V O L U M E 71, N U M B E R 8 P H Y S I C A L R E V I E W L E T T E R S 23 A U G U S T 1993

Absence of Self-Averaging in Shattering Fragmentation Processes

D. L. Maslov* Department of Physics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6

(Received 30 December 1992)

We consider fluctuations in both the shattering and normal (nonshattering) fragmentation pro­cesses. The continuum shattering process is shown to lack the self-averaging property: the rms fluctuation in the total number of the finite-size fragments is larger than the mean value. The dis­crete shattering process exhibits strong fluctuations at the intermediate times when the total number of fragments is much greater than 1 but smaller than the number of the monomers contained in the primary particle. For the normal process, the fluctuations are small and self-aver aging exists.

PACS numbers: 82.20.-w, 05.40.-fj, 05.60,+w

The process of fragmentation is evident in a variety of physical phenomena: cosmic showers [1], polymer degra­dation [2], collision [3] and phonon [4] cascades in solids, fracturing of solids [5] and clusters [6], and many oth­ers. Despite the fact tha t the first works on fragmen­tation appeared before this century [7], this field still presents surprises to those who study it. For example, it has been discovered only recently [2] tha t if the breakup time decreases as the process goes on (i.e., as fragments get smaller), the relevant conserved quantity (total mass, total energy, etc.) is accumulated by the zero-size frag­ments (from now on, by "size" we mean mass, energy, etc.). This type of fragmentation process ("shattering") can be thought of as a process opposite to the gelation transition [8] tha t occurs in coagulating systems. It is worth noting tha t the shattering effect, as well as the vast majority of the other results in the theory of frag­mentation, has been obtained by considering only mean quantities (e.g., the mean value of the total number of the fragments N).

However, it has been known for a long time tha t the simplest binary fragmentation process with the breakup time independent of the fragment size (the Yule-Furry process [1]) represents a classical example of a non-self-averaging system: At late times, the rms fluctuation 6N of N is equal to the mean value (N), no matter how large (N) is. The reason for such a behavior is tha t the memory of the fluctuations tha t occurred at the early stages of the process is not erased as the process goes on. This lack of self-averaging makes fragmentation similar to such stochastic phenomena as diffusion-limited aggre­gation (DLA) [9] and spinodal decomposition (SD) [10], and, in a less direct way, to conductance fluctuations in mesoscopic samples [11]. It is worth noting tha t in DLA and SD, self-averaging can be absent for some quanti­ties, but present for others [e.g., the growth probability distribution in SD is a non-self-averaging quantity, while the domain-size distribution is (temporal) self-averaging [10(c)]]. In fragmentation, non-self-averaging is of a more global character—it is exhibited by the simplest quantity which one can think of (the total number of fragments) and does not go away as more complicated stochastic

variables are considered. Also, the linearity of the prob­lem allows for an explicit solution, which makes this sys­tem very attractive for study.

Motivation for the study of the Yule-Furry process originated in the study of cosmic showers [1], where the decay times of the relativistic particles are almost inde­pendent of their energies. However, from a general point of view, the Yule-Furry process is only a marginal case between two wide classes of fragmentation: one (shat­tering) for which breakup times increase with increas­ing fragment size and the other (normal or nonshatter­ing) with decreasing breakup times. Examples of these classes are known in different fields of physics [2-7], The fluctuations for both of these classes have been given at­tention only recently in Ref. [12], where the fluctuations in the fragment-size distribution function for a contin­uum model were studied [13]. Unfortunately, the fluc­tuations in AT, which are more readily observed in an experiment, cannot be obtained from the fluctuations of the distribution function (they can be obtained only from the two-point correlation function, which is very difficult to calculate). Also, any fragmentation process cannot last forever—there is always a smallest fragment tha t is indivisible in the framework of the given problem (e.g., a monomer in the polymer chain degradation). This is why a discrete model tha t explicitly incorporates the presence of a fragmentation threshold is a better approximation of these processes than continuum models.

In this Letter we consider fluctuations of N in both the shattering and normal fragmentation processes. For the continuum shattering process we show tha t the rel­ative fluctuation R — 6N/{N) of the total number of nonzero-size fragments is greater than 1 and grows with time. We also study numerically the fluctuations in the discrete analog of the shattering process and show that R reaches a maximum at a time comparable to the breakup time of the primary particle and then decreases. At its maximum, R can be smaller or greater than 1 depending on the number of the monomers contained in the primary particle NQ. At times later than the breakup time, R is small and self-aver aging is recovered.

To obtain analytical results, we adopt the method of

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© 1993 The American Physical Society

V O L U M E 71, N U M B E R 8 P H Y S I C A L R E V I E W L E T T E R S 23 A U G U S T 1993

the first-instant master equation [14,15]. This method enables one to obtain a closed hierarchy of the backward kinetic equations for both (N) and 6N without considering the intermediate equations for many-point correlation functions. Define the quantity P(x\N,t) to be the conditional probability of having N fragments of any size at t ime t provided tha t there was one particle of size x at time to. The evolution P is described by the first-instant master equation [14,15]

dtP{x\N1t)=YJ / dyw(x\y)P(y\N'yt)P(x-y\N-N\t)-P(x\N,t)/T(x), (1) N<=OJO

we assume that w is independent of y and, hence, has the form w = [xr(x)]~1. For the sake of simplicity we take r in the power-law form r oc xa. If a > 0 , it is a shattering process; if a < 0 , it is a normal process; the marginal case a = 0 corresponds to the Yule-Furry process.

Equations (4) and (6) are simplified greatly by noting that the dimensionless quantities (N) and (N2) depend only on the scaling variable £ = t/r(x). Making use of the explicit form of r , Eq. (4) is reduced to

where dyw(x\y) is the probability per unit t ime tha t the fragment of a size x splits into two fragments of sizes y and x — y, respectively. The breakup time r is defined by

l / r ( x ) = / dyw(x\y). (2) Jo

Equation (1) is derived by tracing the initial particle, which can either split into two fragments tha t subse­quently produce two independent cascades containing N' and N — N' fragments, respectively (first term on the right-hand side), or avoid the breakup (second term).

It is convenient to introduce the generating function

G{x\Z, t) = J2 eZNP(x\N, t) (3) N

and, using Eq. (1), to obtain the equation for G. From the latter one readily obtains equations for (N(x\t)) — dG/dz\z=1,

dt{N) = Lt{{N)}, (4)

where L+ is an adjoint (" backward " ) collision operator [14] acting on the initial value of the fragment size

tf{f(x\...)} = -f(x\...)/r(x)

+ f dyw(x\y)[f(y\...) + f(x-y\...)}, Jo

(5)

and for (N2(x\t)) = d2G/dZ2\z=i,

dt(N2)=L^{{N2)} + Q(x\t), (6)

where

Q(x\t) = 2 f dyw(x\y)(N(y\t))(N(x - y\t)). (7) Jo

Equations (4)-(7) reflect the general property of the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy for a fragmentation process [15]: an equation for fcth mo­ment contains ( 1 , . . . , k)th moments but does not con­tain (k -f l , . . . ) t h ones. Thus, in principle, the higher moments can be found recursively, if the lower ones are known.

It follows from the previous studies [2-4] tha t it is the dependence of w(x\y) on x tha t determines the behavior of the process while the dependence on the size of the daughter fragment y is of less importance. From now on

dt(N)=CH(N)}, (8)

where

r°°^ dn £ t { r } = _ r ( 0 + 2\a\ra / % - Q r ( ? 7 ) , (9)

a = 1/a, and the two sets of integration limits in the integral over 77 correspond to the cases a > 0 and a < 0, respectively. Similarly, Eq. (7) is reduced to

3 € < 7 V 2 ) = £ t { ( J V 2 ) } + Q ( 0 , (10)

where

Q(0 = 2 / ' dr,(N(tr,-))(N[£(l - »,)-]>. (11) ./o

The specifics of the shattering case (a > 0) stem from the fact tha t , formally, the breakup t ime of the zero-size fragments is zero: r (0) = 0. Thus an infinite number of the zero-size fragments is produced after some finite time. This corresponds to the appearance of the singular term of the form 6(y)/y in the fragment-size distribution function [16(b)] (or to the apparent violation of the con­servation law, if this term is not explicitly written down [2,16]). In this case, the well-defined quantity is the total number of the finite-size fragments. Therefore, a regular solution to Eq. (8) is to be understood as the total num­ber of finite-size fragments. Differentiating Eq. (8) with respect to £, we reduce it to the second-order differential equation

£<9f (TV) + (£ - a)dz(N) + a(N) = 0. (12)

For a > 1, the boundary conditions (N(0)) — 1 and (TV) > 0, V£ determine uniquely the solution of Eq. (12),

(JV(0> - * ( a , - a , - 0 - C £ a * ( l + 2a, 2 + a, - £ ) ,

(13)

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V O L U M E 71, N U M B E R 8 P H Y S I C A L R E V I E W L E T T E R S 23 A U G U S T 1993

where 3>(a, c, z) is the confluent hypergeometric function, C = 4 c o t ( 7 r a ) r ( 2 a ) / r ( a ) 3 ( l + a ) , and T(x) is the gamma function [17]. There is no regular solution for 0 < a < 1, because in this case the process speeds up so rapidly tha t even the number of nonzero-size particles becomes infinite [18].

If we substitute (N) given by Eq. (13) into Eq. (10), the latter determines (N2) of the finite-size fragments. To evaluate (TV2) we note tha t at late times Eq. (13) gives (N) oc £ 2 a exp( — £) (the decreasing of (N) with time means tha t the finite-size part of the distribution is depleted at the expense of the zero-size part) . Sub­stituting this form into Eq. (11) and evaluating the integral over rj by the saddle-point method, we see that the source term Q decays much faster than (N): Q oc ^ 4 a " 1 / 2 e x p ( - 2 a + 1 C ) . Thus, the general solution of Eq. (10) is determined mainly by the partial solution of the homogeneous equation, which can be chosen as (AT). Finally, we have (N2) = (N) + O ( e x p ( - 2 a + 1 0 ) and R grows exponentially with time: R « (AT) -1/2 oc £~ a exp(£ /2) ^> 1. Thus, the fluctuations dominate over (AT). The absence of self-averaging is more pronounced in the shattering process than in the Yule-Furry process, where R = 1. The reason is tha t the shattering process speeds up as time increases so each subsequent genera­tion of fragments has a shorter lifetime than the previ­ous generation. Thus, the influences of the early long-lived stages of the process are more important than in the Yule-Furry process, where all generations have the same lifetime. It is useful to represent the time depen­dence of the higher moments of a random variable r(t) in a form (rk) oc (r)9^k) [10(c)]. The case g(k) = k ("gap scaling") corresponds to the Gaussian distribution and, hence, is self-averaging. DLA [9] and SD [10] are known to exhibit multiscaling [g(k) •=£ fc], i.e., the distribution is characterized by a continuum set of exponents. In our case (r = N) it can be shown tha t {Nk) « (N) for late times, and, thus, g(k) ~ 1. This case can be charac­terized as anti- Gaussian: although the time dependence of all moments is described by the same functional form (exponential), the fluctuations dominate strongly.

Now we turn to the case of normal fragmentation (a < 0). In this case r(0) = co and the fragment-size distribution function is regular for all sizes. To find (AT2) we introduce a new function v(£) = (A"2(£))£1_/3, where (3 = — I/a > 0. The equation for the Laplace transform v(s) is obtained from Eq. (10)

-(s + l)dsv = 0(1 + 2/s)v + Pis), (14)

where V is the Laplace transform of the function V = £0Q and Q is given by Eq. (11). The solution to Eq. (14) is

s "»='*»( i + f^S)) ' <i6> where v0(s) = T(/3)(l + s)^s~20. The behavior of v(£)

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as £ —> oo is determined by the behavior of v(s) as s —*• 0. To find the asymptotic form of V as s —> 0, we reduce Eq. (8) again to Eq. (12) (a —> —(3), which gives (N) = <3>(—/3,/3, — £) (in accordance with the result of Ref. [2]). At late times (N) » T{(5)T{2f3)~1i2^. Using this result, we find V(s) « T( l -f /3)5~ ( 1 + 3 / 3 ) . Expanding Eq. (15) about s « 0 we finally obtain (A^2) « (AT)2 ("gap scaling"). Thus, in the normal case the fluctuations are negligible and they decrease as the total number of frag­ments increases; i.e., self-averaging takes place. Again, the reason is in the size dependence of r . The normal process slows down as time increases and there is enough time for the fluctuations tha t occurred at the early stages to wash out.

It is useful to overview the evolution of the contin­uum fragmentation process as a function of the exponent a. For a < 0, we have slow (power-law) growth of (N) (oc £2/lal) and the fragment-size distribution function ex­hibits no singularities. In this case the fluctuations are small (R <C 1). As \a\ decreases, the growth becomes faster and R increases. As a reaches zero (the Yule-Furry process), the transition from the power-law growth of (AT) to an exponential one [(N) oc exp(£/r) [1]] occurs. This is accompanied by the drastic increase in the fluctuations (R = 1), but the fragment-size distribution function still remains regular. When a becomes positive, the distribu­tion function splits into regular (finite-size) and singular (zero-size) parts. {N} is defined only for the finite-size part, while the number of the zero-size fragments is in­finite. The finite-size part is depleted at the expense of the zero-size one and (N) decreases exponentially at late times. At these times, the fluctuations in the regular part are abnormally large (R^> 1), while for those quantities that are defined for the singular part (e.g., the mass or energy accumulated in it), the fluctuations can be shown to be small [19].

As we have said above, the specifics of the shattering case are determined by the fact tha t r (0) = 0. However, this fact is a specific feature of the continuum model in which the fragmentation does not stop even for the zero-size fragments. This is why it is interesting to study the discrete process, which always has a smallest indivisible unit—a "monomer," Unfortunately, analytical calcula­tions do not lead to the tractable results in the discrete model. To carry out numerical calculations, it is con­venient to note tha t the Green's function of Eq. (6) is simply the mean fragment-size distribution function (n) (this fact was also employed in Refs. [12,15]). Thus (A"2) can be found by numerically performing the convolution of the Green's function and the source term Q in Eq. (6). Analytical expressions for (n) in the discrete case have been found for some specific forms of r in Ref. [2] and we have used these results in our calculations. Since the shattering case is of special interest, we shall consider only those functional forms for r tha t would have led to the shattering in the continuum model.

Figure 1 shows the relative fluctuation R (left y axis)

V O L U M E 71, N U M B E R 8 P H Y S I C A L R E V I E W L E T T E R S 23 A U G U S T 1993

for the discussion of the concept of multiscaling. This work was supported by the NSERC of Canada.

&

t FIG. 1. Mean fractional total number of fragments

J\f = (N)/No (right y axis) and relative rms fluctuation 6N/(N) (left y axis) as functions of time for the model r(k) = k(k + 1). Different curves correspond to the differ­ent values of No, as shown in the plot.

and the fractional number of the fragments M = (N)/NQ (right y axis) as a function of time [in units of T(NQ)] for r(k) = fc(fc-f-l). When t « 0 there is, essentially, only one particle in the system and fluctuations are small. As time increases, R increases rapidly and reaches a maximum at t < T. The value of R at the maximum depends on the size of primary particle and for A"o > 10 it is greater than 1. Thus, in the vicinity of the maximum, the fluctuations dominate. As time goes on, R decreases and approaches the value R = 1 at some time t*. R < 1 at times t > t* and self-averaging is gradually recovered. Thus, t* has a meaning of the self-averaging onset time. If A o is large enough, the number of fragments produced by the time £*, (N(t*)), is much greater than 1. For example, for AT0 = 100, (N(t*)) « 40. Had we been dealing with a self-averaging system, we would have expected tha t R would be small already at t < t*, when 1 < (N) «C (N(t*)). In our case, R > 1 at t < t*. This is an intrinsic feature of the non-self-averaging fragmentation process. Figure 1 suggests that , if No is large enough, the quantities t* and J\f* are independent of Ao and, hence, are the universal characteristics of the process [for a given r(k)). This pre­diction can be easily tested in Monte Carlo simulations or in experiments on, e.g., polymer degradation.

In conclusion, we have shown tha t both continuum and discrete shattering fragmentation lack self-aver aging. The absence of self-averaging is an important property to know for real or numerical experiments. It means that averaged quantities can be measured with a reasonable accuracy only in a large number of independent experi­ments, but not in longer experiments on a single system. To the best of our knowledge, this issue has never been raised with respect to various experiments on fragmenta­tion (cf. Ref. [7]), and we believe tha t the results of this paper could be useful in view of the ongoing and future experiments.

The author is indebted to S. E. Esipov for numerous discussions, to E. Levinson for a careful reading of the manuscript and valuable comments, and to M. Plishke

* On leave from Institute of Microelectronics Technology, Chernogolovka, Moscow District, 142432, Russia.

[1] S. K. Srinivasan, Stochastic Theory and Cascade Pro­cesses (American Elsevier, New York, 1969).

[2] (a) E. D. McGrady and R. M. Ziff, Phys. Rev. Lett. 58, 892 (1987); mathematical aspects of this problem are also discussed in (b) A. F. Filippov, Theory Probab. Its Appl. (USSR) 4, 275 (1961).

[3] P. Sigmund, in Sputtering by Particle Bombardment, edited by I. R. Behrisch (Springer, Berlin, 1981).

[4] Y. B. Levinson, in Nonequilibrium Phonons in Non-metallic Crystals, edited by W. Eisenmenger and A. A. Kaplyanskii (North-Holland, Amsterdam, 1986).

[5] S. Redner, in Statistical Models for the Fracture of Dis­ordered Media, edited by H. J. Herrmann and S. Roux (Elsevier, New York, 1990), Chap. 10, and references therein.

[6] M. F. Gyure and B. F. Edwards, Phys. Rev. Lett. 68, 2692 (1992).

[7] For an excellent review of fragmentation, see D. E. Grady and M. E. Kipp, J. Appl. Phys. 58, 1210 (1985).

[8] R. M. Ziff, J. Stat. Phys. 23, 241 (1980). [9] See, for example, C. Amitrano, A. Coniglio, and F. di

Liberto, Phys. Rev. Lett. 57, 1016 (1986); P. Meakin, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. L. Leibowitz (Academic, New York, 1988), Vol. 12; A. Coniglio and M. Zanetti, Europhys. Lett. 10, 575 (1989).

[10] See, for example, (a) A. Sadiq and K. Binder, J. Stat. Phys. 35, 617 (1984); (b) A. Milchev, K. Binder, and D. W. Heermann, Z. Phys. B 63, 521 (1986); (c) C. Roland and M. Grant, Phys. Rev. Lett. 63, 551 (1989); (d) C. Yeung and D. Jasnow, Phys. Rev. B 42, 10 523 (1990).

[11] P. A. Lee, A. D. Stone, and H. Fukuyama, Phys. Rev. B 35, 1039 (1987).

[12] S. E. Esipov, L. P. Gor'kov, and T. J. Newman, J. Phys. A 26, 787 (1993).

[13] The anomalous statistical properties of shattering frag­mentation processes have been noticed, but not studied in detail, in Ref. [2(b)].

[14] J. Lewins, Importance, the Adjoint Function (Pergamon, Oxford, 1965).

[15] M. M. R. Williams, Proc. R. Soc. London, Ser. A 358, 105 (1977); J. Phys. D 13 351 (1980); I. Pazsit, ibid. 20, 151 (1987); D. L. Maslov, Zh. Eksp. Teor. Fiz. 95, 1076 (1989) [Sov. Phys. JETP 68, 619 (1989)].

[16] (a) N. R. Corngold, Phys. Rev. A 39, 2126 (1989); (b) D. L. Maslov, Phys. Rev. A 41 , 7070 (1990).

[17] McGrady and Ziff [2(a)] have calculated (JV(f)) by using a different method. For a > 0 their result corresponds only to the first term on the right-hand side of Eq. (13). However, for £ ^> 1 and 1/2 < a < 1 this term is neg­ative and, hence, cannot represent the total number of fragments. For a < 0 our result coincides with that of Ref. [2(a)].

[18] Analysis shows that for 0 < a < 1 the distribution func­tion of nonzero-size particles [2(a)] has a nonintegrable singularity at x —• 0-K

[19] D. L. Maslov (unpublished). 1271