Correlation functions and relaxation properties in chaotic dynamics and statistical mechanics

Volume 144, number 6,7 PHYSICS LETTERS A 12 March 1990

CORRELATION FUNCTIONS AND RELAXATION PROPERTIES IN CHAOTIC DYNAMICS AND STATISTICAL MECHANICS

Massimo FALCIONI i Dipartimento di Fisica, Universit,~ "'La Sapienza", P. le Aldo Moro 2, 1-00185 Rome, Italy

Stefano ISOLA 2 Dipartimento di Fisica, Universit& di Firenze, Largo Fermi 2, 1-50125 Florence, Italy

and

Angelo VULPIANI 3 Dipartimento di Fisica, Universit?, de L'Aquila, Piazza dell'Annunziata 1, 1-67100 L'Aquila, Italy

Received 18 September 1989; revised manuscript received 3 January 1990; accepted for publication 9 January 1990 Communicated by A.P. Fordy

We give a derivation, for chaotic systems, for a general fluctuation-response relation for which the van Kampen critique does not hold. Moreover we discuss the connection between relaxation properties and correlation functions.

The aim of what follows is to point out a problem which plays an important role both in statistical mechanics and in the ergodic approach to dynamical systems: the understanding of the relaxation properties of a system and its response to external perturbations in terms of the invariant features of its asymptotic probability distribution. The relevance of this relation is well evident: it allows one to reduce "non-equilibrium" properties (i.e. relaxation and response) to "equilibrium" ones (correlation functions). A remarkable example can be found in the linear response theory of statistical mechanics, which links the response to an external field to correlation functions computed at equilibrium (Green-Kubo formulas) [ 1 ]. Another example where this relation holds is given by inviscid hydrodynamics with chaotic behaviour [2 ]. On the other hand, for typical geophysical systems (viscous fluids), although this

INFN Sezione di Roma. 2 Dipartimento di Matematica e Fisica, Universit/t di Camer-

ino, Italy. 3 INFN Sezione di Roma and GNSM-CISM Uniter di Roma.

relation is considered as roughly valid, no general ar- guments are yet available for its complete justifica- tion [ 3 ]. Finally, some authors even claim that in fully developed turbulence there is no relation between equilibrium fluctuations and relaxation toward equilibrium [ 4 ].

Usually, the derivation of the fluctuation-response relation is obtained from a perturbative approach and/or linearization of the difference between the trajectory of the perturbed system and that of the unperturbed one. On the other hand, it is well known that chaotic systems exhibit sensitive depen- dence on initial conditions, i.e. the distance between two nearby trajectories increases exponentially in time. This raised severe criticism about the validity of these derivations [ 5 ]. In some cases however a macroscopic description, for which the above criticism does not hold, has been given [ 6 ].

In this Letter we present a derivation of the fluctuation-response relation which seems to have a wide generality and, in particular, contains all known results for stochastic processes and chaotic inviscid hydrodynamics. To obtain the desired result, the basic

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assumption is the existence of an invariant probability measure (for which some "absolute continu- ity" type conditions are requested) together with the mixing property.

We consider the general setting in which a time evolution x( t ) = f i x ( 0 ) (where the time t may be an integer or a real number) acting on a manifold ~ is given. In physical contexts f t is either a flow gen- erated by an ordinary differential equation (as, for example, in Hamiltonian systems) or a map (obtained, for example, from a Poincar6 section in phase space), and J¢ is either P'~ or a torus or an interval. However, as will be clear in the following, we do not limit ourselves to deterministic evolution laws, i.e. stochastic differential equations are not ruled out by our treatment.

As a starting point, we assume that we are given an ergodic probability measure/ t which is invariant under time evolution, i.e., for any measurable function A (an observable) the following identity holds:

u(A) = J U(dr)A (x)

= ~ u (dx )A ( f t x ) = la(Aof ' ) . ( 1 )

Moreover, we demand that the measure/z be an absolutely continuous measure, that is

# ( d r ) = p ( x ) d r , (2)

where d r is the Lebesgue measure and p is a non-negative measurable function such that f p ( x ) d r = 1. Whenever truly chaotic behaviour is observed we expect this condition to be true (at least along suitable subregions of the phase space). As a matter of fact, the time average

T

( A ) = lim 1 f AOrtxo ) dt T ~ o o 1 . /

0

evaluated for suitable initial points Xo, defines a physical measure which is expected to be absolutely continuous along unstable directions (the directions where we get information from the separation of nearby points). Incidentally, it can be shown that, for Axiom A dynamical systems, this physical measure is equivalent to the asymptotic measure selected as a "zero-noise" limit (E ~ 0) of a stationary probability measure/~, under small random perturbations of the time evolution ( f t). Moreover, this asymp-

totic measure is characterized by having conditional probabilities which are absolutely continuous with respect to Lebesgue measure on the unstable mani- folds [ 7 ]. However, it is believed that this picture is applicable to more general chaotic dynamical systems as well [ 8 ]. In the following we shall assume the ergodic identity/~(A) = (A) .

The last condition we need is something stronger than ergodicity and makes the system have nice stochastic properties. Let us introduce the correlation function for the observables A and B:

Can(t) = ( A ( x ( t ) )B(x(O) ) ) - ( A ) ( B )

= l~( (Aof t)B ) - #(A )#( B ) . (3)

Notice that the first expression is the one actually used in computing Can from experimental time se- ries. This function yields a measure of the irregular- ity of the motion which is different from the one provided by the characteristic exponents. Suppose the average ( A ( f ' x ) B (x) ) factorizes at large time separation, that is CAn(t)~O when t--*~. Then the functions A ( x ( t ) ) , B (x ) becomes statistically independent for large t, or, in other words, the evolution becomes progressively independent of where the system began. This property is usually referred to as mixing.

Now, let us show how correlation functions provide useful information about the relaxation of the system toward the invariant probability measure # (we refer to/~ as to the "equilibrium distribution").

Consider a situation where at time t = 0 the state of the system is described by a "non-equilibrium distribution" Vo such that

vo(dx) =h (x ) /~ (dx ) , (4)

where h is a non-negative/z-measurable function for which f h ( x ) # ( d x ) = 1. Without loss of generality, we can set

h ( x ) = 1 + g ( g ) , (5)

where the function g(x) satisfies I g l < l and f g ( x ) # ( d r ) =0. We may study the evolution of the measure vo setting

v t ( A ) = v o ( f - tA) , (6)

so that the mean value of A(x) as a function of time is given by

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t" /. = j A = J A Or

=/ t (X) + / t ( ( A o f ' ) g ) , (7)

where (5) has been used. Therefore we have

(A>t = (A> +CAg(t) , (8)

where we have used the notation ( >t = ut() . I f mixing takes place, i.e. CAg(t)~0 when t--. oo for any choice of the function A, then in this limit we have relaxation toward equilibrium: <A > t ~ <A >.

Now we can use eq. (8) in order to obtain a fluctuation-response relation. Let us assume, for simplicity, that our dynamical system lives in R" so that a set of local coordinates is given by the n-tuple x = (Xl .... , x , ) . Consider a perturbation 8x(0) in the variables x at the initial time t= 0. Such a perturbation corresponds to an impulsive "force" added at t = 0 to the evolution equations. The initial non- equilibrium distribution u0 for the system thus perturbed must satisfy the following relation,

vo(dx) = d / ~ ( x - ax(0) ) . (9)

In addition, let us write the equilibrium probability distribution in the form

/t (dx) =exp[ -S (x ) ] dx , (10)

where S(x)=-logp(x) (see eq. (2) ) . Using (4), (9) and ( 10 ) we have

h(x) = 1 + Z OS(x(O) ) axj(O) Oxj

+ 0 ( I 8x(O) 12) . (11)

Comparing this expression with (5) we obtain the function g for this particular setting. Then, from (8) and ( 11 ) we have

( a x , ) , = ( x , ) , - ( x , )

(x,(t) OS(x(O))~ = ~ / ax:(O)+O(lax(O)12)"

Finally, if we let

ro(t)= (x~(t) OS(x(O) )/Oxj)

be the mean response of the variable x~ due to a small change of the variable xj, that occurred a time t be- fore, we obtain

(Sx~)t= E rijSx:(O), (12) J

where only the lowest-order terms are kept. One can immediately realize that in classical statistical mechanics eq. (12) stands for linear response theory [1]. It is also worth noticing that in (12) is con- tained the well-known fluctuation-response relation for stochastic processes [9] (for a review see also ref. [10] ) as well as the results for inviscid hydrodynamics [ 2,3 ]. Moreover the same relation has been obtained in discrete one dimensional systems [ l l ].

Unfortunately, in general cases, we do not know what the function S looks like, so that it is not clear what the functions r o represent. In other words we do not know which correlation function is associated with a given quantity and therefore, at this point, eq. (12) provides only qualitative information. Note that in those problems where/~ is Gaussian, as in equilibrium statistical mechanics or in some particular stochastic diffusion equations, r o is but the time-correlation function for the variables x~ and xj and pre- cise identification becomes possible.

We remark that (12) holds for general chaotic dynamical systems if we add some small random noise, in order to guarantee the validity of the absolute con- tinuity condition ( I0 ) . From an "experimental" point of view, this "beneficial" noise originates from different possible sources. For example in computer experiments it is provided directly by the round-off errors of the machine, whereas in physical experiments imprecision in the measurements plays an es- sential role [ 12 ].

We note that the above derivation does not suffer from van Kampen's criticism [ 5 ]. His main point is that linearity of the microscopic behaviour is en- tirely different from macroscopic linearity, so that the correctness of solving the equations of motion to first order in the external fields would be uncontrollable.

However, in obtaining eq. (12) no such approx- imation has been made. The only linearization is done at the initial time and implies nothing but the smallness of the impulsive perturbation. Neverthe- less van Kampen's argument constitutes a relevant remark on the practical possibility to compute the response directly from the microscopic motion (e.g. by means of a molecular dynamics procedure) [ 13 ].

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In order to gain some further insight on how things work, we have performed a numerical test with two models: the H6non-Heiles [ 14] model and the Orszag-McLaughlin [ 15 ] model.

A basic time evolution x (t) is followed during the entire course of the run. At time t= t~ the variable xj is perturbed by a "kick" of amplitude 5xj(tl) = g j ( q ) - x j ( h ) = ~ and the deviations 8x~(t, q) =gi( t , t~)-x~(t, t~) are computed, as the integra- tion of the two solutions proceedes, until a pre- scribed time t2= t~ + z. At time t= t2 the variable x i is again perturbed, another sample 8xt(t, t2) is computed, and so forth. This procedure is repeated N times and the (normalized) mean response function is then evaluated as

--VI N 8xi(t, tk) (13) r6( t ) U ~ ~ '

where the index j in Gj indicates that the variable xj has been formerly perturbed. According to eq. (12) and the successive discussion, ~ is chosen small enough to guarantee linear response. We stress that the errors in computing rij(t) increase faster than exp(2t), where 2 is the largest Lyapunov exponent. Indeed, an estimation of the error due to the finite sample size, in a numerical experiment with N "kicks", is given by

C (~x2)~/2 ~ C x / ~ ~ 18x(0) I exp[ ½L(2)t],

(14)

where C is a constant depending on the correlation time and L (2) is the generalized Lyapunov exponent [ 16] of order 2. Generally one finds ½L(2) >2. This exponential growth makes the practical estimation of the response a non-trivial computational task.

The H6non-Heiles [ 14] model is a non-linear Hamiltonian system, originated by astronomical problems, with the following Hamiltonian,

H=½(q~ +q~ +p~+p~)+q~q2_! 3 3q2. (15)

We have performed our computations in the chaotic zone of the energy surface H = 0.125. In fig. 1 we show the normalized autocorrelation function of q~ and its auto-response function (computed with N = 20 000 ). Agreement is good up to t ~ 25. The disagreement for

1.o ~ . . . . I ' ' ' I . . . .

05

00 /

- 0 . 5

-1.0 0 10 20 30

Fig. 1. Normalfzed autocorrelation function (full line) and auto- response function, versus time, of the variable q~ of the H6non- Heiles model, computed on the chaotic zone of the energy surface H= 0.125. The error bars on the points of the auto-response function represent the statistical errors.

larger time delay is related to the non-linearity of the equations of motion, which leads to a non-quadratic behaviour of S in qi-

The Orszag-McLaughlin [ 15 ] model is defined by the following evolution equations:

dx, dt =Xi+lXi+2 -~Xi-lXi-2 --2Xi+lXi_l , (16)

with i= 1, ..., rn and X~+m=X~. These equations are analogous to the inviscid hydrodynamical ones be- cause: (a) they involve only quadratic interactions; (b) they possess a quadratic invariant and (c) the Liouville theorem is valid. In fig. 2 we show the normalized autocorrelation functions of the x I variable, with the corresponding auto-response functions (computed with N = 10000) for m = 5 , 10, 20. The agreement is quite good and increases with the number of degrees of freedom m. This may be related to the ergodicity of the system (16) on the invariant hypersphere ~ ' = ~ x 2 = const, and to the central limit theorem, from which we expect that the probability distribution of each x~ variable is close to the Gauss- ian one, for "large" m.

To terminate this Letter, we give some further considerations that might hopefully provide a better understanding of the intimate connection between

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1 . 0 0

0 . 7 5

0 . 5 0

0 . 2 5

0 . 0 0

- 0 . 2 5

- 0 . 5 0 0

I . . . . I . . . . I . . . . I '

a

. - _ . . . . ~

1 2 3 4 5

t

1 . 0 0 [ . . . . [ . . . . [ . . . . I '

0 . 7 5 b

0 . 5 0

0 . 2 5

0 . 0 0 -

- 0 . 2 5 -

- o . ~ . . . . I . . . . I . . . . I . . . . I . . . . I , , 0 1 2 3 4 5

t

1 . 0 0

0 . 7 5

0 . 5 0

, i , , , , , , , , , , , , , , , , , ,

. . . . I . . . . I . . . . I . . . . I . . . . I , 1 2 3 4 5

t

0 . 2 5

0 . 0 0

- 0 . 2 5

- 0 . 5 0 0

relaxation properties and correlation functions in the context o f dynamical system theory. Let us consider, for simplicity, the case o f maps f (i.e. the time t is chosen to be an integer n). Then the time evolution o f the non-equilibrium probability density (5) can be rephrased by means o f the functional iteration

h , , ( x )=~h , ,_~(x )=A:"h(x ) , (17)

where h, (x) = h ( f - ' x ) and L# (the Perron-Froben- ius operator) maps a suitable function g onto

1 ( ~ g ) (x) = ~ g(y) (18)

y:f(y)=x I det Df (y) I

(Dfdenotes the derivative o f f ) . In a sense, eq. (17) provides a generalization o f the Liouville equation to more general (non-Hamil tonian) systems.

Let us restrict ourselves to the simplest situation where L# has a discrete spectrum. In this case we may introduce eigenvalues 2k and eigenfunctions Ok(X) such that

~ k ( x ) =2kq~k(X) • (19)

Then q~o(X) = p ( x ) and 20= 1. Moreover, it is easily seen that ergodicity requires 12kl ~< 1, ~,kSk 1 and (~k) =0 , for k ~ 0 . Whereas the mixing property is expressed by the fact that 12kl < 1 for k ¢ 0 , namely the eigenvalues different from 20= 1 are located in- side the unit circle in the complex plane. Now consider again the situation where the state o f the system is described by a non-equilibrium distribution as in (4), with probability density h o f the form (5). Furthermore assume that coefficients ( a l . . . . . aM) and (ill .... , tiM) exist such that the functions g and A are uniquely expanded as

M M

g ( x ) = ~ Otk0k(X), A ( x ) = ~ flkOk(X). (20) k = l k = l

Using these expansions and noticing that ~ pre- serves the Lebesgue measure, we can write

~1 Fig. 2. (a) Normalized autocorrelation function (full line) and auto-response function, versus time, of the variable x~ of the Orszag-McLaughlin model, computed on the surface ~'= t x, 2 = 3.0, for m = 5. The error bars on the points of the auto-response function represent the statistical errors. (b) The same as in (a), for m= 10. (c)The same asin (a), for m=20.

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M

< A > , , - ( A > = ~ Ak~,~,, (21) k = l

where

3k = akilk (0~)

Consider now the autocorre la t ion function o f A. Again, we can write

M

CAA(n)-- (A)E= ~ g2k2~,, (22) k ~ l

where

Ok=il~(0~>.

( F r o m (22) we see that i f A is or thogonal to some Ok, i.e. ilk=O, then 2k will not influence the behaviour of C ~ (n) . This might be a cr i ter ion for deal ing with the del icate p rob lem of the observable-depen- dence of the decay o f the correlat ion funct ions.) Compar ing (21) and (22) , we easily realize that whenever the coefficients Otk are not zero (i.e., for sui table forms o f the pe r tu rba t ion ) , the behav iour of CaA (n ) and that of ( A ) n are influenced by the same eigenvalues.

We conclude by stressing that, in s i tuat ions as general as those we have considered, a f luc tua t ion- re - sponse relat ion always holds. This could be helpful in the choice o f appropr ia t e indica tors for the s tudy o f re laxat ion propert ies , even though only quali ta- t ive in format ion can be extracted, i f the invar iant probabi l i ty d i s t r ibu t ion is not known (not even in

approx imat ive form) . In a for thcoming paper we shall discuss this problem, in the context o f geophysical fluids.

We would like to thank G. Ciccott i and B. Hol ian for useful discussions.

References

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T.L. Bell, J. Atmos. Sci. 37 (1980) 1700. [4] H.A. Rose and P.L. Sulem, J. Phys. (Paris) 39 (1978) 441. [5] N.G. van Kampen, Phys. Norv. 5 (1971) 279. [6] H. Grabert, P. Htinggi and P. Talkner, Phys. Lett. A 66

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R. Graham, Z. Phys. B 26 (1977) 397. [10] P. Htlnggi and H. Thomas, Phys. Rep. 88 (1982) 207. [ 11 ] S. Grossmann, Z. Phys. B 57 (1984) 77. [ 12 ] D. Ruelle, Phys. Lett. A 72 ( 1979 ) 81. [ 13 ] G. Ciecotti and G. Jacucci, Phys. Rev. Lett. 35 ( 1975 ) 789. [ 14] M. Hdnon and C. Heiles, Astron. J. 69 (1964) 73. [ 15 ] S.A. Orszag and J.B. McLaughlin, Physica D 1 (1980) 68. [ 16 ] H. Fujisaka, Prog. Theor. Phys. 70 (1983) 1265;

R. Benzi, G. Paladin, G. Parisi and A. Vulpiani, J. Phys. A 18 (1985) 2157; G. Paladin and A. Vulpiani, Phys. Rep. 156 (1987) 147.

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Correlation functions and relaxation properties in chaotic dynamics and statistical mechanics