Proc. Natl. Acad. Sci. USAVol. 90, pp. 9393-9397, October 1993Physics

Poincare resonances and the limits of trajectory dynamics(large Poincare systems/singular invariants/persistent interactions/broken time symmetry/complex irreducible spectral representation)

T. PETROSKYt AND I. PRIGOGINEtttCenter for Studies in Statistical Mechanics and Complex Systems, The University of Texas, Austin, TX 78712; and tInternational Solvay Institutes, CP231,1050 Brussels, Belgium

Contributed by L. Prigogine, April 27, 1993

ABSTRACT In previous papers we have shown that theelimination of the resonance divergences in large Poincaresystems leads to complex irreducible spectral representationsfor the Liouville-von Neumann operator. Complex means thattime symmetry is broken and irreducibility means that thisrepresentation is implementable only by statistical ensemblesand not by trajectories. We consider in this paper classicalpotential scattering. Our theory applies topersistent scattering.Numerical shnulations show quantitative agreement with ourpredictions.

Section 1. Introduction

A basic problem in modem physics is the elucidation of thetime paradox. On all levels in nature, we find evolutionarypatterns. In contrast, the traditional formulation of laws ofnature does not distinguish past from future. Our goal has,therefore, been to formulate laws ofphysics that include timesymmetry breaking. This goal has been realized for classes ofdynamical systems such as chaotic maps or large Poincaresystems (LPS) (1-6). For these systems, the Brussels-Austingroup derived a complex irreducible spectral representationfor the operators associated to their time evolution (such asthe Liouville operator or the Perron-Frobenius operator).Here, complex means that time symmetry is broken, andirreducible means that we deal with a statistical descriptionthat cannot be reduced to trajectories in classical dynamics(or to wave functions in quantum mechanics). The irreduc-ibility of the spectral representation leading to a probabilistictheory can be considered as the signature of chaos. LPS(classical or quantum) are chaotic systems in this generalizedsense. Here chaos is due to Poincare resonances that lead todiffusive processes. Such a formulation is only possible ingeneralized function spaces in which unitary operators mayadmit eigenvalues of modulo different from unity. In addi-tion, our formulation unifies fields of physics that appearquite separate in the usual presentation: dynamics (leading tothe evolution of ensembles), statistical mechanics character-ized by the introduction of "collision operators," and ther-modynamics (microcanonical distribution).To clarify the meaning ofour approach, we consider in this

paper the simple case of potential scattering. Scatteringcorresponds to a dynamical process. Still scattering is alsothe basic mechanism through which irreversibility appears instatistical mechanics (Fokker-Planck or Boltzmann equa-tions) (7). We shall show that dissipative effects arise forpersistent scattering, where we have to go beyond thetraditional S-matrix-type approach. Classical scattering canbe described in terms of Moller "states" 'I',) constructed incomplete analogy with quantum theory [this method wasintroduced by Resibois (8) and extensively used by one oftheauthors (9)]. More precisely, we deal here with Moller

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operators, as they are defined here in phase space and not inthe Hilbert space of quantum theory. Of special interest arethe Moller states I1D,), corresponding to the value k = 0 ofthe Fourier index k. A close relation exists between I Io,) andthe invariants ofmotion for scattering systems. We stress thatthese invariants are asymptotic invariants, valid only in thelimit of large volumes L3 -- o, where the scattering processoccurs. We next consider ensemble averages of Poincareinvariants and their time derivatives. We show that theensemble averages associated to delocalized distributions(independent ofx) decay through diffusive processes and thatthis decay is linear in time. These processes are not includedin trajectory dynamics.We come, therefore, to a most exciting situation, as we

may test the existence of diffusive processes for LPS in theframe of classical dynamics but beyond the trajectory for-mulation. Numerical simulations quantitatively confirm ourpredictions. This result shows that classical dynamics mustbe reformulated on the level of ensembles when extended toLPS.Let us first indicate the difference between "transient" and

"persistent" scattering. For transient scattering the distri-bution function p(x, p, t) is localized in the configurationspace and can be normalized to one,

fdp fdx p(x, p, t) = (27r)3/2fdp dkpk(P, t)8(k) = 1, [1]

where Pk(P) is the Fourier transform of p(x, p).We then consider the situation where p(x, p, t) has a finite

limit for lxl -m oo. The interaction with the potential istherefore persistent. For this situation the Fourier compo-nent of the distribution function is singular function at k = 0with a delta function singularity (9-11).

Pk(P, t) = pd(p, t)8(k) + pj(p, t), [2]

where pL is the nonsingular part of the distribution functionat k = 0. The distribution function cannot be normalized tounity, as a square of the delta function appears in Eq. 1. Forbox normalization, the delta function can be understood as aweighted Kronecker symbol An(k) - Qk,O, where fl(L/2Xr)3 with volume L3. The square of the delta function isthen 82(k) = fln(k), which diverges in the limit of largevolumes fQ om. We shall use this interpretation in Section3. The nonnormalizable situation can be realized, for exam-ple, by considering potential scattering involving many in-dependent identical particles that interact only with a singlepotential. We then consider the so-called "thermodynamiclimit" corresponding to a finite concentration c = NIL3 withan infinite number of incident particles N -X on and 13X* onand consider a reduced distribution function.

Abbreviations: SPI, singular Poincare invariants; LPS, large Poin-care systems.

9393

9394 Physics: Petrosky and Prigogine

Note that observables such as the Hamiltonian also have adelta singularity in the Fourier representation (see Eq. 3). Itis therefore natural to include distribution functions that arealso singular functions. The eigenvalue problem of the Li-ouville operator LH involving singular functions has quitespecific features. There appears a nonunitary similitudebetween LH and the "collision operator" 6 (2) which has acomplex spectrum. We recover in this special case the resultof our general theory associated to "subdynamics" (12-15).The singular part of the eigenfunctions satisfies a closedeigenvalue problem associated to 6 (in the lowest order it isprecisely the well-known Fokker-Planck operator), while theregular part becomes a functional of the singular part.The method given here can be easily extended to the

problems studied in statistical mechanics. An example is thewell-known Lorentz model in which a light particle interactswith an ensemble of heavy masses. We shall study thisexample as well as the extension of our results to quantummechanics in subsequent papers (T.P., D. Driebe, and Z. L.Zhang, unpublished work). The aim of this paper is to give anoverall view of our results. Proofs and details will be pub-lished separately.

Section 2. Classical Moller States and SingularPoincari Invariants (SPI)

Let us consider classical potential scattering for a unit massm = 1. We assume that the Hamiltonian is given by

H(x, p) = Ho(p) + AV(x) = fdk ( 6(k) + AVk)e [3]

The parameter A is a dimensionless coupling constant. Forsimplicity we assume short-range repulsive forces (i.e., nobounded motion).The Liouvillian is given by the Poisson bracket LHf i{H,

f} and generates the evolution of the probability distributionfunction p(x, p, t) of ensembles in phase space,

p(t) = eCLHtp(O). [4]

The Liouvillian is a hermitian operator in the Liouville space(9). In this space the inner product ofphase functions (x, pl g)-g(x, p) and (fix, p) _fc.C.(x, p) is defined in the "bra-ket"notation (9) by (j g) f dx f dp (flx, p)(x, pl g).For the unperturbed Liouvillian LO LHO = -ipa/lax the

spectrum is given by ref. 9:

Lolk, v) = (k.v)ik, v), [5]

where (x, p1k, v) = (21r)-3/25(p - v)exp(ik-x). For k = 0 wehave the invariants of motion LoJO, v) = 0.The completeness and orthonormality relations are

fdvfdkik, vXk, vl = 1, (k, vik', v') = 6(k - k')6(v - v'). [6]

The perturbed Liouvillian Lv is given in ref. 9;

d v-v)(k, vILvlk', v') = Vk-k'(k - k')

W

' . [7]

The matrix element is a "distribution" (as it contains aderivative of the 8-function).Let us consider the eigenvalue problem for the total

Liouvillian LH. To this end we introduce classical "off-shell"Moller states in analogy with Moller's scattering states for theHamiltonian in quantum scattering theory (16-18)

1i(Dk,v(z)) = 1k, v) + ~~T(z)ik, vT),

z - L[8]

where z is a complex number with Imz # 0. The "off-shell"T-matrix is defined by

1T(z) = ALv + ALv T(z).

z -Lo

We also have

(1, ulLHicIk,,(k.v + z)) = (k.v)(l, ul4k,(k v + z))

z+

- +Tz-1u Tluk,v(k-v + z),

[9]

[10]

where Tl,u;k,v (1, ulTik, v).Let us consider the "on-shell" limits z -+ ±iO of the Moller

states, ik,v,) i84a,v(k-v ± iO)). The phase functions Ok,5(x,p) are well defined (as velocity distributions):

ID,(x, p) = (2r)-3/2eik-x6(p - v)

I(2 r)-3/2 dl e"1x Ti,p;k,v(k-V +iO)\~TT;J k-v-Ilp ±iO

[11]

Then the Moller states become the eigenstates of the Liou-villian:

(x, piLHI4?,) = (k-v)(x, pl',). [12]

For this case we have also the usual completeness andorthonormality conditions for the on-shell Moller states.Of special interest are the Moller states for k = 0 that are

related to invariants of motion. As mentioned, (4o,v(z*)ix, p)is still a distribution involving the velocity. Therefore, weconsider the phase-space quantity If(z),

If(X, p; z) = (21r)3/2fdv fv) (4o,,v(z*)ix, p)

=f(p) + dk eikx _ k-f(p) + O(A2). [13]

kp+ za opIn the limit of z --* ±iO, If(z) are invariants of motion,

If(X(t), p(t); +iO) = 0. [14]

The standard perturbation analysis for the Hamilton- Ja-cobi equation using the generating function F(P, x) shows (19)that the new momenta P+ coincide with If(x, p; ±iO) forf(p)= p (unpublished work). We call If(±iO) "singular Poincardinvariants" (SPI), as the Fourier components of the invari-ants are singular at resonances k-p = 0 (refs. 9 and 20; see alsorefs. 21 and 22).

In the above construction of SPI we have considered ascattering process that occurs in infinitely large space. Wecan obtain the same result, starting with the standard boxnormalization approach and taking the limit of large volumesQ- oo. It is important to stress that the invariants If areasymptotic invariants valid only in this limit.

Section 3. Limits of Trajectory Dynamics andNumerical Simulations

In the previous section we have constructed SPI through theon-shell Moller states in the phase-space representation. As

Proc. Natl. Acad Sci. USA 90 (1993)

Proc. Natl. Acad. Sci. USA 90 (1993) 9395

SPI are singular at the resonances, we want to see theinfluence of resonances on the invariants in more detail.From Eq. 10 we have (for k = 0)

z(1, uILHI(DO,v(Z)) = ~~Tl,u;o,,v(z).z -1u

[15]

In the (1, u) representation the on-shell Moller states aredistributions in wave vectors 1, as well as in velocities. Thedistributions in I are usually handled by introducing localizedtest functions (the wave packets of traditional quantumscattering theory). However, we want to go here beyond theusual S-matrix theory. In Eq. 15 a discontinuity appears at 1= 0. For I # 0 we obtain zero in the limit z -- +i0. On thecontrary for I = 0 we have a nonvanishing contribution for theon-shell Moller states. As this effect appears only for I = 0,it is of the order 1/fl and can only be observed whenassociated to delocalized situations (see below). We verifiedthis result by numerical simulations.Let us consider in more detail the effect of the time

variation of ¢0,5(z) on the Poincare invariant If(x, p; +iO). Todo so we introduce the integral of If(z) over the phase spacewith a weight function g(x, p),

[If(t; z)]g dx dp [e+iLHtIf(x p; z)]g(x, p) [16]

The application of trajectory dynamics (Eq. 14) would lead to

dlim - [if(t; z)]g = 0. [171

Z-++io dt

However, Eq. 17 is correct only if gk(p) has no deltafunction singularity as in Eq. 2; this corresponds to thesituation where the function g(x, p) is a localized function inconfiguration space. In contrast, if g does not depend on x,e.g., g = go(p), then the time derivative (Eq. 17) does notvanish due to the resonances. To see this, we first note that(4N,¢,l exp(-iLHt)Lv = (4o,¢ILv because we can apply Eq. 12for a localized interaction V(x). Therefore, also taking intoaccount Eq. 15, we have for all times t

dlim - [If(t; Z)]g0

z-*+io dt

= -i lim fdk du dvf(v)(Do0,v(z*)iLHJk, u)S(k)go(u)z-.+ioJJ J

= -i du dvftv)(0, vIT(+i0)I0, u)go(u) [18]

= A2IdpfAp) dkIVVkI2k- lrS(k-p)k - go(p) + 0(A3).J J O~ ~ ~~pap

We obtain a diffusion process breaking time symmetry. (Theright-hand side is even in the momentum p, the equation istherefore not invariant for t -* -t and p -+ -p.) [We consid-ered here z --+i0. We have a similar equation for z -- -i0 butwith the opposite sign. This corresponds to the existence oftwo semi-groups. This situation has been discussed in ourpaper on the Friedrichs model (1)]. This effect is a nonlocaleffect requiring therefore an extended formulation of classicaldynamics involving ensembles. The origin of this effect is therole of resonances that lead to singularities.As seen from Eq. 18, To,v;o,u(+iO) is in the lowest approx-

imation in A, a second-order derivative operator in themomentum. The time evolution of [If(t; z)Igo cannot be

expressed by trajectory dynamics but only as the result ofdiffusive processes in momentum space, as described by theFokker-Planck operator. Following kinetic theory we shallalso call this matrix the "collision operator" 8 (2), which isrelated to the Liouvillian by

3,v;o,u= To,v;o,u(+io) = (4(DvILHIO, U)

= (0, VILHI o4,u) [19]

The collision operator acts only on the subspace of thedelocalized states with k = 0. Eq. 19 is the basic formulawhich relates dynamics of LPS to nonequilibrium statisticalmachanics.A close relation exists between the evolution of [If(t;

+iO)]g0 in Eq. 18 and the evolution of the ensemble average(f(p)) of quantityf(p) for a given initial distribution functionp(x, p, 0) = go(p). Indeed, they coincide in the asymptotictime limit (unpublished work). As the right-hand side of Eq.18 is constant, the delocalized superposition of the singularinvariants varies strictly linearly in time. Remarkably, thedissipative evolution associated with this linear t contributionalready starts from t = 0 [because the singular invariants If

(0)belongs to H subdynamics in our complex spectral represen-tation (unpublished data)]. In contrast, the time evolution ofthe ensemble average (f(p)) starts with t2, as easily under-stood using a short time expansion.We have done numerical simulations for potential scatter-

ing. [We thank K. H. Wen and Z. L. Zhang for the numericalsimulations and will present more details about the simula-tions (T.P., I.P., K. H. Wen, and Z. L. Zhang, unpublishedwork).] We have solved the Hamilton equations ofmotion fortrajectories and also the Liouville equation for ensembles.Thus we have calculated the evolution of the singular invari-ants (Eqs. 13 and 16) both for trajectories and for delocalizedensembles. We have approximated the singular invariants tofirst order in A. This leads to an error of order A2. Also wehave approximated the Fourier integral by Fourier series, byputting the system into a large box of volume Ld with theusual periodic boundary conditions. The Fourier spectrum istherefore discrete; i.e., k = nAk, where n e(ni, . . ., nd) isan integer-vector, and Ak 21r/L. We have also replaced theintegration of the wave vectors and the Dirac delta function,respectively, by a weighted sum fk-=1 and a weightedKronecker symbol if(k)- 6,, [for l-Ld/(2ijr)d]. In thelimit of large volumes the spectrum becomes continuous andthey reduce respectively to the integration fdk and to thedelta function 8(k) (23).There is a characteristic time scale tb= (IvAk)-1. We

impose the condition: t << tb. We may then consider thespectrum as continuous. We also replace the infinitesimalquantity z = +iO by z = +ie, where e is a finite small positivenumber. As the result the delta function Tr6(wkk') is approx-imated by the Lorentzian distribution el(w2, + e2). To obtaina consistent approximation ofthe delta function, there shouldhave enough number of discrete states around the peak oftheLorentzian. Therefore, our expressions have to be under-stood in the continuous limit Ak -- 0 and E -* O+ with thecondition for the small parameters: E >> vJAk.Numerical simulations for scattering in a two-dimensional

configuration space, i.e., d = 2, were completed. We thencompared the numerical results to our theoretical predictionsfor increasing values of the volume.The Hamilton equations of motion have been solved by the

standard fourth-order Runge-Kutta method. We have chosena Gaussian potential given by AV(x, y) = AVOexp[-(x2 +y2)/4a2] with a - 1. A typical time step was At = 0.01, andAVo = 10-3 was chosen for this simulation. We also chose the

Physics: Petrosky and Prigogine

9396 Physics: Petrosky and Prigogine

0

-1.0 x 10-4

< -2.0 x 10-4

1-1

-3.0 x 10-4

-4.0 x 10-4

0 5.0 x 10-5 1.0 x10-4 1.5 X 10-4 2.0x104 2.5 x104lNolume

FIG. 1. Time variation of the singular invariant AIt -IfAt) - IAO)vs. L2.

ratio e/(IvlAk) = 10. We have verified that the energy of thesystem is conserved up to 12 digits.

In Fig. 1 we plot the numerical results for the time variationof the singular invariant AIt If(t) - If(O) forf(v) = v. in Eq.13 vs. L-2. This figure corresponds to the case of the initialcondition (x, y;px, py) = (0.4, 1.41; -0.45, -1.40) and to timet= 5.0. For small volumes the error involved in the numericalsimulations exceeds the theoretically predicted error (AVo)2= 10-6. By increasing the volume, the numerical error due tothe volume dependence decreases, and AI, reaches thepredicted order of magnitude. This result shows that thesingular invariant is indeed an asymptotic invariant.We have next numerically solved the Liouville equation for

delocalized ensembles go(v) and calculated the time evolutionof the ensemble average of the singular invariant in Eq. 16 inthe (k, v) representation. As the Liouville equation is a partialdifferential equation, the numerical integration is more dif-ficult. In the numerical integration we have to deal withfour-dimensional discretized variables (kx, kg, vx, vy), whichrequires a large memory.

In Fig. 2 we plot the time evolution of A[If], [If(t)]g -

[If(O)]g in Eq. 16 for f(v) = vx and for the delocalizeddistribution g = go(u), given by

/7q 2 1 1go(u)

Ux) Uxo)2 + 772 (u - uo)2 +U [20]

We chose the parameters, q = 0.5, (uxo, uyo) = (0.5, 0.5), AWo= 0.1 and the volume L2 = (37,T)2. The time step of theintegration is At = 0.001. The numerical simulations showthat it changes linearly in time. As the ensemble theorypredicts, the linear dependence starts at t = 0.

In Fig. 3 we plot the rate ofchange A[Ifl,/At at the first stepof the iteration of the numerical integration vs. L -2. Byincreasing the volume size, the numerical value approaches

\-2.50 x 10- \

z,-5.00 x 10i-

-7.50 x 10- \

-1.00 x 1o-6

-1.25 x 10-6

FIG. 2. Time evolution of A[If(t)Ig [If(t)Ig - [If(O)Ig in Eq. 16for f(v) = v. and the delocalized distribution (Eq. 20).

FIG. 3. Change of A[If<t)]g/At at the first step of the iteration ofthe numerical integration vs. L-2.

the limiting value -4.0 10-4. The theoretical value of the

rate of change can be estimated by the right-hand side of Eq.18, which gives the value -(6.0 +0.1) x 10-4. Taking intoaccount the discretization procedure, this agreement is quitesatisfactory.

In conclusion these simulations prove the existence ofdiffusive effects in the frame of classical dynamics but whichcan only be observed in conjunction with delocalized ensem-bles.

Section 4. Complex Irreducible Spectral Representation

Our principal conclusion is that the Moiler states 4F-, are nomore zero eigenstates of the Liouvillian for persistent scat-tering. We therefore must reconsider the eigenvalue problem

LHIF (a)) = ZalF (a)). [21]

We sketch here only the main results; details will be forth-coming.To solve the eigenvalue problem, let us first note the

relation that can also be derived from Eq. 8,

(4DO,u(z*)ILHI(Do,v(z)) = (0, uIT(z)10, v)

+ (0, uT(z) T(z)I0 V

[22]

In the on-shell limit this gives (see Eq. 19),

lim (4Do,u(z*)ILIi14o,v(z))= (4k,uILHIO,v) = (0, uIO1O, v). [23]z-*+iO

This expression leads to a "similarity" relation between LHand 0 involving the transformation A,

(0, ulA-'LHAIO, v) = (0, ulO, v). [24]

A and its inverse operator connect the unperturbed states tothe Moller states as I'Iov) = AIo, v) and ('t vl = (0, vIA-1,respectively. (To introduce A and the inverse operator A-',some care is required to deal with the resonance divergence,as well as the normalization of the Moller states.) Theoperator A is a nonunitary operator, as it transforms thehermitian operator LH into the disipative operator 0, whichhas a complex spectrum. The nonunitarity comes from thesame mechanism of the discontinuity at I = 0 in Eq. 15 dueto the resonance singularity. The relation (Eq. 24) is inagreement with our previous approach through the subdy-namics theory (2). The similitude between LHand 0 involvingthe nonunitary operator A expresses the relation betweendynamics and probabilistic processes in a striking way.

-5.0 x

-1.0 x

4-1.x1-11III

-2.0 x

-2.5 x I4 x 10-'

lNolume

-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Proc. Natl. Acad. Sci. USA 90 (1993)

t

Proc. Natl. Acad. Sci. USA 90 (1993) 9397

Section 5. Concluding Remarks

LH AIO, v) = A6O1, v), [25]

leading to the relation between the eigenstates If(a)) of thecollision operator Olf(a)) = zIJfta)) and the eigenstates IF(a))of the Liouvillian (Eq. 21):

If(a)) = du|b, u)Fg(u; a), and

JF(a)) = Alf(a)) = fduI4s )Fdg(u; a), [26]

where Fod(u; a) satisfies the eigenvalue equation (with Ea =

ZaQl),

IJdu d,v;o,.Fg(u; a) = faFo(v; a). [27]

[To obtain these relations, we have used the relation AO =

(0) (0)A P 0, where P-l-1 f duAO, u)(0, ul is a projection operator

(0) '(0)for the delocarized component, (p)2 = P , in the box nor-malization formalism.] As the collision operator 00,V;o,. is welldefined in the limit of large volumes, and Fod(v; a) are alsowell defined in this limit.

Since VI\+u have a delta function singularity, the eigen-states IF(a)) also have a delta function singularity:

(k, vlF(a)) = Fg(v; a)8(k) + Fj(v; a). [28]

Moreover, the nonsingular part Fk is a functional of thesingular component Fd, as one can see from Eq. 26.For weak-coupling interactions the eigenvalue problem

(Eq. 27) of the collision operator reduces to the eigenvalueproblem for the Fokker-Planck operator in Eq. 18. It is wellknow that the Fokker-Planck operator for central forces forthree-dimensional case has the same structure as the orbitalangular momentum operator in quantum mechanics (11).Therefore, the eigenstates ofthe Fokker-Planck operator aregiven in terms of the spherical harmonics Fd(v; a) = w-18(Ivl- w)Y (6, 4) for a = (w, 1, m). The eigenvalues are thengiven by = -iA2Aw-3l(1 + 1), where A = 47r5 fr dq q3IVq12.For I = m = 0, Eq. 26 leads to the microcanonical distributionfor the equilibrium mode of the Liouvillian. We also see thatthe eigenstates ofthe Liouvillian break the time symmetry, asthe nonvanishing eigenvalues are imaginary numbers. [Thesolution of our eigenvalue problem can also be tested bycomputer simulations. This has already been done for thequantum case which is numerically easier to handle (T.P.,I.P., K. H. Wen, and Z. L. Zhang, unpublished data).]For the class of delta singular functions the only possible

invariant of the motion is the microcanonical distribution. Allother invariants are destroyed due to the resonances. There-fore, potential scattering is nonintegrable in conjunction withpersistent scattering.As is a well-defined finite number in the continuous limit

flQ-+, the eigenvalue z,, is an infinitesimal quantity. We haveverified the relation za = fl-1a by numerical simulation; thevolume dependence corresponds to the fact that the systemcannot reach equilibrium over a finite time scale because ofa single scattering. We need repeated scattering. We haveverified this statement for a simple model of a many-bodysystem-i.e., the Lorentz gas. We then obtain a finite eigen-value for the Liouvillian that is proportional to the concen-tration c of the scatterers (unpublished data).

For a long time, classical dynamics appeared a closed sub-ject. This, however, is no more so when nonintegrablesystems in the sense of Poincard are considered. We can nowgo beyond Poincare's negative statement (nonexistence ofinvariants analytic in the coupling constant) for classes ofdynamical systems, such as LPS. For these systems, we haveconstructed singular Poincard invariants (which are analyticin the coupling constant but involve distributions). They arehowever only asymptotic invariants. In conjunction withdelocalized distributions they decay according to diffusivemodes. That shows that there exist nonlocal effects inclassical mechanics which can only be described on the levelof ensembles. These effects can be studied by computersimulations. They show that a more general formulation ofclassical mechanics is needed when dealing with LPS. Thissignals the appearance of another form of chaos as we haveto go beyond the trajectory description.For persistent scattering that corresponds already to a LPS,

there appear specific singularities in the distribution functions(see Eq. 2). The appearance of such singularities is quitegeneral. They appear as well in quantum theory and instatistical mechanics, where they lead to the existence ofreduced distribution functions in the thermodynamic limit (9).For this type of distribution functions we must consider othersolutions of the Liouville equation that are not reducible totrajectory dynamics (or to wave functions in quantum me-chanics). For this class of situations we obtain a unifiedformulation of laws of physics that is intrinsically statisticaland irreversible. In contrast to the traditional perspective, thelaws of physics express "possibilities," rather than "certi-tudes."

This paper has benefited from much constructive criticism anddiscussion from Drs. I. Antoniou and S. Tasaki. We thank Dr. K. H.Wen and Mr. Z. L. Zhang for the numerical simulation. We aregrateful to Dr. H. Hasegawa and Prof. E. C. G. Sudarshan forinteresting remarks. We also acknowledge U.S. Department ofEnergy Grant DE-FG05-88ER13897, Robert A. Welch FoundationGrant F-0365, and European Community Contract PSS 0661 forsupporting this work.

1.2.3.4.5.6.7.

8.9.

10.11.

12.13.14.15.16.

17.

18.

19.

20.21.

22.23.24.25.26.

Petrosky, T., Prigogine, I. & Tasaki, S. (1991) Physica A 173, 175-242.Petrosky, T. & Prigogine, I. (1991) Physica A 175, 146-209.Hasegawa, H. H. & Saphir, W. C. (1992) Phys. Lett. A 161, 471-482.Hasegawa, H. H. & Saphir, W. C. (1992) Phys. Rev. A 46, 7401-7423.Antoniou, I. & Tasaki, S. (1993) J. Phys. A 26, 73-94.Antoniou, I. & Tasaki, S. (1993) Int. J. Quantum Chem., in press.George, Cl., Mayne, F. & Prigogine, I. (1985) Adv. Chem. Phys. 61,223-299.Resibois, P. (1959) Physica 25, 725-732.Prigogine, I. (1962) Non-Equilibrium Statistical Mechanics (Wiley, NewYork).Prigogine, I. & Balescu, R. (1959) Physica 25, 281-323.Balescu, R. (1963) Statistical Mechanics of Charged Particles (Wiley,New York).Prigogine, I., George, Cl. & Henin, F. (1969) Physica 45, 418-434.George, Cl. (1973) Physica 65, 277-302.De Haan, M., George, Cl. & Mayn6, F. (1978) Physica A 92, 584-598.Petrosky, T. & Hasegawa, H. (1989) Physica A 160, 351-385.Grosjean, C. C. (1960) Formal Theory ofScattering Phenomena (InstitutInteruniversitaire des Sciences Nucleaires, Brussels), Monographie No.7.Goldberger, M. L. & Watson, K. M. (1964) Collision Theory (Wiley,New York).Newton, R. G. (1982) Scattering Theory of Waves and Particles (Spring-er, New York).Goldstein, H. (1980) Classical Mechanics (Addison-Wesley, Reading,MA).R6sibios, R. & Prigogine, I. (1960) Bull. Cl. Sci. Acad. R. Belg. 46,53-60.Prigogine, I., Grecos, A. P. & George, Cl. (1976) Proc. Natl. Acad. Sci.USA 73, 1802-1805.Petrosky, T. (1987) J. Stat. Phys. 48, 1363-1372.Hugenholtz, N. M. (1957) Physica 23, 481-532.Van Hove, L. (1955) Physica 21, 517-540.Petrosky, T. & Prigogine, I. (1988) Physica A 147, 439-460.Haubold, K. (1964) Bull. Cl. Sci. Acad. R. Belg. 50, 991-1008.

We also have

Physics: Petrosky and Prigogine