PHYSICAL REVIEW C 72, 057602 (2005)

Zipf’s law in multifragmentation

X. Campi∗ and H. KrivineLaboratoire de Physique Theorique et Modeles Statistiques† Bat. 100, Universite de Paris XI, F-91405 Orsay Cedex, France

(Received 22 August 2005; published 30 November 2005)

We discuss the meaning of Zipf’s law in nuclear multifragmentation. We remark that Zipf’s law is a consequenceof a power-law fragment size distribution with exponent τ � 2. We also recall why the presence of such adistribution is not a reliable signal of a liquid-gas phase transition.

DOI: 10.1103/PhysRevC.72.057602 PACS number(s): 25.70.Pq, 05.70.Jk, 64.60.Ak

The search for reliable signatures of the liquid-gas phasetransition in nuclear multifragmentation is, both theoreticallyand experimentally, one of the major issues of this field ofphysics. The empirical observation that the size distributionof heavier clusters generated in various processes satisfies theso-called Zipf’s law [1], has raised interest and curiosity. Thiswas first pointed out by Ma [2] in the framework of the isospin-dependent lattice-gas model and was more recently seen innuclear fragmentation data [3,4].

In the present context, Zipf’s law1 states that the mean size(mass or charge) s(r) of the largest, second-largest · · · r-largestclusters decreases according to their rank, r = 1, 2, · · · , n, as

s(r) ∼ 1/rλ, (1)

with λ � 1.The examination of the above-mentioned numerical simu-

lations [2] and experimental data [3,4] shows that a fairly goodagreement with approximation (1) is indeed obtained when theexponent is λ � 1. This happens when other observables reachextreme values [maximum value of the moments of the clustersize distribution (csd), minimum of the effective τ parameterfit of the csd, maximum fluctuation of the largest fragmentetc.]. This seems to be the origin of the suggestion [2] thatthe fulfillment of Zipf’s law is a good signal of the liquid-gasphase transition.

Power laws appear widely in many domains, ranging fromnatural sciences to economics and sociology [5–7]. Theirorigin is often controversial. Such distributions reveal the lackof a typical scale. In physics one finds them, for example,in the vicinity of critical points where correlation lengthsdiverge. The value of the exponent λ is, in principle, afingerprint of the underlying phenomenon.

The aim of this report is to point out that the finding of Zipf’slaw is nothing but a consequence of the power-law shape ofthe csd with exponent τ � 2. More precisely, both exponentsare connected through the formula λ = 1/(τ − 1).

The proof of this statement is straightforward. Let s be thesize (mass or charge) of the clusters and s(r) be the size of the

∗Electronic address: campi@ipno.in2p3.fr†Unite de Recherche de l’Universite de Paris XI associee au CNRS

(UMR 8626).1In its original formulation, Zipf’s law concerns the rank of the

frequency of words in a text.

cluster of rank r. If the csd is a power law

Pr[s ∈ (s, s + ds)] ∼ ds/sτ , (2)

by integrating over s, one obtains the probability of finding acluster of a size larger than S:

Pr[s > S] ∼ 1/S(τ−1). (3)

We now take S = s(r), where s(r) is the average size ofclusters of rank r. This is a strictly decreasing real function ofr, hence without degeneracy. Using randomly choosen clusters,the event E “the cluster has a size larger than s(r),” is identicalto the event “the cluster has a rank less than r.” Arranging inascending order the ranks from 1 to n, the probability of E is

Pr(E) = Pr[s > s(r)] = r − 1

n − 1∼ r. (4)

On the other hand, probability (3) now gives

Pr(E) ∼ 1/s(r)τ−1.

Therefore

r ∼ 1/s(r)τ−1,

and, from approximation (1),

λ = 1/(τ − 1). (5)

Similar proofs can be found in the literature [see forexample Ref. [8], in which the same arguments are used toprove that if the ranking follows approximation (1), then thecsd is necessarily a power law].

The above formulas are strictly valid for infinite samplings.We checked numerically that these remain accurate for finitesamplings. We proceeded as follows. We generate partitionsof an integer number N, with the condition that the mean csdis a power law of given exponent τ (i.e., each part of N istaken as a cluster size s). From these random numbers s, weconstruct the function s(r). As expected, for large N (N >

1000), the function s(r) is very close to a perfect power law ina large domain of r and Eq. (5) is satisfied within numericaluncertainties in the fits of the power laws. For N � 100 and2 �τ � 3, Eq. (5) is fulfilled within a few percent. In general,s(1) lies above the best-fit curve. This is due to the finite size.Indeed, in the domain of s contributing to s(1), the csd deviatesfrom a power law. For the same reasons, for larger values ofτ the agreement is less good. Figure 1 shows an example of

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1 10 100r10

−3

100

s(r)

s10

−3

100

n(s)

FIG. 1. The csd n(s) (upper panel) and mean size distributions(r) of clusters of rank r (lower panel), corresponding to a power-lawcsd with τ = 2.2. The size s of the clusters is generated from randompartitions of N = 100 with the constraint that the mean csd is a powerlaw with exponent τ = 2.2 (see text). The straight lines are best fits,with slopes τ = 2.2 and λ = 0.86, respectively.

s(r) calculated from a power law csd with τ = 2.2. A best fitwith approximation (1) gives λ � 0.86.

It is interesting to see what happens if the csd is nota power law. For example, for an exponential distribution∼ exp(−αs), following the same reasoning, one finds s(r) ∼− log r . Various theories of cluster formation [9,10] offer otherinteresting examples. Close to their critical point, the csdbehaves as

n(s) ∼ s−τ f (sσ ε), (6)

where f is a scaling function, ε is the distance to the criticalpoint and τ and σ are two critical exponents. Right atthe critical point, the scaling function f (0) = 1. In threedimensions τ � 2.21 for the lattice gas model with clustersdefined according to the Coniglio-Klein prescription [9,11]and τ � 2.18 for the percolation theory [10]. As expected,(approximate) Zipf’s laws have been observed [2,12] in thevicinity of the corresponding critical points.

We present below some results for a bond-percolationcalculation [10] with N = 100 occupied sites. In Fig. 2 weshow, as a function of the distance to the critical bondparameter ε = pc − p, the evolution of λ (upper panel) and theχ deviation of s(r) with respect to two power-law fits (lowerpanel). The continuous curve (squares) is the best fit of s(r)

−0.2 −0.1 0.0 0.1 0.2

ε10

−3

10−2

10−1

100

χ2

0.0

0.5

1.0

1.5

2.0

λ

FIG. 2. The slope parameter λ [approximation (1)](upper panel)and the χ 2 deviation from a power law of s(r) (lower panel) asfunctions of the distance ε to the percolation critical point (see text).The continuous curve (squares) corresponds to the best fit with λ freeand the dashed-dotted curve (crosses) to λ = 1 [2].

with two parameters c/rλ. The minimum of χ occurs whenλ � 0.90 for a slope τ � 2.16 of the corresponding csd, inagreement with Eq. (5) within 4%. The dashed-dotted curve(crosses) corresponds to the fit with fixed λ = 1 (i.e., a strictZipf’s law, as done in Ref. [2]). This fit, which violates Eq. (5)by 26%, giving an incorrect localization of the critical point,shows as expected a larger χ2. Similarly, for the lattice-gasmodel with Coniglio-Klein clusters one expects, according toapproximation (1), λ � 0.83 [2].

Before summarizing, we would like to add a comment onthe observation of power-law csd and Zipf’s laws. As men-tioned before, various theories of cluster formation (Fischerdroplets [13], lattice gas [9], Lennard-Jones’s fluids with Hill’sclusters [14], percolation [11]), predict power-law csd’s withτ � 2 at the corresponding critical points. However, such csd’salso appear elsewhere (for example, in the supercritical regionof the lattice gas and realistic Lennard-Jones fluids).

In summary, (approximate) Zipf’s law is just a mathematicalconsequence of a power-law csd with exponent τ � 2. Suchdistributions appear at the critical point of many theories,but also elsewhere. In consequence, we conclude that theobservation of a Zipf’s law is neither a new and independentsignal of a critical behavior, nor an unambiguous signal of athermodynamical phase transition.

We thank Marc Mezard for a helpful discussion.

[1] G. K. Zipf, Human Behavior and the Principle of Least Effort(Addisson-Wesley, Cambridge, MA, 1949).

[2] Y. G. Ma, Phys. Rev. Lett. 83, 3617 (1999); Eur. Phys. J. A 6,367 (1999).

[3] Y. G. Ma et al. (NIMROD Collaboration), Phys. Rev. C 71,054606 (2005).

[4] A. Dabrowska et al., Acta Phys. Pol. B 35, 2109(2004).

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[5] D. L. Turcotte, in Fractals and Chaos in Geologyand Geophysics (Cambridge University Press, New York,1993).

[6] D. Sornette, in Critical Phenomena in Natural Sciences, 2nd ed.(Springer, Heidelberg, 2003), Chap. 14.

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[11] A. Coniglio and W. Klein, J. Phys. A Math. Gen. 13, 2775(1980).

[12] M. S. Watanabe, Phys. Rev. E 53, 4187 (1996).[13] M. E. Fisher, Physics (N.Y.) 3, 255 (1967).[14] X. Campi, H. Krivine, and N. Sator, Physica A 296, 24 (2001);

Nucl. Phys. A681, 458c (2001); X. Campi, H. Krivine, andJ. Krivine, Physica A 320, 41 (2003).

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