Fujihara Akihiro

Nakagawa Teruhiro

Ohtsuki Toshiya

Fujihara, Akihiro

Ohtsuki, Toshiya

Nakagawa, Teruhiro

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・327Two-component sandpile model : self-organized criticalityof the second kindAkihiro Fujihara1,∗ Toshiya Ohtsuki2, and Teruhiro Nakagawa21 Department of Information and Communication Systems Engineering,Faculty of Engineering, Chiba Institute of Technology,2-17-1 Tsudanuma, Narashino, Chiba 275-0016, Japan2 Field of Natural Sciences, International Graduate College of Arts and Sciences,Yokohama City University, 22-2 Seto,Kanazawa-ku, Yokohama 236-0027, JapanTwo-component sandpile model is investigated numerically and theoretically. Monte Calrosimulations have been performed to show that both avalanche size and lifetime obey thepower-law distribution whose exponents are approximately equal to 1.5 and 2.0, respec-tively. This result indicates that the system exhibits SOC. A mean-ﬁeld theory has alsobeen considered to understand the origin of SOC in the model. We ﬁnd that the systemapproaches a steady critical state and its universality class is diﬀerent from that of exist-ing one-component sandpile models. This reason comes from the conservation law of thenumbers of two diﬀerent sands in the local toppling rule which enables to coexist an in-ﬁnite number of stable states which plays a role of dissipation. Instead of double controlparameters in one-component sandpile models, a rate constant of dissipation is removed intwo-component models. Consequently, we ﬁnd a novel SOC model and its diﬀerent class inthe sense that the more conserved quantities result in the less control parameters.∗Electronic address: akihiro.fujihara@p.chibakoudai.jp横浜市立大学論叢自然科学系列　2017年度：Vol.66 No.1・2・328I. INTRODUCTIONAccording to a vast amount of studies on fractals, 1/f noise, and so on,a lot of non-equiliblium systems in nature reportedly exhibit a critical statewhere the power-law distribution is observed. However, it has not been deeplyunderstood why a huge variety of distributions spontaneously comes to thecritical state. Self-organized criticality (SOC) proposed by Bak et al. [1–4]has provided a reasonable understanding for the emergence of criticality usingvarious sandpile models. These models are cellular automata to move sandswith threshold dynamics governed by local toppling rule. The domino eﬀectof toppling sands to release local stresses in the system induces a sequence oftopplings called an avalanche, whose size and lifetime distributions obey thepower-law distribution without tuning any parameter. A number of works onSOC have been done numerically. Several analytical results have also beenobtained. Dhar [5, 6] has found an exact solution in the abelian sandpile modeland has shed light on the static behavior, such as the number of total recurrentstates and hight correlation functions. However, its dynamics is still obscure,that is, the power-law distributions of avalanches still have not been derivedanalytically.Though the sandpile models have no control parameters by appearance,Vespegnani and Zapperi [7, 8] have found that they do have hidden controlparameters: a slow dissipation rate and a slower addition rate of sands. Toexplain this, they proposed a mean-ﬁeld theory to consider the state equations,Two-component sandpile model : self-organized criticality of the second kind29i.e.,∂∂tρκ = fκ(ρa, ρc, ρs), (κ = a, c, s), (1)where ρa, ρc, and ρs are the state densities of active, critical, and stable states,respectively. Imposing the conservation law of the number of sands given bythe local toppling rule, it is concluded that the models become critical in thedouble limit, i.e.,ϵ, δ → 0, ρa = δ/ϵ→ 0, (2)where ϵ is a dissipation rate and δ an addition rate. In other words, ρa isan order parameter and ϵ, δ are control parameters and SOC is achieved byrough tuning of the unapparent parameters ϵ, δ around zero. It is found thatmost of SOC models [9, 10] belong to the same universality class. We call thisclass SOC of the first kind. It seems interesting whether other kind of SOCexists, or equivalently, whether the condition of the double limit (2) are strictlythe necessary condition of SOC ? In this letter, we give an answer to thesequestions.Tsuchiya and Katori [11] have proved rigorously that SOC breaks down inthe abelian sandpile model when the toppling rule of sands is non-conservative.Here, we pay attention to the number of conservation laws. Does anythingnew happen if the number of conservation laws is increased? How about therobustness of SOC? To answer these questions, we consider two-componentsandpile models which deal with two kinds of sands. This means the modelshave two kinds of conservation laws. Firstly, numerical simulations are carried横浜市立大学論叢自然科学系列　2017年度：Vol.66 No.1・2・330out to check power-law behavior. Secondly, a mean-field theory is investigatedto examine the essence of the model.II. MODEL DESCRIPTIONTwo-component sandpile model is a cellular automaton defined on a regularlattice. A pair of two non-negative integers, h(x) ≡ (i, j), is assigned to eachsite, x, on the lattice, where i, j(≥ 0) denote the numbers of sand A and B. Ateach time step, one unit of sand A or B is added to a randomly chosen site ata relative ratio (0 <)rAB(< 1) of sand A to sand B. We can consider severaltypes of toppling rule for the time evolution of the system. For example, thetoppling occurs when(a) i ≥ ith or j ≥ jth,(b) i ≥ ith and j ≥ jth, or(c) i+ j ≥ kth (i, j > 0) where ith, jth,where kth is a certain threshold value. If the number of sands at a site reachesthreshold, sand A and B on the site topple one by one to randomly chosennearest neighbor sites until the number of sands at the site becomes less thanthe threshold. In these rules, sand A and B topple jointly. For example, if a sitehas n grains of sands A and no sands B, and thenm (≤ n) new grains of sands Bare added, it ends up with n−m grains of sands A and no sands B. These rulessatisfy two kinds of conservation laws of local toppling. Avalanche continuesunless the numbers of sands in all the sites becomes less than the threshold.Two-component sandpile model : self-organized criticality of the second kind31FIG. 1: Stable and unstable states in the rule (b).Among three rules of toppling, the AND rule (b) is the most interesting becauseof the presence of infinite stable states as shown in Fig. 1.Stable states hs and unstable states hu are defined byhs = {(i, j) | 0 ≤ i < ith or 0 ≤ j < jth}, (3)hu = {(i, j) | i ≥ ith and j ≥ jth}. (4)The stable states possibly absorb all the sands added into the system. Thismeans that all avalanches inevitably stop without introducing artificial bound-ary dissipation, which is indispensable in all one-component sandpile models.Rules (a) and (c) cannot remove the boundary dissipation because the numberof stable states is finite under these rules. Hereafter, we consider the rule (b)with periodic boundary conditions and the threshold is fixed to ith = jth = 1.横浜市立大学論叢自然科学系列　2017年度：Vol.66 No.1・2・332III. NUMERICAL RESULTSFirst, we numerically investigate avalanche size S and its lifetime T . Here, Sis deﬁned by the total number of topplings in an avalanch, and T is done by thetotal time steps in an avalanche. The cumulative distribution function (CDF)of avalanche size, D(S), and lifetime, D(T ), in the one-dimensional lattice arecalculated by numerical simulations and they are shown in Figs. 2. We canobserved that both CDFs obey the power-laws distirubtion,D(S) ∼ S−τ , D(T ) ∼ T−α, (5)where the power-law exponents are approximately τ ≃ 0.5 and α ≃ 1.0. Equiv-alently, the probability distribution function (PDF) is drived byP (S) ∼ S−(1+τ), P (T ) ∼ S−(1+α). (6)The exponents in PDF becomes 1 + τ ≃ 1.5 and 1 + α ≃ 2.0. They arefound to be very close to the critical exponents of branching processes [12].We carried out numerical simulations in 2− and 3−dimension and obtainedalmost the same distributions in CDF and PDF. These results imply the modelhas meanﬁeld-like characteristics [13]. We also ﬁnd that varying rAB from0.5 to 0.1 does not aﬀect the power-law tails. This means that the modelends up with spontaneously going to a SOC state robustly in the long timelimit without tuning rAB. It becomes evident that the two-component sandpilemodel exhibits SOC. It should be noted that because of the absence of boundarydissipation, the power-law tails become extended inﬁnitely as the number ofadded sands N increases even in a ﬁnite system.Two-component sandpile model : self-organized criticality of the second kind33log10D(S)log10S(a) N = 105 N = 106 N = 107 N = 108 Slope : -0.50 2 4 6 8 10-8-6-4-20log10D(T)log10T(b) N = 105 N = 106 N = 107 N = 108 Slope : -1.00 2 4-8-6-4-20FIG. 2: Double logarithmic plots of CDFs of avalanche size (a) and lifetime (b). Thelattice size L = 104 and the number of added sands N = 105, 106, 107, and 108,respectively. Sands A and B are added at equal rate rAB = 0.5.横浜市立大学論叢自然科学系列　2017年度：Vol.66 No.1・2・334IV. MEAN-FIELD THEORY AND ITS RESULTSNext, we construct a mean-ﬁeld theory of the two-component sandpile modeland compare to that of the one-component models [7, 8]. We introduce sets ofprobability densities in stable states, X, and unstable states, Z, asX = {X(i,j) | 0 ≤ i < 1 or 0 ≤ j < 1}, (7)Z = {Z(i,j) | i ≥ 1 and j ≥ 1}. (8)Rate equations for X and Z give an inﬁnite system of ﬁrst-order nonlinearordinary diﬀerential equations. However, it is diﬃcult to handle the inﬁnitedegrees of freedom. To avoid this diﬃculty, we deﬁne the following reduced sixvariables.X∗ = {X0, XA, XB}, Z∗ = {Z0, ZA, ZB} (9)withX0 ≡ X(0,0), Z0 ≡∞∑i=1Z(i,i),XA ≡∞∑i=1X(i,0), ZA ≡∞∑i=2∞∑j=1Z(i,j),XB ≡∞∑j=1X(0,j), ZB ≡∞∑i=1∞∑j=2Z(i,j).Two-component sandpile model : self-organized criticality of the second kind35For obtaining a closed set of equations for X∗ and Z∗, we assume the Poissondistribution for X(i,0) and X(0,j) asX(i,0)XA=µi−1A(i− 1)!e−µA (i ≥ 1), (10)X(0,j)XB=µj−1B(j − 1)!e−µB (j ≥ 1) (11)where µA ≡ δAt, µB ≡ δBt are the mean values of each Poisson distributionat time t. We also define δA and δB as the rate constants of additions of eachsand. Therefore, the rate equations for the reduced variables are described bydX0dt= −2X0(Z0 + ZA + ZB) + Z0−(δA + δB)X0, (12)dXAdt= (X0 −XA)(Z0 + ZA + ZB) + ZA+δAX0 − δBXA, (13)dXBdt= (X0 −XB)(Z0 + ZA + ZB) + ZB+δBX0 − δAXB, (14)dZ0dt= (αAXA + αBXB − 2Z0)(Z0 + ZA + ZB)− Z0+αBδAXB + αAδBXA, (15)dZAdt= {(1− αA)XA + Z0}(Z0 + ZA + ZB)− ZA+(1− αA)δBXA, (16)dZBdt= {(1− αB)XB + Z0}(Z0 + ZA + ZB)− ZB+(1− αB)δAXB, (17)where αA(t) ≡ X(1,0)/XA = exp(−δAt) and αB(t) ≡ X(0,1)/XB = exp(−δBt).横浜市立大学論叢自然科学系列　2017年度：Vol.66 No.1・2・336 X0 XA and XB Z0 ZA and ZBTimeProbability densities0 20000 40000 60000 80000 10000000.20.40.60.81FIG. 3: Time evolution of the reduced six variables at δA = δB = 0.0001. Initial valuesare set to X0 = 1.0 and XA = XB = Z0 = ZA = ZB = 0.0. At the steady state,X0 = Z0 ≃ 0, XA = XB ≃ 0.5, and ZA = ZB ≃√δA/2 = 0.005.A numerical solution for the equation system (12)-(17) are plotted in Fig. 3.We find that in the long time limit, the system goes to a steady state whichis supposed to be SOC. The steady state can be examined analytically. Whent → ∞, then αA, αB → 0 and X0, Z0 → 0. Therefore, the following relationsare derived from Eqs. (12)-(17).XA =ZAZA + ZB + δB, XB =ZBZA + ZB + δA. (18)Two-component sandpile model : self-organized criticality of the second kind37Suppose that ZA + ZB ≫ δA, δB and ZA : ZB = δA : δB, we obtainZA =√2δ12Bδ32A(δA + δB)32, ZB =√2δ12Aδ32B(δA + δB)32. (19)Equations (19) show that the model goes to a critical state (ZA = ZB = 0)in the limit δA, δB ∼ δ → 0. Out of two conditions (2) for SOC of the ﬁrstkind, our model can successfully remove one of them with respect to the con-trol parameter of dissipation ϵ. Furthermore, the order parameters behave asZA, ZB ∼ δ1/2 when the control parameters go to zero δ → 0. These two char-acteristic features of the two-component model, that is, the existence of thesteady state without dissipation and the asymptotic behavior to the criticalstate in δ → 0, distinctively diﬀer from conventional SOC models. This indi-cates a strong evidence that the system belongs to a diﬀerent universality class.Consequently, we are successfully able to ﬁnd the new mechanism and class ofSOC, which we call SOC of the second kind.V. DISCUSSIONWe have investigated two-component sandpile models with the AND rule (b)with periodic boundary conditions mainly on 1−dimensional lattices. Extentionof the sandpile model to two-component one is essential to generate an inﬁnitenumber of stable states which substitute for boundary dissipation. Therefore,the model becomes more natural because the dissipation is usually introducedartiﬁcially only for stopping avahanches. Simulation results indicate the systemexhibits SOC.横浜市立大学論叢自然科学系列　2017年度：Vol.66 No.1・2・338The mean-ﬁeld theory conﬁrms that the model goes to the steady and criticalstate belonging to the diﬀerent universality class from existing one-componentmodels. Here, only rough-tuning of rate constants of additions δ is enough toinduce SOC. We can successfully construct the new SOC where the system doesnot have any artiﬁcial dissipation mechanism. Consequently, the two conditions(2) are not the necessary condition for SOC, which is the answer to the questionmentioned previously.It is interesting to view SOC models from the standpoint of the numberof conserved quantities. Whe the model has no conserved quantities, such ascontact processes, show the power-law distribution strictly at the critical point.Therefore, ﬁne-tuning of control parameters is required. When the model hasone conserved quantity, such as one-component sandpile models, they becomeSOC in rough-tuning of two control parameters, an addition and a dissipationrates of sands, around zero. The two-component sandpile model with twoconserved quantities show SOC in rough-tuning of only one control parameter,an addition rate of sands, around zero. It could be concluded that the moreconservation laws result in the less control parameters.Sandpile models are considered to have some relevance to power laws ofearthequake magnitude distributions, called Guthenberg-Richter law [14]. One-component models have to introduce some artiﬁcial boundary dissipations tostop the avalanche. However, the surface of the earth has no apparent boundary.At this point, it is more natural to apply two-component sandpile models toearthequake magnitude distributions. Furthermore, an earthquake could betriggered by multiple physical quantities, such as elastic energy and stress ofTwo-component sandpile model : self-organized criticality of the second kind39earth’s crust. If an earthquake takes place only when both the energy and stressreach the threshold simultaneously, the distribution of earthquakes would bewell described by the SOC of the second kind. In this paper, we express theprocess in terms of a sandpile. Obviously, the sand is merely a symbol and itcould be any kind of substances which trigger the same dynamics. In additionto earthquakes, therefore, a wide variety of SOC phenomena is triggered bymultiple factors and it would belong to SOC of the second kind. This pointshould be made clear for future work.[1] P.Bak, C.Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987).[2] P.Bak, C.Tang, and K. Wiesenfeld, Phys. Rev. A 38, 346 (1988).[3] H. J. Jensen, Self-Organized Criticality, (Cambridge University Press, Cambridge,1998).[4] D. Sornette, Critical Phenomena in Natural Sciences: Chaos, Fractals, Selforga-nization and Disorder: Concepts and Tools, (Springer, Berlin, 2000).[5] D. Dhar, Phys. Rev. Lett. 64, 1613 (1990).[6] D. Dhar, Physica A 263, 4 (1999).[7] A. Vespignani and S. Zapperi, Phys. Rev. Lett. 78, 4793 (1997).[8] A. Vespignani and S. Zapperi, Phys. Rev. E 57, 6345 (1998).[9] S. S. Manna, J. Phys. A 24, L363 (1991).[10] B. Drossel and F. Schwabl, Phys. Rev. Lett. 69, 1629 (1992).[11] T. Tsuchiya and M. Katori, Phys. Rev. E 61, 1183 (2000).[12] T. E. Harris, The Theory of Branching Processes, (Dover , New York, 1989).[13] P. Alstrom, Phys. Rev. A 38, 4905 (1988).[14] D. L. Turcotte, Rep. Prog. Phys. 62, 1377 (1999).

Two-component sandpile model : self-organized criticality of the second kind

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Two-component sandpile model : self-organized criticality of the second kind

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