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Eigenmoments for Multifragmentation

arXiv:hep-ph/9306239v1 7 Jun 1993Eigenmoments for MultifragmentationB.G.Giraud and R.PeschanskiService Physique Th´eorique, DSM-CE Saclay, F91191 Gif/Yvette, FranceAbstract: Linear rate equations are used to describe the cascading decay of an initialheavy cluster into fragments. Using a procedure inspired by the similar, but continuouscase of jet fragmentation in QCD, this discretized process may be analyzed into eigen-modes, corresponding to moments of the distribution of multiplicities.

The orders of thesemoments are usually noninteger numbers. The resulting analysis can be made time inde-pendent and is applicable to various phenomenological multifragmentation processes, inwhich case it leads to new approximate finite-size scaling relations for the spectrum offragments.In this work we consider binary fragmentation processes where any fragment withmass number k breaks into fragments with mass numbers j and k −j, j = 1, 2...k −1,with a probability wjk per unit of time.It is assumed that wjk is time independent.By definition, wjk is symmetric if j is replaced by k −j, naturally.Let Nj(t) be themultiplicity of fragment j at time t in a process initiated from the decay of a cluster A,namely Nj(0) = δjA.

The model under study is described by the following set of linear,first order differential equations,dNjdt= −cjNj +AXk=j+1wjkNk, j = 1, ...A,(1)withcj =j−1Xℓ=1wℓj2 . (2)With components Nj, j = 1, ...A, for a column vector |N >, the system, Eqs.

(1), obviouslyboils down to d|N > /dt = W|N > with a triangular matrix W. For the sake of clarity weshow here W when A = 4,W =0w12w13w140−w12/2w23 = w13w2400−(w13 + w23)/2w34 = w14000−(w14 + w24 + w34)/2. (3)1

The general solution of Eqs. (1) is obviously a sum of exponentials whose rates of decayin time are the diagonal matrix elements −cj, the trivial eigenvalues of the triangular W.(We notice that any increase of the dimension A leaves intact the preexisting eigenvaluesand only adds new ones.

)This matrix W has a remarquable property, namely a fixed left (row-like) eigenstateM1, whose components M1j = j, j = 1, ...A, do not depend on the wℓk’s. This comes fromthe symmetries of W demanded by the conservation of the total mass M1 = PAj=1 jNj,with dM1/dt =< M1|W|N > .

The corresponding eigenvalue is, naturally, −c1 = 0.Moreover, it is clear that the other eigenvalues, −cλ, λ = 2, ...A, induce a triangularmatrix of left† eigenstates Mλ, namely the components Mλj vanish when j < λ.The higher λ, the more the corresponding (bra-like) eigenstate is only probing heavyfragments in the multiplicity distribution N . This is reminiscent of a hierarchy of mo-ments Mq = PAj=1 jqNj, when the exponent q increases.

Even though moments do notmake a strictly triangular rearrangement of the information contained in N , they rep-resent a natural continuation of the first eigenvector M1, and this letter will show thatthere is a practical connection between moments and eigenvectors. Indeed, besides trian-gularity, there is another argument indicating the interest of moments in the solution ofrate equations like Eqs.(1).

In field theory models of jet fragmentation at high energy,similar evolution equations appear as solutions of perturbative QCD; they are continu-ous, since the quantity which fragments is the energy-momentum of quarks and gluons,before their transformation into the observed hadrons. These equations are well knownas Gribov-Lipatov-Altarelli-Parisi (GLAP) equations[1].

As a function of the fraction xof energy-momentum, the equations for the fragmentation function N(x) can be exactlydiagonalized by moments Mq =RdxxqN(x). In this case, all values of q ≥1 are admitted,as a consequence of the continuous character of the equations.

Our task, in this letter,is to examine the effect of discretization on this field-theoretical result. Note that such adiscretization have been introduced long ago[2], but only for a numerical approximation ofthe continuous equations.2

In the following, we will consider “triangularly redefined” momentsMλ =AXj=λjq(λ)Nj,(4)where the set of exponents q(λ), λ = 2, ...A, may contain nonintegers. The time derivativeof such a redefined moment isdMλdt=AXj=λjqAXk=j+1wjkNk −cjNj=AXk=λkqNkd(λ, k),(5)withd(λ, k) =kXj=λ jkqwjk −12kXj=1wjk,(6)where we have used an interchange of indices j and k in the double summation, and alsothe convention that diagonal rates wkk are identically vanishing.

The double summationis sketched on Fig.1, displaying the weight matrix wjk.It turns out that, in practice, there exists special values q(λ) for which the coefficientsd(λ, k) happen to be, at least approximately, independent of k when k is large enough.Then, they can be factored out in formula (5) and they appear as eigenvalues for the corre-sponding eigenvector Mλ. Moreover, simultaneously, since we know the exact eigenvalues(see Eq.

(2)), one gets d(λ, k) ≃−cλ. This will be shown both analytically in a “continu-ous” limit (i.e.

large matrices) and numerically for various models with finite sizes A. Infact one is led to consider first the discrete models which admit the field-theoretical typeof equations in the continuous limit - let us call them the scale-invariant case, since nodependence on the matrix size appears explicitely, and then the more general situation.1. The scale-invariant caseWhen A is large, λ finite, and k large but smaller than A, then the ratio x = j/kcan be considered as a continuous label, 0 ≤x ≤1, in Eq.(6).

Moreover, with positivevalues of q, an extension of the first summation in Eq. (6), Pkj=λ, into a summation Pkj=1brings a weak contribution from 0 < x < λ/k.

From QCD, where x corresponds to thefragmentation of momentum and a scale-invariant property of the transition weights is3

valid, one may consider a large class of models settingwjk = ϕ(j/k)/k = ϕ(x)/k,(7)where ϕ is any suitable function of the scaling variable x. More precisely, because ofthe symmetry necessary for wjk, a large class of legitimate models correspond to wjk =[f(j/k) + f(1 −j/k)]/(2k) = [f(x) + f(1 −x)]/(2k).

Hence, for large values of k, bothsummations in Eq. (6) amount to the discretization of an integrald(λ) =Z 10dx[f(x) + f(1 −x)](xq −1/2)2=Z 10dx[f(x) + f(1 −x)][xq + (1 −x)q −1]4,(8)where dx replaces 1/k.

It will be noticed that d(λ) does not depend any more on k. It stilldepends on λ via the exponent q(λ), naturally. Note also that the “splitting” function fcan be general, even with some singularity at both ends of the integration domain, providedthe integral itself converges.There may also be a continuous limit for cλ if λ becomes large.

Indeed, according toEq. (2), cλ →R 10 dxϕ(x)/2, if 1/λ amounts to dx and if this integral converges.

In such acase, the spectrum of W accumulates into a quasi degeneracy. However, it is important torealize that the convergence of the cj’s is not required.

On the contrary, it is quite possiblethat the limiting continuous model does not exist, leading to an infinite hierarchy of q(λ).We shall meet such cases later on. Moreover, for low values of the label λ, this continuouslimit is not in order.We now notice that ϕ(x) is a semi-positive definite function since wjk, a transitionrate, cannot become a negative number.

Hence d(λ) is a monotonically decreasing functionof q. It vanishes for q = 1, as expected from the conservation of M1.

According to Eq. (5),the time evolution of a triangular moment Mλ becomes very simple if d(λ) can be identifiedwith the eigenvalue, −cλ.

Hence, for each λ, we consider the exponent q(λ) which is theunique solution of the consistency equationZ 10dx[f(x) + f(1 −x)][1 −xq(λ) −(1 −x)q(λ)] = 4cλ. (9)The discrete set of solutions qλ of this equation, when the integer label λ runs from1 to A, define the “eigenmoments” of the theory, namely those moments whose time4

evolution is (almost) proportional to just one exponential exp(−cλt), rather than a mixtureof such exponentials. It will be noticed that, since the sequence of coefficients cλ increasesmonotonically, the sequence of solutions q(λ) is also a monotonically increasing sequence,starting from q(1) = 1 with c1 = 0.Set temporarily f(x) = 1/xβ, with β = 1.

This case is reminiscent of the QCD evolu-tion equations for gluons whose kernel contains the same singularity at small x [1]. Thenwjk = [1/j + 1/(k −j)]/2, and one finds easily that c2 = 1/2, and that the correspondingsolution of Eq.

(9) is q(2) = 2. One also finds that c3 = 3/4, and that the correspondingsolution of Eq.

(9) is q(3) = 3. More generally, one finds for Eq.

(9) the solution q(λ) = λ.This definitely suggests that integer moments form an infinite sequence of eigenmomentsfor that choice of f, f(x) = 1/x. In agreement with this hint, ones finds easily from Eq.

(6)that the sequence of coefficients d(2, 20) = −0.475, d(2, 21) = −0.476,...d(2, 49) = −0.4898,d(2, 50) = −0.4900,... converges towards −c2. Just to give another example, the sequenced(5, 20) = −0.980, d(5, 21) = −0.983,...d(5, 49) = −1.016, d(5, 50) = −1.017,... convergestowards −c5 = −1.042.

And so on for all the moments Mλ, which thus generate excellentapproximations to eigenvectors when the exponents q(λ) are just integers.This argument is independent from the normalization of f, since Eq. (9) is homo-geneous with respect to trivial multiplications of f by an overall constant.It may beinteresting to note that the diverging sequence of values for q(λ) might be related to thedivergence of the eigenvalue sequence at infinite λ.A similar result can be observed if β ̸= 1, but now the solutions of Eq.

(9) do notcorrespond to integer exponents.For instance, with β = −0.5, one finds c2 = 0.177,q(2) ≃3.13, c3 = 0.232, q(3) ≃5.26, c4 = 0.259, q(4) ≃7.4, c5 = 0.275, q(5) ≃9.6,...The convergence of the coefficients d towards the eigenvalues is still surprisingly good. Forinstance d(2, 20) = −0.176, and d(5, 20) = −0.275.This discussion (for the scale invariant case) is illustrated on Figs.2.

Here, ratherthan asking whether a moment may behave like an eigenvector, we consider the reversequestion: given an eigenvector Mλ, does it happen that the components Mλj induceeffective moments Mq(λ)? Namely, is there an exponent q(λ) compatible with Mλj ∝5

jq(λ), j > λ? In Figs.2, for instance, the components of the second (λ = 2) and fourth(λ = 4) eigenvectors are displayed as functions of the fragment size.

It will be stressedthat they are almost linear in a Log-Log plot in a large interval starting from the maximalchoosen value A = 30. These figures, Figs.2, show the structure of moments depending onthe parameter β.In table I we show the comparison between the actual eigenvalues cλ, (for λ = 2 and 4)with those obtained through Eq.

(9) after the determination of the effective values q(λ) fromFigs.(2). The agreement is pretty good, except perhaps for β ≃2, which is at the borderlineof convergence of the integral in Eq.(9).

It can be thus claimed that various choices of ϕin Eq. (7) make it possible to find eigenmoments via the continuous limit and solutions ofEq.(9).

It will be noticed that this continuous limit is closely linked to the denominatork in Eq. (7), since this denominator induces the needed measure dx, independently of theoverall scale given by A.2.

Scale-dependent casesIn a more general situation, wjk is not compatible with Eq.(7). This will in generallead to an introduction of the overall scale A in the problem, and corresponds to caseswhere the scale invariance of the weights is not preserved.

Let us illustrate this by thefollowing instance:wjk = [(j/k)−β + (1 −j/k)−β]/(2kα),(10)where α may be different from 1.To be specific, but as an example of more general value, we display on Fig.3 theLog-Log plot Mλj versus j, for λ = 2, A = 50, β = 0 and various values of α. From thisfigure, one realizes on this simple example that the diagonalization by eigenmoments isobtained for α ≤1, while for α > 1 there is a clear distorsion of eigenvectors with respectto moments.

It is interesting to interpret this phenomenon analytically by inspecting themodification which occurs with the choice of Eq. (10) for the weights.

Instead of Eqs. (5-6),one finds the following,dMλdt=AXk=λkq−α+1Nkd(λ, k),d(λ, k) =kXj=λ jkq¯wjk −12kXj=1¯wjk,(11)6

where ¯w corresponds to rescaled weights with α = 1. Following the previous discussion,based on the existence of a continuous limit d(λ, k) ≃d(λ), one is led to the approximateconsistency equation:−d(λ) ≃Aα−1c(λ),(12)where the renormalisation factor Aα−1 takes care of the initial values of the moments.The occurrence of the A−dependent factor is the signal of the lack of scale invarianceof the fragmentation dynamics when α ̸= 1.

Note that Eq. (12) uses the assumption thatthe eigenvalues do not change substantially between q and q −1 + α.

It can only be anapproximation.In Table II, we display the different values obtained for the q(2) for various values ofα, obtained by the fitted slopes at the origin for the curves obtained in Fig.3. We comparethem to those obtained from Eq.

(12) when the input are the actual exact eigenvalues c2.The agreement is here also quite satisfactory, except in the region when α > 1. One mightassociate this phenomenon to the well-known[3] fact that a shattering transition takes placeat finite time, the conservation of mass being broken in the continuum limit.

A specialstudy of this case is in order for the future.The class of models which can be analyzed by eigenmoments is thus larger than theclass described by Eq.(7). It must be noted, however, that the “eigenorders” q(λ) are notuniversal, but clearly model-dependent.As an application of the properties of eigenmoments, let us consider the problem of3-dimensional bond percolation in a finite-size square lattice.

This model seems to givea successful description of nuclear multifragmentation, when a heavy ion receives enoughexcitation energy to form a highly unstable state and decays into several fragments[4]. Thestatistics of fragment numbers and sizes seem to follow predictions of a percolation modelin which each lattice site is populated by a nucleon, and the percolation parameter p,namely the survival probability for bonds, varies between 0 and 1.

There is no obvioustime scale in this model, hence a comparison of its predictions with those of linear rateequations models requires the use of eigenmoments in order to obtain an intrinsic timescale from the evolution of such eigenmoments.7

For this purpose, we remark that, if they are identified as eigenmoments, the Mq’s arelinked by linear relations in Log-Log plots, and their explicit time dependence disappears.We are thus led to display in the same way the moments obtained from the percolationmodel, choosing for instance M2 for reference, see Fig.4. Different moments are displayed(with q = 1, 1.5, 2, 3, 4, 5) and show the interesting feature of a quasi-linear dependence forthe values q = 3, 4, 5, given the fact that for q = 1 (mass conservation) and q = 2 (referencescale) the linear dependence is fixed.

It is clear from this figure that the quasi-linear formis obtained between p = 1 and p = pc, where pc is the critical value above which, in thecontinuous limit, an infinite percolation cluster is formed. Indeed, the figure shows thedominant contribution of the cluster of largest mass to the averaged moments.

This largestcluster is, for finite size problems, the representative of the infinite cluster when p ≥pc.Notice that the moments implied by the rate equations are the full moments, includingthe largest fragment, while in usual analyses of percolation models[4], scaling properties areinvestigated with moments modified by the subtraction of the largest cluster. Moreover, insuch traditional analyses of percolation, the reference time scale is generally given by themoment M0 or a similar variable related to the multiplicity of fragments.

The comparisonand compatibility of our approach with such analyses is an open problem of some interest[6].In conclusion, from this first study on the percolation model, we obtain a hint thatlinear rate equations could provide a time dependent description of multifragmentation.But it is difficult at this stage to obtain informations on the set of eigenorders q(λ) whichcould be associated with percolation. The existence of scaling relations between momentscan be proven in the vicinity of pc for percolation through finite-size scaling[4].

Our resultis compatible with this and, furthermore, involves the whole region 1 > p > pc. In therepresentation provided by rate equations, we have obtained scaling relations valid forshort time scales, while the previous general results, see for instance Ref.

[7], involve longtime scales only. An open problem is to connect both analyses for a general system ofequations.Acknowledgments R.P.

thanks Xavier Campi for his patient explanations on nuclearmultifragmentation and for providing the authors with the suitable percolation program.8

Thanks are due to G´erard Auger, Brahim Elattari, Pierre Grang´e, Hubert Krivine, EricPlagnol and Jean Richert for fruitful discussions.†Footnote The right-hand-side, ket-like eigenstates define also a triangular matrix,apparently unrelated to the matrix of bra eigenstates, except for trivial biorthogonalityrelations. Up to now we have been unable to find a practical use of these ket-like eigen-vectors.References[1] G. Altarelli and G. Parisi, Nucl.

Phys.126 (1977) 297. V.N.

Gribov and L.N. Lipatov,Sov.

Journ. Nucl.

Phys.15 (1972) 438 and 675. For a review and references, Basicsof perturbative QCD Y.L.

Dokshitzer, V.A. Khoze, A.H. Mueller and S.I.

Troyan (J.Tran Than Van ed. Editions Fronti`eres, France, 1991.

)[2] P. Cvitanovic, P. Hoyer and K. Zalewski, Nucl. Phys.

B 176 (1980) 429. [3] E.D.

Mc Grady and Robert M. ZiffPhys. Rev.

Lett. 58 (1987) 892.

[4] X. Campi, Phys. Lett.

B 208 (1988) 351, and contributions to the the Proceedings ofVarenna 1990 and 1992 Summer Courses of the International School of Physics EnricoFermi. [5] For a general review on percolation: D. Stauffer, Introduction to Percolation Theory(Taylor and Francis, London and Philadelphia, Penn.

1985. )[6] On a phenomenological ground in relation to nuclear multifragmentation, the problemhas been raised by: J. Richert and P. Wagner, Nucl.

Phys. A517 (1990) 299.

[7] Z. Cheng and S. Redner, J. Phys. A: Math.

Gen. 23 (1990) 1233.9

Figure captionsFig.1: Discrete rate equations: Double SummationThe double summation range for truncated moments Mλ is represented by the hatchedtriangle. White dots: diagonal weigths wjj are zero.

Black dots: non-diagonal weigthswjk. The interchange of indices j and k in the description of the hatched triangle leads toEqs.

(5-6).Fig.2: Eigenvectors: scale-invariant case (α = 1)The components of the rate equation eigenvectors Mλj are displayed as functions of j in aLog-Log plot. The chosen weights correspond to Eq.

(10) with α = 1, and different valuesof β. The curves correspond to a smooth interpolation (dark line), resp.

extrapolation(dashed line), of the exact eigencomponents for a system of size A = 30. Fig 2-a: λ = 2 ;Fig 2-b: λ = 4.

The curves are used for the determination of the effective values q(2), q(4),see Table I.Fig.3: Eigenvectors: scale-dependent cases (β = 0)Same as Figure 2-a but for weights following Eq. (10) with β = 0 and different values of α.The curves give the eigenorders listed in Table II.Fig.4 Percolation analyzed with the “M2 time scale”Relative strengths of moments Mq, q = 0, 1, 1.5, 2, 3, 4, 5, as functions of M2 in a Log-Logplot.

Data taken from 3-d bond percolation on a 6∗6∗6 lattice. The corresponding valuesof the bond survival probability p are shown on the horizontal axis.

Its critical value ispc = .25. Full lines: moments.

Dashed lines: contributions of the largest cluster. Dashed-dotted line: the reference moment M2.

Notice that a linear behaviour is approximatelyobtained for 0 ≤p ∼< pc and q = 3, 4, 5.Table CaptionsTable I: For α = 1, λ = 2, 4 and different values of β, the effective exponents q(λ) fromFigs.2-a,b and the corresponding values for −d(λ), see Eq.(9). The latter are comparedwith the exact eigenvalues cλ.Table II: For β = 0, λ = 2 and different values of α, comparison of the measuredeffective eigenorders (obtained from Fig.3) with those predicted from Eq.

(9).10

βq(2) −d(2)c2q(4) −d(4)c4−22.4.061.0626.1.11.11−1.52.7.09.096.5.14.14−.53.1.18.187.4.26.2603.25.257.1.38.38.52.6.36.355.8.57.5712.51.504.2.94.921.51.5.79.712.81.81.621.1.8511.74.62.7Table Iαq(2)measuredq(2)predicted01.031.04.51.171.22.61.261.33.81.641.713.031.15.75.21.22014Table II11

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