Fluctuation-response relations in systems with chaotic behavior

Fluctuation-response relations in systems with chaotic behavior G. F. Carnevale Scripps Institution of Oceanography, La Jolla, California 92093

M. Falcioni Dipartimento di Fisica, Universith ‘Za Sapienza, ” P. le A. More 2, 00185 Roma and INFN, sezione di Roma, Italy

S. lsola Dipartimento di Fisica, Universitb di Firenze, L. go Fermi 2, 50125 Firenze and Dipartimento di Matematica e Fisica, Universitb di Camerino, Camerino, Italy

R. Purini Istituto di Fisica dell’Atmosfera-CNR, P. le L. Sturzo 31, 00144 Roma, Italy

A. Vulpiani Dipartimento di Fisica, Universitb de L’Aquila, Via Vetoio 67010 Coppito, L’Aquila and INFM, Unitb di Roma, and INFN, sezione di Roma, Italy

(Received 5 October 1990; accepted 17 May 199 1)

The statistics of systems with good chaotic properties obey a formal fluctuation-response relation which gives the average linear response of a dynamical system to an external perturbation in terms of two-time correlation functions. Unfortunately, except for particularly simple cases, the appropriate form of correlation function is unknown because an analytic expression for the invariant density is lacking. The simplest situation is that in which the probability distribution is Gaussian. In that case, the fluctuation-response relation is a linear relation between the response matrix and the two-time two-point correlation matrix. Some numerical computations have been carried out in low-dimensional models of hydrodynamic systems. The results show that fluctuation-response relation for Gaussian distributions is not a useful approximation. Nevertheless, these calculations show that, even for non-Gaussian statistics, the response function and the two-time correlations can have similar qualitative features, which may be attributed to the existence of the more general fluctuation-response relation.

I. INTRODUCTION

The fluctuation-response (FR) relation has a long his- tory of application in all branches of classical and quantum statistical physics. This is a relation between the average linear response of the system to external perturbations and the average two-time correlations. Consider a system with m dynamical variables represented by Xi, i= 1, m. By the average linear response, G,?(T), we mean the average response, after a time 7, of the variable x6 to first order in the perturbation of the variable Xj at t = 0. The FR relation provides a formula for this response, which can be written, in generalized form,1,2 as

G&l = (xi(Wjb(0) I ). (1)

Here, we adopt a notation in which the full set of variables {x} is represented by x when no index is attached. We will give the formula for fj in the next section; its structure depends on the “natural” probability density, p(x) (i.e., the probability density reached asymptotically for t -+ 00 > .

If the probability density is Gaussian, the function fj is linear in the variables x(O), and so the FR relation is a simple linear regression in terms of second-order two-time moments. In theoretical turbulence studies, the simple version of the FR which is appropriate only for Gaussian statistics, has had some success in closure calculations,3’4 although this fails for small scales in high-Reynolds num-

ber flows due to intermittency.4 In climate modeling, this linear regression model has been shown to have some pre- dictive capability,4V5 and in plasma physics it has found success in certain low-dimensional models subject to the restriction that the initial distributions are Gaussian.6 These previous investigations raised several interesting questions worth further investigation. The moderate suc- cess5 of the linear regression model with a 25mode truncation of the Navier-Stokes equation poses the question of how useful it would be for, say, a five-mode truncation or for the celebrated three-mode Lorenz equations.7 As could be anticipated on theoretical grounds and from experience, the stationary statistics of these low-dimensional models do not obey the linear regression model well at all. Perhaps a somewhat more interesting question is how does the use- fulness of this model improve with an increase in the number of dynamical modes. Conservative systems in which the energy is quadratic in the dynamical variables should obey the linear regression model better if the number of dynamical variables is large. This follows from the fact that the statistics of the trajectory from any initial condition will be well approximated by a Gaussian distribution in phase space the larger the dimension of that space.8 We shall expand on this with a specific example. For more general situations, little can be added at this point. To test the validity of the FR for chaotic systems with a small

2247 Phys. Fluids A 3 (9), September 1991 0899-8213/91/092247-08$02.00 @ 1991 American Institute of Physics 2247

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number of degrees of freedom, we have investigated the Lorenz model and a five-mode truncation of the Navier- Stokes equation.’

For completeness, we will review the derivation of the FR relation in Sec. II. We will also address some of the criticisms and difficulties in regard to the derivation. Fur- ther, we will display explicitly the linear regression equations suitable for Gaussian statistics. In Sec. III, we present the results from the numerical experiments, which illustrate the failure of the linear-regression relation for three important chaotic models of relevance to fluid dynamics. These examples also show that the strength and time scales of the fluctuations are similar to those of the response functions, which we believe is related to the existence of the generalized FR.

II. DERIVATION OF THE FLUCTUATION-RESPONSE RELATION

Consider a representation of a dynamical system by the discrete set of ordinary differential equations

kf= Vi(X). (2)

Although more general problems in which V is stochastic and/or depends explicitly on time could also be intro- duced, this present representation will suffice for our pur- pose. These equations evolve a set of initial conditions x(O) to produce the solution X(T), which is then implicitly a function of the x(0). We will consider only the response to perturbations made in the initial conditions. At least for- mally, we can write for the perturbed solution, x’(t), the Taylor expansion in the perturbation SXi( 0) :

X[t~)=Xit~.) + C a-q(r) -6Xj(O) + 0(&X2).

j axj(O) (3)

The first-order term in the perturbation defines the response function

(4)

This is also known as the two-point linear response function since it gives the response of the system at time t=r to linear order in the perturbation at time t = 0 (for simplic- ity we take r > 0 throughout).

The response function is a function of the initial data, that is, it depends on a particular trajectory. Its evolution for particular initial conditions is governed by an evolution equation simply derived by differentiation of Eq. (2) with respect to x2 The boundary conditions, which follow from Rq. (3), are gc(O) = 66, where 6, is the Kronecker delta.

Now consider the average response function. By aver- aging over the initial distribution, we obtain

Gijtd=kii('r))= j- ~p[xOlM% (5) J

where dx=dmx represents integration over all m variables. After an integration by parts, the rhs of Rq. (5) becomes

-I a+(o) 1

Xi(T) ax,(o) dx(0). (6)

Here and throughout, we assume that p vanishes sufficiently rapidly as Ix(O) 1 + CQ so that all boundary terms in the integrations by parts may be neglected. For a nonvanishing probability distribution, we can write

pLWN I =expC - Six(O) II,

and hence

(7)

asbto)i G@(T)= J xi(T) axj(o) ~[~(O)ldx(o)~ (81

or equivalently

GijtT)s Xi(T) (

as[xto) 3 ax,to) a

J 1

That is, the average linear response function is the same as the two-time correlation between xi( r.) and

w-0) I fjCxO) s ax,(o) * (10)

This derivation of the FR is subject to the criticism given by van Kampen” that the derivation relies on the formal Taylor series expansion which may not converge for values of r of interest. Chaotic systems exhibit sensitive dependence on initial conditions, i.e., the distance between two nearby trajectories increases exponentially in time. Thus the first coefficient in the Taylor expansion that was used increases exponentially in time,’ ’ and the convergence of the series may fail, This and other arguments101’2 lead one to suspect the validity of this expansion. Here, we give an alternate derivation in a simpler form than has been presented previously.‘2 First, we note that a perturbation of the values of the Xi at time t = 0 results in a new probability distribution given by

p’[xm,ol =p[W) - WO),Ol. (11) The average (Xi( 7) ) ‘, where ( * * *>’ indicates the average over the perturbed distribution, is given by

(Xi(T))‘= J xi(r)p’[xtO),O]dX

= s xi(T) (~0 + Spo)dX. (12)

Here we have adopted a shorthand notation with po=&(0),OJ and Spo=p’[x(0),O] - p[x(O),O]. Now, ifpo is nonvanishing, we can rewrite this result in terms of averages over the unperturbed distribution

Gi(~))‘=(XitT)) + (Xi(~)+@/&* (13)

If we write p. as in Eq. (7) and assume it is differentiable, then for the infinitesimal perturbation 8x( 0), we obtain precisely the same result as in Eq. (9). Note that this derivation avoids the van Kampen” criticism since the Taylor expansion of the dynamical variable is not used. However, both derivations require that the distribution be

2248 Phys. Fluids A, Vol. 3, No. 9, September 1991 Carnevale et al. 2248


nonvanishing and differentiable, In any event, van Kam- pen’s argument constitutes a relevant remark on the prac- ticality of numerically computing the response directly from the microscopic motion.13 We shall see this explicitly below where we include a calculation of the statistical error in the measurement of the response function.

Equation (9) is the result of linear response theory14 in classical statistical mechanics. It also applies for stochastic processes (for a review, see also, Ref. 15), as well as for spectrally truncated inviscid hydrodynamics.“‘619 Unfor- tunately, we may not know the function S, so that, although we know the response function can be related to two-time correlations, we do not, in general, know which correlations to associate with a given perturbation. In the case of canonical equilibrium of a Hamiltonian system, S is simply proportional to the Hamiltonian, so knowledge of the Hamiltonian immediately closes the relation.’ Further- more, in all cases where p is known to be Gaussian, as is the case in some stochastic diffusion equations, Gii is simply proportional to the two-time correlation function for the variables xi and X~ However, the type of problem in which we are most interested here is the chaotic system of low order such as represented by the Lorenz equations. In those cases, it is the invariant measure which is of interest but for which an analytical expression is lacking. Further, we need to assume that the natural measure p is nonvanishing so that the function S = - In p is well-defined and that S is differentiable for the validity of Eq. (9). These requirements are not fulfilled by systems whose densities vanish on regions of phase space and/or whose trajectories lie on attractors of fractional dimension. If p vanishes on a portion of phase space but the geometry is sufficiently simple, it may be possible to transform to a restricted coordi- nate system in which the phase space does not include regions of vanishing p. However, in general, this will not be possible, and worse, the distribution may not be differentiable. Either derivation presented above fails if the natural measure of the system is not differentiable. However, this problem may be avoided if we take noise into consider- ation. From the physical point of view a certain degree of noise in the system always exists. This may be due to external influences or thermal noise. It has the effect of se- lecting a natural measure from an infinity of possible invariant measures.20 However, this noise may also be regarded as smoothing and spreading the natural measure so as to satisfy the nonvanishing and differentiability requirements. In numerical calculations this noise is provided by round-off errors. Thus we may expect to observe a relation between fluctuation and response in which the distribution to use is an effective distribution that is a smoothed version of the natural measure. In van Kampen’s resolution of the problem in the derivation of the FR relation, he invoked the stosszuhlunsutz of molecular chaos, which yields a differentiable distribution. In a sense we resort to the same fix here.

Note that the derivations given above do not rely on the statistics being stationary, and the results implicitly depend on the origin of time. Our interest in this paper, however, will be with perturbations away from a stationary

ensemble and the eventual relaxation back to equilibrium. In general, the perturbed density need not relax to the same initial distribution or to any stationary state. How- ever, certain dynamics called mixing always statistically relax to a unique stationary density. The three systems discussed below are all believed to be mixing. By definition, mixing implies that all two-time correlations eventually factor, that is, for all functions A and B one has

(A[x(t+7)lB[x(t)l)-t(A)(B), (14)

in the limit r+ CO, where now the notation ( * * * ) will refer only to averages over the invariant measure. In other words, the sets x ( t + T) and x(t) become statistically in- dependent as r--r CO. This relaxation of correlations yields a measure of the irregularity of the motion which is, in general, different from that provided by the characteristic ex- ponents.

Mixing ensures that the mean response function goes to zero as r+ 0~). To see this, consider the large r limit of Eq. (13):

tXi(T))‘+(Xi(T)) + (xi(~))(~~lJfJO)~

Note that

(15)

~Sp~po) = J (p; - poMx=O, (16)

since both p and p’ have the same normalization. Thus for mixing systems

(xi(Q))‘+ bi(7)) (17)

and the response must vanish as r+ CU. Finally, we discuss the Gaussian case explicitly. For a

zero-mean Gaussian distribution, we have

s=; & xpg IX) 1, i

(18)

where

uij= (Xixj> 9 (19)

and so

G&T) = c (xib-)xk(0))ukj’ k

(20)

or

(Xi(T)Xj(O))= : Gi,k(T)Ok,p (21)

This is the explicit form of the linear regression model. For nonzero-mean variables, the left-hand side of (2 1) is simply replaced by

(xi(T)xj(“)) - (xi(O)) (xj(O) )9 (22)

with the corresponding replacement for (T. There are many simple systems (e.g., linear) for which

it is possible to construct a stationary Gaussian ensemble. As a more relevant nonlinear example, we note that in the canonical ensemble for inviscid truncated two-dimensional flow S is linear in the kinetic energy and entropy and, hence, is quadratic in the velocity amplitudes.‘“” Thus a



canonical equilibrium in that case satisfies the linear regression model exactly, no matter how few degrees of freedom are maintained. Also, there is a class of analogous problems with random external forcing, where again the Gaussian distribution is maintained (even for nonstation- ary statistics). In these cases, the randomness is intro- duced’4P21 by assignment of the initial data or through random external forcing. Here, however, we are more interested in the case in which the randomness is intrinsic to the system. That is, we consider systems which are chaotic, and we consider one trajectory and assume that the average in time over the behavior of any one trajectory can be approximated by the “ensemble” average. In these cases, it is a more difficult matter to construct examples in which the Gaussian distribution is exact.

III. NUMERICAL EXPERIMENTS

Here, we examine three numerical examples which point out the failure of the linear regression model for low-order chaotic systems but also show evidence of the existence of the more general FR. We gave the first example previously in Ref. 12. It is based on the Orszag- McLaughlin mode122 and nicely illustrates one condition under which the linear regression is valid in the limit of a large number of variables. It is represented here again for completeness along with a new graph showing explicitly the rate of statistical error growth for this case. The second example is based on the Lorenz model7 We investigated two very different chaotic attractors which occur in this model. Finally, we investigated a truncated model of 2-D Navier-Stokes equations.

In each case, we compute the correlation functions (Xi( t + r)xj( t) ) and the linear response Gii( T) . We make a direct comparison between the measured response and the “Gaussian prediction” given by Eqs. ( 19) and (21). The ensemble average correlations in (21) are computed only indirectly as time averages. For these systems, it is reasonable to invoke the assumption of ergodicity.22 Thus we compute averages according to

1 T (A)= lim -

s T o ~IxO)l& (23)

T-co where time is discretized for the numerical computation.

The response function Gii( r) is computed numerically as follows. A basic time evolution x(t) is followed during the entire course of the run. At time t= tl, the variable Xj is perturbed by a “kick” of amplitude

GXj(t;t*)‘Xi(t) -Xj(t)=E* (24)

As the integration of the two solutions precedes, the devi- ations Sxi(t;tl) are computed, until a prescribed time t2 = tl + r. At time t = t2, the variable Xj is again perturbed, another sample SXi(t;tz) is computed, and SO forth. This procedure is repeated N times and the (normalized) mean response function is then evaluated as

IV G,(d=; z 6X&T + ~a;fa 1 a-l e ’

(25)

where the index j in Gij, indicates that only the variable xi has been perturbed at each kick. Here, E is chosen small enough to guarantee that Sxj(t;tl) is just the linear response. We note that the errors in the computed G,$r) increase faster than exp(Ar), where il is the largest Ly- apunov exponent. Indeed, an estimation of the error due to the finite sample size in a numerical experiment with N “kicks” is given by23

e-*[((Sx)z) - (&x)2]1’2~exp[L(2)t/2],

where L( 2) is the generalized Lyapunov exponent.23 Gen- erally, one finds L(2)/2 is greater than the largest Ly- apunov exponent. 23 Thus in accord with van Kampen’s criticism, this fact makes the practical estimation of the response a nontrivial computational task. In our figures showing the response function, we will also display the statistical error for G&r) computed according to

6xi”z ta’ta)rB Gf(r)]1’2. (27)

A. The ~~~Zag-~cLaug~~~n model

The Orszag-McLaughlin2* model is defined by the following evolution equations:

Ri=Xi+*Xi$*+Xi-,Xj-2-2Xi+~Xi-,, c-w

for i= l,...,m and xi + ,,, = xP These equations are analogous to the inviscid hydrodynamical equations because (a) they involve only quadratic interactions; (b) they possess a quadratic invariant; and (c) the Liouville theorem is valid.

For m)5 the dynamical system (28) shows chaotic behavior and the “energy,”

(29)

appears to be the only isolating integral. Accordingly, the microcanonical distribution should be an appropriate probability density:22

p(x) =JQ(E - &$, (30)

where JV is a normalization factor, E. is the given energy of the system, and S is the Dirac delta function. This density does not satisfy the conditions for the generalized FR since the function S is not defined. One approach would be to transform to a new set of coordinates defined only on the energy surface and consider the FR in those coordinates. Here, however, we use this model to illustrate a different point. Recall that the microcanonical ensemble can be replaced by the canonical distribution asymptotically as the number of variables in the system tends to infinity if the energy is additive,s Namely, for large m, we can replace p by the canonical distribution

p(x) =Jtre-@, (31)

where Jy‘ is another normalization constant and

P - ‘=@f). (32)



This is a Gaussian distribution for this model. Thus, if the number of degrees of freedom is sufficiently large, the Gaussian FR in the form (20) will be approximately valid. Figure 1 shows the normalized autocorrelation functions of the x1 variable, with the corresponding autoresponse functions (computed with N = 10 000) for an increasing number of variables: m = 5, 10,20. The average behavior here is such that oti = 6&c$, which is in accord with the energy being simply the sum of the X: but no cross terms. Thus the formula (20) reduces to

Gij(d = (xj(~)x~(OH/a~,~ (33)

In Fig. 1, we display examples for i=j. For m = 5 the agreement with Eq. (33) is very poor when compared to the cases m = 10 and m = 20. The validity of the Gaussian approximation improves with increasing number of variables.

Figure 2 shows the exponential statistical growth of the error of the autoresponse function in the case m = 5. This is in qualitative agreement with the theoretical prediction given by Eq. (26).

B. The Lorenz model 0.25

The Lorenz model7 consists of three nonlinear ordinary differential equations, that are obtained by a three- mode truncation of a modal expansion of the equations governing the convection in a fluid layer heated from below (usually referred to as the Rayleigh-BCnard problem). The equations are

0.00

-0.25

-0.50 \ 0 1 2 3 4 5

&=a(x* - x,), Ji2= -x1x3 + TX1 - x2,

(34) 1.00

f3=x,x3 - bx3, 0.75

The parameters o and 6, which are related to the Prandtl number and to the wave number of the convective structure, respectively, are maintained fixed at the values (T = 10 and b = 8/3. The parameter r (the reduced Rayleigh number R/R,) plays the role of control parameter. We concen- trate our attention on two different values of r, which cor- respond to motions on two very different strange attractors. The first value of r that we consider is r = 28, for which there is numerical evidence for the existence of a strange attractor St. This attractor originates from an in- verted subcritical bifurcation (occurring at r = 24.74), which makes unstable the previously stable stationary state represented by the two fixed points at

0.50

0.25

0.00

-0.25

-0.50

[ * Jm,h Jm,r- 11. (35)

The motion on S, is representable as the sum of two qualitatively different behaviors: a spiral motion around the left and right foci and a jump process between the two sides of the attractor. As r is increased, the attractor S1 deforms slowly, until rz 145. Then, after a complicated sequence of bifurcations, a stable periodic orbit appears at r = 148.4. The second regime of r that we consider is the one occurring for r> 166.07, where the periodic orbit no longer exists and the system displays a new aperiodic strange attractor,

FIG. 1. (a) Normalized autocorrelation function (full line) and autoresponse function versus time difference r, for the variable xi of the Orszag- McLaughlin model, computed on the surface E = 6, for m = 5. The error bars on the points of the autoresponse function represent the statistical errors computed, as in Eq. (27). (Taken from Ref. 12.) (b) The same as in (a) for m = 10. (c) The same as in (a) for m = 20.

S2, with intermittent behavior.24 The motion on S2 is a type of intermittent behavior consisting of laminar periods (in which the orbit comes close to the “ghost” of the periodic orbit) randomly interrupted by turbulent bursts.

1 .oo

0.75

0.50

0.00

-0.25

-0.50 0 1 2 3 4 5

1.00

0.75

0.50

I I 84 1 I I I I I I I I I I I I I I I I I I- O 1 2 3 4 5

2251 Phys. Fluids A, Vol. 3, No. 9, September 1991 Carnevale et a/. 2251


FIG. 2. Statistical error (crosses) computed, as in Eq. (27), versus time delay T, for the case (a) of Fig. 1. The solid line represents error growth at the rate given by the largest Lyapunov exponent.

In Figs. 3 (a) and 3(b), we report the normalized autocorrelation functions together with the autoresponse functions (computed by means of 30 000 kicks), for the variables x1 and x2, respectively, on the attractor Si (r = 28). Figures 4(a) and 4(b) show the same functions for the same variables as above, but now they are computed on the attractor S2 (r = 166.1)) and for the response functions we calculated an average over 120 000 kicks. (Note that the value of the maximum Lyapunov exponent for the example on S, is zO.9 and for that on S2 is ~0.6. The time scales associated with the evolution of the response and correlation functions seem to be somewhat smaller than these. )

We see from Figs. 3 and 4 that response and correlation have some qualitative behavior in common (characteristic correlation time, oscillations). In particular, the similarity of qualitative changes occurring in the correlation and response functions when we pass from St to S, deserves attention. These similarities may result from the fact that the response must be related to some two-time correlation of the problem. l9 Nevertheless, in both of these cases, we have found that the linear regression based on Gaussian statistics [Eq. (2 )] fails badly. This is illustrated, perhaps most dramatically, by the case for attractor S,. In Fig. 5, we show again the autocorrelation (solid curve) and autoresponse function (dot-dashed curve) for the variable xi from Fig. 3 (a). The response function is to be compared with the prediction for G1t (dashed curve) based on the assumption of Gaussian statistics and computed from formula (20).

C. The five-mode model of NavierStokes equations

Here, the Reynolds number, Re, is the only control parameter. There are systematic studies of Eq. (36) (for a review see Ref. 9). In those studies all of the following behaviors have been observed with increasing Reynolds number: (i) stable fixed points; (ii) Hopf bifurcations to periodic cyclic orbits; and (iii) periodic/aperiodic/chaotic orbits.

Here we consider a truncated, spectral model for two- In Fig. 6, we show some results for the case dimensional turbulence in a doubly periodic domain. The Re = 30.85, which is in the chaotic regime. We show the velocities are expanded in the Fourier series taking into average autoresponse function (error bars only), the auto- account the incompressibility condition. Only five modes correlation function (solid line), as well as the Gaussian are retained corresponding to wave vectors k, = ( 1,1 ), prediction (dashed) for the autoresponse [i.e., Eq. (20)] k2 = (3,0), k, = (2, - l), k, = (1,2), and k, = (0,l). Af- for the variable x1. It is clear that, as for the Lorenz model,

0 0.2 0.4 0.6 0.8 1

1.0

0.5

0.0

-0.5

0 0.2 0.4 0.6 0.8 1

FIG. 3. (a) Normalized autocorrelation function (full line) and autoresponse function (error bars), for the variable x, of the Lorenz model, versus the time delay 7. This is an example of evolution on the attractor S, (see text). (b) The same as in (a) for the variable x9

ter some resealing and with forcing only in wave vector k3, the equations representing the evolution of the Fourier velocity amplitudes become

A*= - 2.7~ + 4X2X3 + 4X&, i>= - 9X, + 3x,X3,

As= - 5x3 - 7X1X2 + Re, k4= - 5X4 - X1X5, (36)

ks= - xs - 3X,X&



0

5.0

2.5

0.0

-2.5

-5.0 :

0 0.2 0.4 0.6 0.6 1

FIG. 4. (a) Normahzed autocorrelation function (full line) and autoresponse function (error bars), for the variable x1 of the Lorenz model, versus the time difference r. This is an example of evolution on the attractor S, (see text). (b) The same as in (a) for the variable xr.

the relationship between the mean response and the prediction based on Gaussian statistics is very poor.

IV. CONCLUDING REMARKS

We have emphasized that any system with a smooth invariant probability distribution has a fluctuation-response relation, This relation shows that the relaxation toward the invariant distribution relies on the details of that distribution. Thus an explicit statement of the FR for a given system requires a knowledge of the invariant measure, which, unfortunately, is lacking in most cases of interest. Nevertheless, the existence of such a relation may be useful, even if only qualitatively, in choosing the appropriate variables for the study of the relaxation properties of chaotic systems.

For the case of Gaussian statistics, the fluctuation-response relation takes on the particularly simple form of the linear-regression model. We have pointed out that this is valid only in the case of Gaussian statistics, and we have emphasized this point by providing several examples where this Gaussian model fails. Furthermore, we have discussed

o.25 0.00

k’\,, .“.-.+,,/ _ ;; I;; \

i ,‘<, /’ \ / -._.j /

I ” “I” 1 ’ I ‘I-

, .J’ /

/’ : \ . . /’ --_ - \ I’

0.00 - ,‘<, /’ \ / -.-.’ / \ /

-0.25 - ' 1' N/ 1 \ / -0.25 ' 1' N/ 1

I- i

-0.50 - 0 0.2 0.4 0.6 0.0 1

FIG. 5. The autoresponse function (dot-dashed line) from Fig. 3(a) is compared to the predicted autoresponse function (dashed line) obtained with the Gaussian approximation, that is, linear regression model given by Eq. (20). The autocorrelation (solid line) is also included.

an example of a system for which the Gaussian approximation becomes a good representation of the fluctuation- response relation as the number of dynamic variables increases. In this case, the Orszag-McLaughlin model, it seems that the validity of the Gaussian approximation is due to the asymptotic behavior of the microcanonical ensemble, with an additive quadratic energy, as the number of variables increases. However, we have not been able to address this issue more generally, and the question for what types of chaotic dynamics will the Gaussian approximation become valid as the number of variables increases remains open.

2

0

-2

i

0 0.5 1 1.5

FIG. 6. Normalized autocorrelation function (solid line), autoresponse function (error bars)and the predicted autoresponse function (dashed line) obtained with the Gaussian approximation, that is, linear regression model given by Eq. (20), for the variable xi of the f ive-mode model of Navier-Stokes equation, in the chaotic regime (Re = 30.85).



ACKNOWLEDGMENTS

GFC thanks the IBM European Center for Scientific and Engineering Computing ECSEC), the Institute for At- mospheric Physics (IFA-CNR), and the University of Rome for their hospitality during the period of this research. RP thanks the “Comitato Ambiente ed Habitat” of the Italian National Research Council (CNR).

This research has been supported at various stages in part by National Science Foundation Grants No. OCE 89- 11858 and No. ATM 89-14004, Office of Naval Research Grant No. NOO014-89-J-1155, the DARPA/ONR Univer- sity Research Initiative Grant No. N0014-86-K-0758, and by the Institute for Atmospheric Physics (IFA, Rome).

‘R. H. Kraichnan, Phys. Rev. 113, 1181 (1959). ‘U. Deker and F. Haake, Phys. Rev. A 11, 2043 (1975). ‘R. H. Kraichnan, J. Fluid Mech, 47, 513 ( 1971). ‘C. E. Leith, J. Atmos. Sci. 32, 2022 (1975). “T. L. Bell, J. Atmos. Sci, 37, 1700 (1980); T. L. Bell, in Turbulenceand

Predictability in Geophysical Fluid Dynamics and Climate Dynamics, Proceedings of the International School of Physics, Enrico Fermi, ed-

ited by M. Ghil, R. Benzi, and G. Parisi (North-Holland, New York, 1985).

6A. E. Koniges and C. E. Leith, Phys. Fluids 30, 3065 ( 1987). ‘E. N. Lorenz, J. Atmos. Sci. 20, 130 (1963). *A. I. Khinchin, Mathematical Foundations of Statistical Mechanics

(Dover, New York, 1949). 9C. Boldrighini and V. Frances&in& Commun. Math. Phys. 64, 159

(1979). ‘ON. G. van Kampen, Phys. Norv. 5, 279 (1971). “J. P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57, 617 (1985). “M. Falcioni, S. Isola, and A. Vulpiani, Phys. Lett. A 144, 341 ( 1990). 13G. Ciccotti and G. Jacucci, Phys. Rev. Lett. 35, 789 ( 1975). 14R. Kubo, M. Toda, and N. Hashitsure, Statistical Physics (Springer-

Verlag, Berlin, 1985), Vol, 2. “P, Hanggi and H. Thomas, Phys. Rep. 88, 207 ( 1982). 16R. H. Kraichnan, J. Fluid Mech. 67, 155 (1975). “P. D. Thompson, Phys. Fluids 25, 1159 (1982). ‘*G. F. Camevale, Phys. Fluids 25, 1547 ( 1982). 19H. A. Rose and P. I. Sulem, J. Phys. (Paris) 39, 441 (1978). 2oD. Ruelle, Phys. Lett. A 72, 81 (1979). 2’J. Lee, Physica D 24, 54 ( 1987). **S. A. &zag and J. B. McLaughlin, Physica D 1, 68 (1980). 23G. Paladin and A. Vulpiani, Phys. Rep. 156, 147 (1987). 24P. Manneville and Y. Pomeau, Physica D 1, 219 (1980).



Fluctuation-response relations in systems with chaotic behavior

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