plane for ~2 = 0. The initial conditions were Ij (0) = 3.0, 12(0) = 1.0, o~ (0) = 0.743, and 02(0) 0.0. The parameters k and p were ki2~ llOandpl.0.

0

ii

20

I

ii

30

Iii

40

50

2 from the asymptotic form of (19) compared to Fig. 2. D/k the numerical evaluation (points labeled by N) for 10 ~ k ‘~ 50, p = 1.0, e = 0.5. In this plot K = 2ire2k has the range 15.7 ~ ~ 78.5. The deviation of the numerical results from the analytic form is less than 4% and is consistent with the relatively small number of initial values of 80 96 in each angle.

—

step, the larger the number of oscillating integrands. In this problem each oscifiating integrand can be given by an ordinary Bessel function. For p = 0 the discretized volume preserving mapping arising from the hamiltonian (15) is the Chirikov standard mapping [4] with parameter K. Indeed, for p = 0, (19) reduces to the results of Rechester and White [5] including a correction, the J 2 term, pointed out by Cohen [6]. It is important to 2(K) note that our derivation ofD(1 0, k) contains no assumed randomness added to the hamiltonian equations of motion. A non-zero diffusion coefficient arises from the intrinsic stochasticity of the motion. (External random forcing can be incorporated into our representation for P(z, tI w, t0) in a straightforward fashion. This will be discussed by us in the near future [7] .) In fig. 2 we compare the asymptotic form (19) for p = 1.0 to a direct numerical evaluation of D(10 , k) from its definition (16). Returning to the original flow, namely e = 0, is accomplished by the following observations. When 0, we require the diffusion coefficient ~(‘o, k) given by -~

D~(10,k)

lim ((I(Ne) Ne—~=

—

~

/2N.

(20)

0

From (19) we see this has the form (21) withy = ~ and C~) a dimensionless function. We have evaluated ~7) (y, p) as a series in y near y = 0, and require its value at y = oo The standard techniques for making such an extrapolation [81 need knowledge of a large number of terms in the series we have begun in (19J. The result of determii~ing these andincarrying the de3/2(b(p), CL) dimensionless. This wqrk will be terms reported anotherout paper. We sired here extrapolation wifi yieldD = k note only the change in k dependence from the diffusion of actiQns in ~he mapping. Our general technique has provided us with a tool for learning the behavior of physical quantities in the limit 381

Volume 82A, number 8

PHYSICS LETTERS

20 April 1981

where the hamiltonian, eq. (1), is very non-integrable. Most interesting among these physical quantities are those, like our diffusion coefficient, which characterize the intrinsic stochasticity of the chaotic orbits of the system. These are not amenable to calculation by perturbation theory in k; they must vanish in every order since diffusion or intrinsic stochasticity is not a small-k phenomenon. Passage to the original flow from our discretized phase space volume preserving mapping requires some labor and may be subtle; still our results fmd direct application to mappings known to accurately approximate a physical flow. Finally by use of these methods one may hope to learn much about more complicated intrinsically stochastic motions, such a fluid turbulence at large Reynolds number with no artificial external forcing or externally imposed random boundary or initial conditions. References See the lectures in: Topics in nonlinear dynamics, ed. S. Jorna (American Institute of Physics, New York, 1978). J. Ross and J.G. Kirkwood, J. Chem. Phys. 22 (1954) 1094. B. Jouvet and R. Phythian,Phys. Rev. Al9 (1979) 1350. B.V. Chirikov, Phys. Rep. 52 (1979) 263. A.B. Rechester and R.B. White, Phys. Rev. Lett. 44 (1980) 1586. [6]R. Cohen, private communication. [7] H.D.I. Abarbanel and J.D. Crawford, Strong coupling expansions for nonintegrable hamiltonian systems, LBL Report #11887 (Nov. 1980). [1] [2] [3] [4] [5]

[8J E. Mjolsness, H.A. Rose and D.H. Sharp, Adv. Appl. Math. 1(1980) 22.

382