Ohtsuki Toshiya

Sekiyama Makoto

Yamamoto Hiroshi

Yamamoto, Hiroshi

2885

Sekiyama, Makoto

2886

Ohtsuki, Toshiya

2887

Yokohama City University Repository / 横浜市立大学学術機関リポジトリ

横浜市立大学論叢自然科学系列　2017年度：Vol.66 No.1・2・353Asymptotic Power-law Index in Aggregation and ChippingProcesses with Power-law Aggregation KernelHiroshi Yamamoto∗Center for Natural Sciences, Kitasato University,1-15-1, Kitasato, Sagamihara, 228-8555Makoto SekiyamaGraduate School of Nanobioscience,Yokohama City University, 22-2, Seto,Kanazawa-ku, Yokohama, 236-0027Toshiya OhtsukiDivision of Science, International College of Arts and Science,Yokohama City University, 22-2, Seto,Kanazawa-ku, Yokohama, 236-0027The asymptotic power-law behavior of a distribution function P (X) for X clusters is ana-lyzed for aggregation and chipping processes with power-law kernel of aggregation processes,K(n,m −→ n +m) = n−ψm−ψ/A2. An exact value of non-integer power-law index is ob-tained, that is, P (X) ∼ 1/Xα, α = n2 + 1 − ψ in the case ψ < 0.5. Numerical simulationsagree well with this result.∗Electronic address: hyamamot@kitasato-u.ac.jp横浜市立大学論叢自然科学系列　2017年度：Vol.66 No.1・2・354I. INTRODUCTIONAggregation process can be found in various phenomena in both nature andsocial sciences[1–4]. These are nonequilibrium systems, thus not described byGibbs distribution like thermal equilibrium system. In systems where onlyaggregation mechanism is operative, all the clusters in system eventually co-agulates one condensed cluster. On the other hand, with some fragmentationmechanism the system evolves, in many cases, a power-law distribution[5]. Wehave already investigated the discrete aggregation-chipping processes with theconstant kernels for both the aggregation and the chipping[6–8], the case wherethe chipping kernel has a power-law form[9], and the case with the constantchipping kernel[10]. In these papers, we have reported that the asymptoticpower-law cluster size distribution P (X) ∼ 1/Xσ emerges when the weight ofthe chipping kernel has a power-law form C(n) ∼ 1/nσ and the case 2 > σ > 1.In the case 0 < σ ≤ 1, our analysis suggests that there is no asymptotic power-law solution, and in the case σ = 0 the chipping kernel becomes a constant, theexponential distribution appears. On the other hand, in the case σ > 2, stillthe power-law distribution holds, but also the condensed ”big” cluster appears.These results suggest that even the chipping kernel has strong influences forthe resultant cluster size distribution, and for the actual systems that mostlyhave the complex kernels of the aggregation and the chipping processes, furtheranalysis is required to understand the resultant distribution.In this paper, we consider the case where the aggregation kernel has a power-law form, and analyze the asymptotic power-law behavior of the cluster sizeAsymptotic power-law index in aggregation and chipping processes with power-law aggregation kernel55distribution.II. THEORYNow we introduce a weighted-aggregation and chipping process, described asfollows: initially N clusters are set where each cluster has a certain number ofunits, according to an initial distribution. The unit process consists of two parts,weighted-aggregation and chipping. First, two clusters are chosen accordingto a certain aggregation kernel K(X,Y −→ X + Y ) (X and Y clusters) areintegrated and make one big cluster (X + Y cluster). Secondly one unit ischipped oﬀ from a randomly chosen cluster which has more than 2 units (Z(≥ 2)cluster) and become a cluster. In an one unit process, the total number ofthe clusters and the units are conserved (X, Y , Z clusters are changed into(X+Y ), (Z−n) and n clusters). The whole process proceeds by repeating theunit process.In the present paper we consider the case where the aggregation kernelK(n,m −→ n+m) has a power-law formK(n,m −→ n+m) = n−ψm−ψ/A2, (1)where 1/A = (Σ∞i=1i−ψ)−1 is the normalization constant. In the limit ψ −→ 0,i.e. the aggregation kernel K(n,m −→ n + m) = 1/A2 thus the two clustersfor the aggregation are chosen freely, a resultant distribution is the power-lawdistribution P (X) ∼ 1/X2.5[7].We analyze here the case ψ is a non-integer and ψ < 0.5. Note that if ψ ≤ 1A would be inﬁnite, thus we need some high-cut for the system size. At a steady横浜市立大学論叢自然科学系列　2017年度：Vol.66 No.1・2・356state, the basic equations of this system are(1− P (1))(1− P (1))A2+P (2)1− P (1) =P (1)2A2(X = 1), (2)∑i+j=Xi−ψj−ψA2P (i)P (j) +P (X + 1)1− P (1)=(2X−ψA+11− P (1))P (X) (X ≥ 2). (3)where αn is a normalization constant,αn =(n−1∑s=11sσ)−1. (4)We now assume that the system has an asymptotic power-law solution,P (X) ∼ 1/Xα. In this case, the first term of the left hand side of (3) can bewritten as∑i+j=Xi−ψj−ψA2P (i)P (j)∼∑i+j=Xi−ψj−ψA21iα1jα=∑i+j=X1A21iα+ψ1jα+ψ. (5)Also, the first term of the right hand side of (3) is,2X−ϕAP (X) ∼ 1A1Xα+ψ. (6)Therefore, using the z-transform ϕ(z) =∑∞X=0 P (X)z−X , the basic equationAsymptotic power-law index in aggregation and chipping processes with power-law aggregation kernel57in z-space has the form near z ∼ 1C1{ϕψr (z) +B(1− z)−(1−α−ψ)}2+ f(z){ϕr(z) +B(1− z)−(1−α)}= C2{ϕψr (z) +B(1− z)−(1−α−ψ)}+11− P (1){ϕr(z) +B(1− z)−(1−α)}+ g(z), (7)where ϕψr (z) is a regular term of the z-transform for P (X) ∼ 1/Xα+ψ, ϕψr (z) isa regular term of the z-transform for P (X) ∼ 1/Xα, B, C1 and C2 are certainconstants, and f(z) and g(z) are certain regular functions of z. Here we usethe singularity of ϕ(z) =∑∞X=0 P (X)z−X near z → 1 is ϕs(z) ∼ (1 − 1/z)β−1for P (X) ∼ 1/Xβ [11],If the equation (7) holds, then the singular terms in (7) must satisfy(1) (1− z)−{2(1−α−ψ)} = (1− z)−(1−α),or(2) (1 − z)−{2(1−α−ψ)} = (1 − z)n where n is an integer, thus this termbecomes a regular term.The condition (1) leads to an equation α = 1−2ψ. When ψ > 1, α < 1 thusthe condition (1) is irrelevant. Therefore, we have the asymptotic power-lawsolution,P (X) ∼ 1/Xα, α = n2+ 1− ψ. (8)When ψ < 0.5 and n = 2, n2 + 1− ψ > 2.0, thus it is clear that the condensedcluster will appear in those cases.横浜市立大学論叢自然科学系列　2017年度：Vol.66 No.1・2・358III. NUMERICAL RESULTSFigures 1 and 2 show the cumulative distributions Cm(X) = Σx≤XP (x) ofthe simulation results with ψ = 0.02 and ψ = 0.03 respectively. The totalnumber of the clusters N is 1000 and the mean value of the cluster size ⟨X⟩ =1000.100 102 104 10610-310-210-1100 Cumulative distribution of P(X)X (-1.48)Cm(X)XFIG. 1: Cumulative distribution of the simulation result for weighted-aggregation andchipping process with the number of the clusters N = 1000, ψ = 0.02 and ⟨X⟩ = 1000.Here the resultant distributions are the power-law ∼ 1/Xα, and the power-law index α is practically the same as 52 − ψ. This is reasonable because whenψ −→ 0, α would be 5/2. Therefore we conclude that there is an asymptoticsolution P (X) ∼ 1/X 52−ψ in the case ψ < 0.5. Also we can see the condensedclusters appear at the right side of those pictures. Apparently, It is required todo simulations with ψ > 0.5, however due to the problem of calculation time, itAsymptotic power-law index in aggregation and chipping processes with power-law aggregation kernel59100 102 104 10610-310-210-1100 Cumulative distribution of P(X)X (-1.47)Cm(X)XFIG. 2: Cumulative distribution of the simulation result for weighted-aggregation andchipping process with the number of the clusters N = 1000, ψ = 0.03 and ⟨X⟩ = 1000.is diﬃcult to make such simulations with the simulation circumstance we havenow.IV. DISCUSSION AND SUMMARYThese results show that the power-law aggregation kernel inﬂuences directlyto the power-law index of the resultant distribution. It is, in some point ofview, obvious, however it also casts another questions.When ψ becomes bigger than 0.5, there is no reason to break this asymptoticsolution until ψ ≤ 1.0. Thus we believe the solution P (X) ∼ 1/X 52−ψ holds0.5 < ψ ≤ 1.0. This means that the condensed cluster would disappear when0.5 < ψ ≤ 1.0, because the power-law index of the resultant distribution α横浜市立大学論叢自然科学系列　2017年度：Vol.66 No.1・2・360would be α < 2.0.What, then, will happen if ψ is larger than 1.0? The simulation resultsshow that the power-law index α of the distribution is α = 52−ψ when ψ < 0.5.However, our analysis shows only that −2(1−α−ψ) must be integer, so it is alsopossible, from the theory, that α = 32 − ψ, α = 7/2 − ψ or another solutions.If, when ψ > 1.0, will the power-law index of the distribution α jump from52 − ψ to 72 − ψ for example, or still hold the solution 52 − ψ? When ψ > 2.5,will the power-law distribution still hold, or break into other distribution likean exponential one? We cannot answer those questions now. However theseresults at least show that the power-law index ψ of the aggregation kernel couldbe another type of the control parameter for the resultant distribution P (X)than the chipping kernel we have reported before, and also could aﬀect theemergence of the power-law distribution or the condensed cluster. It expandsthe possibility of ﬁnding the corresponding phenomena in the real world.In summary, we introduce the weighted-aggregation and chipping model, andanalyze the case that the aggregation kernelK(n,m −→ n+m) = n−ψm−ψ/A2.When ψ < 0.5, our analysis shows there is an asymptotic solution of the prob-ability distribution function P (X) ∼ 1/X 52−ψ. The simulation results agreewith this analysis, but we need to reduce the simulation time for getting moreinformation when ψ is larger than 0.5. These results also show the strong pos-sibility of the singularity analysis, and we believe that this analysis will showAsymptotic power-law index in aggregation and chipping processes with power-law aggregation kernel61more profound features of the aggregation systems.[1] F. Leyvraz, Phys. Rep., 383, 95 (2003).[2] J. Schmelzer, G. Ro¨pke, and R. Mahnke, Aggregation Phenomena in ComplexSystems, Wiley-Vch (1999).[3] H. Takayasu, Phys. Rev. Lett., 63, 2563-2565 (1989).[4] A. K. Giri, Mathematical and numerical analysis for coagulation-fragmentationequations, VDM Verlag Dr. Mu¨ller, Saarbrucken (2011).[5] M. E. J. Newman, Contemporary Physics, 46, 323-351 (2005).[6] H. Yamamoto, T. Ohtsuki, A. Fujihara, S. Tanimoto, K. Yamamoto, and S. Miyaz-ima, Japan J. Indust. Appl. Math., 24, 211-218 (2007).[7] H. Yamamoto, T. Ohtsuki, and A. Fujihara, Phys. Rev. E 77, 061122 (2008).[8] H. Yamamoto, and T. Ohtsuki, Phys. Rev. E 81, 061116 (2010).[9] H. Yamamoto, M. Sekiyama, and T. Ohtsuki, J. Phys. Soc. Jpn., 84, 050443-1 -050443-5 (2015).[10] M. Sekiyama, T. Ohtsuki, and H. Yamamoto, J. Phys. Soc. Jpn., 86, 104003-1 -104003-4 (2017).[11] P. Flajolet and R. Sedgewick, Analytic Combinatorics. Cambridge UniversityPress (2009).

Asymptotic Power-law Index in Aggregation and Chipping Processes with Power-law Aggregation Kernel

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Asymptotic Power-law Index in Aggregation and Chipping Processes with Power-law Aggregation Kernel

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