On correlation function methods for detecting the stochastic transition

On correlation function methods for detecting the stochastic transition

Physica 3D (1981) 644-648 North-Holland Publishing Company ON CORRELATION TRANSITIONt FUNCTION METHODS FOR DETECTING THE STOCHASTIC Giulio CASATI,...

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Physica 3D (1981) 644-648 North-Holland Publishing Company

ON CORRELATION TRANSITIONt

FUNCTION METHODS FOR DETECTING

THE STOCHASTIC

Giulio CASATI, Faust0 VALZ-GRIS fstituto

di Fisicu

Unicersild.

Viu Celoriu

16. 20133 Milano,

Ztufy

and Italo GUARNERI lstituto

Received

di Mutemuticu,

21 November

Unicersitir

di Puriu,

Puvia,

Italy

1980

The use of correlation function methods to predict the onset of chaotic motion in conservative Hamiltonian systems is critically examined. It is shown that microcanonical correlation functions do not, in general, provide a convenient criterion to distinguish between stable and unstable motions.

1. Introduction

transition” The existence of a “stochastic which seems to take place in generic Hamiltonian systems makes it a very important goal to find a practical, effective criterion to distinguish between stable and unstable motions of classical Hamiltonian systems. A number of such criteria have been used so far, proving more or less effective: we quote, e.g., PoincarC’s surfaces of section, equipartition of energy among different degrees of freedom, local instability of motion, etc. [l]. In this paper we would like to discuss the possibility of founding a “stochasticity criterion” on the different behaviour that the autocorrelation of a given phase function is supposed to exhibit, as a function of time, in the regular and in the irregular regimes. That stochasticity affects the decay of correlations is well known: as a matter of fact, this is one of the major reasons why irregular motion is supposed to account for thermodynamic behaviour. In the more precise lantWork

supported

by CNR contract

0167-2789/81/0000-0000/$02.50

No 78.02739.63.

@ North-Holland

guage of ergodic theory, the decay of correlations corresponds to the mixing property [2]; also the exponential character of the decay, for a sufficiently wide class of functions, is quoted as a distinguished stochastic property by Sinai r31.

At the opposite extreme in the ergodic hierarchy, when dealing with integrable systems, one must first distinguish a time-averaged and a microcanonically averaged correlation function (in the ergodic case the two functions coincide). As is well known, the time averaged correlation is a quasi-periodic function of time. The quasi-periodic nature of the motion of integrable systems led many authors [4-61 to the erroneous conclusion that also the microcanonical correlation function will be some nontrivial oscillatory function oscillating about its mean value*. The behaviour of this function was then used to distinguish between stable and *This quasi-periodic behaviour has been contrasted with a possible exponential divergence of correlations for nonintegrable systems [4, 51. Of course this possibility is ruled out by the Schwarz-Holder inequality which, together with invariance of the microcanonical measure, entail IS(T)] G S(O) for any square-integrable f. regardless of the dynamics.

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G. Casati et al./Correfation

function

methods

unstable motion and, in particular, it was used to detect the stochastic transition in the HCnonHeiles system, for which it was found to give the correct value of the threshold energy. It is the purpose of the present paper: i) to show that for generic integrable Hamiltonian systems with N degrees of freedom, the microcanonical correlation function

I

= H=E fh

P,

t)f(q,

P, t +

7) dp

(where dp is the microcanonical measure and the bar denotes complex conjugation; as a consequence of Liouville theorem S(T) will not depend on the arbitrary time t) will decay, for any smooth function f, to its final value as T(‘-~)‘*; this is proved in section 2; ii) to point out that little is known about the precise behaviour of correlations in the chaotic regime; as a consequence, the behaviour of correlation functions can hardly be used to detect the stochastic transition. 2. Decay of the microcanonical function S(7)

correlation

We prove here point (i) in section 1. Let Z, 8 be the action-angle variables for a smooth (Cz) integrable Hamiltonian system with N degrees of freedom, and let f(Z, 0) be a smooth phase function: f(Z, 0) =

2

(1)

c,(Z) eim”,

m

where m is a vector with integral components. A standard analysis gives

S(T) = o(Z)

x1111IGml * eim‘“““6(H(Z)

- E) dNZ, (2)

being the vector of angular velocities the torus labelled by Z.

on

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645

Under the usual assumption that the energy surface H(Z) = E is a compact submanifold of RN and provided that stationary points of the functions (m - w) on the surface H = E (i.e. points where V(m - to) is normal to the surface Z-Z= E) are nondegenerate, we prove that S(T) S(M) = @T(‘-N)‘2) as T-Pm. Thus the microcanonical correlation function decays to its final value as an inverse power of time, the decay being faster with increasing number N of degrees of freedom. Although simple in essence, the proof requires some technicalities. The argument runs as follows: 1) By compactness of the surface H = E, each integral in eq. (2) can be expressed as a finite sum of integrals in RN-’ of the type I-

Jc;(U,)[Ui 0 cp;‘] exp{iT(m * W) o cp;‘(Xl * . * XN-I)} (3) . * .XN-,)dx, . * .dXN-,, x I& 0 (p;'('F(X,

where (Vi, pi) is a system of finitely many charts (local coordinates on the energy surface), {ui} is a subordinate partition of unity and F(Xl,. *. , ‘XN-,) dx,, . . . ,dxN_, IS the microcanonical volume element in local coordinates. 2) Compactness of the surface H = E and nondegeneracy of stationary points imply that the set of such points is finite?. We may then arrange that each qi(Ui) contains at most just one stationary point. We are now in the position of using the stationary phase method. According to well-known theorems [7] all the integrals in eq. (3) with m # 0 are a(~-“) for all positive n if no stationary point is contained in qi(Ui), and are O(T(‘-~)‘~) uniformly in m if one stationary point is contained in pi(Vi). 3) Since the sum in eq. (2) converges uniformly in T, one has S(T) - S(a) = 0(~“-~“*) as tThe set of non-degenerate stationary points is isolated; since the surface Z-Z(Z) = E is compact this set is finite.

G. Casati

646

et ul./Correlntion

function

methods

7 + to, where

S(p)= j-(c,,(I)12S(H(I)- E) dN1. The assumption that a stationary nondegenerate means that det

point z, be

(4)

in a (Xl, *. xN_,) being local coordinates neighbourhood of z. Not every integrable hamiltonian has the above property although “most” of them do possess it in the sense that the set of exceptions only makes up a “small” ensemble in the set of all Hamiltonians H(I). In other words, property (4) is generic, on account of a technical argument*. 1,

3. Comment A few comments are in order: a) While microcanonical correlations decay for both integrable and mixing systems, the asymptotic values are different. Indeed it appears, from the previous section, that S(m) = .f (~(1)(~ dp(I) in the integrable case, whereas in the mixing case S(m) = ((f)l’ = IJ c,,(I) dp(1)(*. b) Harmonic systems are non-generic since condition (4) is clearly violated. Indeed, in this case, eq. (2) together with the fact that o is constant implies quasi-periodicity of S(T).

*For generic integrable systems, the w’s instead of the J’s can be taken as coordinates. By means of a linear change of variables 5 = m * w itself can be assumed to be one of the coordinates. Close to a stationary point of 5, the hypersurface H = E is described by a function relating 5 to the remaining N - 1 coordinates. Now it is known that the set of such functions 5 having only non-degenerate stationary points is a set of the 2nd Baire category when the set of all functions is given a topology in which nearby functions are close together with their derivatives up to 2nd order. (See e.g. Hirsch, Differential Topology (Springer, 1976).) This is precisely the technical meaning of a set being generic.

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trunsifion

Notice also that this quasi-periodic behavior is not at variance with an initial exponential type decay of correlations as the one reported in the interesting paper of Cukier et al. [8] where the momentum autocorrelation function for a mass defect particle in a linear harmonic chain was studied. c) As a consequence of our result, generic integrable Hamiltonian systems have no discrete spectrum apart from the zero eigenvalue. By this we mean that the Liouville operator, as an operator in L2(fi,) (fir the energy surface with microcanonical measure) possesses no proper eigenvalue other than zero. d) Systems possessing the non-generic property of having quasi-periodic correlations do indeed exist. In this respect we would like to discuss a recent paper [5] in which a condition for quasi-periodicity of correlations is found. The authors consider the Fourier transform dG(o) of S(T); this is a positive measure, whose moments sn = J W” dG(o) are formally related to the Liouville operator L via s, = (L”f)(O). Then they consider the determinants so

St

SI

s2 . . . . .&,I

.

A,=

. . . . .

Sn

.

I.. .

I

S”

.

&+I

S2n

and argue as follows: a necessary and sufficient condition for a given sequence {s,} to be the sequence of moments of a unique positive measure dG(w) of the form EC;c,S(w - w,) is that A, 2 0 for all n; in this case S(T) is quasiperiodic. This argument however is not correct. Indeed the stated condition on the A,‘s is necessary and sufficient for a unique measure to exist with the given moments, whether or not it is a superposition of point measures [9]. We are forced to understand that the authors consider every positive measure to be of the type

G. Casati et al./Correlation

function methods for detecting the stochastic

X c,6(0 -w,). For this to be true, the series must be understood to converge in a weak sense; but then S(T) will be a limit of trigonometric polynomials in a correspondingly weak sense: too weak to conclude that S(T) is quasi-periodic.? 4. Discussion; final comment To fully appreciate the difficulty of turning the behaviour of the correlation function into an effective criterion of stochasticity we mention here a few facts about the behaviour of correlation functions in the stochastic case. On this subject the dominating folklore is that correlation functions should decay exponentially in any case. As a matter of fact, the numerical results, analytical estimates and the very few rigorous results available up to now do not allow for definite statements. Numerical computations performed by Alder and Wainwright [lo], pointed out the existence of long time tails in the velocity autocorrelation for a system of hard spheres. They found S(7) - 7-l for hard disks and S(T)- T-~” for hard spheres. A theoretical justification of these results has been given [l 11 and for a review the reader is referred to ref. [12]. It would be interesting to understand the behaviour of correlation functions for stochastic systems in the context of ergodic theory. That is, one would like to establish whether, or not, a definite relation exists between the ergodic properties of a classical system (mixing, positive K-S entropy, homogeneous Lebesgue spectrum, etc.) and the type of decay of correlations. Bunimovich and Sinai 1131 have recently obtained an exact estimation for the velocity autocorrelation function of a particle in a Lorentz gas with a tCaboz and Lonke also give a theorem that a necessary and sufficient condition for the motion on the energy surface & to be stable is that the Liouville operator be self adjoint in I,*(&). The proof of this theorem is not correct: it is thoroughly based on the identification of the spectrum of a self-adjoint operator with its pure-point spectrum.

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647

periodic configuration of scatterers with a unipath: formly bounded free IS(n)] c cos t exp{ - an y}(O< y G 1) (n : number of collisions). The displacement of one scatterer produces a dramatic change in the behaviour of S(n), which now decays as an inverse power of time [14]. Also, numerical computations performed by us indicate that dijferent systems with the same K-S entropy may exhibit both exponential or algebraic decaying correlations [15]. Thus, we feel justified in stating that the problem of what is the “typical” behaviour of correlation functions in stochastic systems is a very difficult one, and that a satisfactory answer lies in the future. It may even prove true that no typical behaviour can be identified. In view of the above the use of correlation functions to detect the chaotic transition looks, at best, premature. In addition, large time computations of S(T) involve severe numerical difficulties. By less computational effort other, more firmly grounded, methods are more efficient. We close this paper by adding a final comment. Almost all stochasticity criteria proposed up to now have been tested by estimating the critical value EC = 0.11 for the stochastic transition in the H&on-Heiles system. Though “experimentally” successful, some of these methods have proven theoretically inconsistent: this is the case of the correlation function method discussed here [4-6] and also of the “Toda criterion” [16] criticized by one of the authors [17]. Thus it seems that great caution is necessary in using numerical computations to support any conclusion on this subject. References [l] There are many review papers on the subject; see for example, B. V. Chirikov, Phys. Rep. 52 (1979) 263. J. Ford, in proc. 1974 Wageningen summer school (North-Holland, Amsterdam, 1979) p. 215. M. V. Berry, in Topics in Nonlinear Dynamics, AIP Conf. N. 46, p. 16. G. Casati, Proc. Internat. Symp. on selected topics in statistical mechanics (Dubna, 1977). See also ref. 6.

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methods for detecting the stochastic

[Z] V. I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics (Benjamin, New York, 1%8). [3] Ya. G. Sinai, Introduction to Ergodic Theory, Mathematical Notes (Princeton Univ. Press, Princeton, 1976). [4] K. C. MO, Physica 57 (1972) 445. [5] A. Lonke and R. Caboz, Physica 99A (1980) 350. [6] M. Tabor, Advan. Chem. Phys. (1980). [7] Specifically we refer to theorems X1-14, XI-15 of Reed and Simon, Methods of Modern Mathematical Physics, Vol. III (Scattering Theory) (Academic Press, New York, 1979). [8] R. I. Cukier, K. G. Shuler and J. D. Weekes, J. Stat. Phys. 5 (1972) 99.

transition

[9] N. I. Akhiezer, The Classical Moment Problem (Oliver and Boyd, Edinburg, 1965). [lo] B. J. Alder and T. E. Wainwright, Phys Rev. Al (1970) 18. [ll] M. H. Ernst, E. H. Hauge and J. M. J. van Leeuwen, Phys. Rev. A4 (1971) 2055. [12] J. R. Dorfman, in Proc. 1974 Wageningen summer school (North-Holland, Amsterdam, 1979) p. 277. W. W. Wood, ib. p. 331. [13] L. A. Bunimovich and Ya. G. Sinai, Commun. Math. Phys. 78 (1981) 479. [14] Ya. G. Sinai: private communication to G. Casati. [15] G. Camporin, G. Casati and I. Guarneri, unpublished. [16] M. Toda, Phys. Lett. 48A (1974) 335. [17] G. Casati, Lett. Nuovo Cimento 14 (1975) 311.