[[abstract]]We present an exact solution of a one-dimensional sandpile model for which sand is dropped along the wall and N=2 grains of sand fall over the neighboring downhill sites when the critical slope is exceeded. The slopes of N consecutive sites organize into a local state. The time evolution of the local states along the spatial direction shows a natural tree structure. As a result, various multifractals can be identified. The spatial two-point correlation function decreases exponentially with a correlation length of the order of the lattice spacing.[[incitationindex]]SCI[[booktype]]紙本[[booktype]]電子

Lee, Shih-chang

We present an exact solution of a one-dimensional sandpile model for which sand is dropped along the wall and N=2 grains of sand fall over the neighboring downhill sites when the critical slope is exceeded. The slopes of N consecutive sites organize into a local state. The time evolution of the local states along the spatial direction shows a natural tree structure. As a result, various multifractals can be identified. The spatial two-point correlation function decreases exponentially with a correlation length of the order of the lattice spacing.SCI紙本電子

Tamkang University Institutional Repository

::::YS::CA::4:"V: ."WVOLUME 67 16 SEPTEMBER 1991 NUMBER 12Exact Solution of a Deterministic Sandpile Model in One DimensionS.-C. Lee, N. Y. Liang, and W.-J. TzengInstitute of Physics, Academia Sinica Taipei, Taiwan I I529, Republic of China(Received 11 April 1991)We present an exact solution of a one-dimensional sandpile model for which sand is dropped along thewall and %=2 grains of sand fall over the neighboring downhill sites when the critical slope is exceeded.The slopes of /V consecutive sites organize into a local state. The time evolution of the local states alongthe spatial direction shows a natural tree structure. As a result, various multifractals can be identified.The spatial two-point correlation function decreases exponentially with a correlation length of the orderof the lattice spacing.PACS numbers: 05.40.+j, 05.70.Jk, 05.70.LnSandpile models originally proposed by Bak, Tang, andWiesenfeld [1] are perhaps the simplest models that cap-ture the essence of many nonlinear dissipative systems.As such, they have attracted the attention of many' physi-cists. In particular, in searching for scaling and univer-sality, Kadanoff, Nagel, Wu, and Zhou [2] have studiedmany models in one and two dimensions numerically.We should try to adhere to their convention and terminol-ogy. Hence, the model we discuss is the "limited, " "non-local" model with N=2 in one dimension. In most of theinvestigations on the sandpile models so far, sand isdropped randomly on the various sites. It is important,however, to distinguish the effect of this "random pertur-bation" from a potential stochasticity that may arise as aresult of the intrinsic nonlinear dynamics. In this respect,we should mention that Wiesenfeld, Theiler, andMcNamara [3] had investigated numerically a two-dimensional "Abelian" model [4] in which sand is addedonly at the center. They concluded that randomness inthe sand drops is not an essential element in order for asystem to show self-organized critical behavior.Consider a one-dimensional lattice and let i =0, . . . , Llabel the lattice sites. The number of sand grains at site iis denoted by h(i). We should focus on the slope variableo(i) =h(i) —h(i+1) and interpret cr(i) to be the num-ber of "particles" at site i. The dynamics can then be de-scribed in terms of the creation and annihilation opera-tors a; and a;. Specifically, dropping a sand grain at sitei, which causes the change h(i) h(i)+1 in the heightvariables and the changes o(i) o(i)+1, cr(i —1) cr(i—1)—1 in the slope variables, is given by the "kineticterm"a;a; —~,while an avalanche triggered at site i, which causes thechanges h(i) h(i) N, h(i + I) h—(i + I )+ I, . . . ,h (i +N) h (i +N) + 1 in the height variables whenh (i) —h (i + 1) & N and the changes cr(i —1) o(i—1)+N, cr(i) a(i) —N —1, a(i+N) o(i+N)+Iin the slope variables when o(i) & N, is described by the"interaction term"(a ) )~at+tva~+'e(a(i) N), —(2)+pa; a;-~, 0~ p; ~ 1, gp; =1, (3)where 8(x) =1 for x & 0 and vanishes otherwise. Weshall investigate the "strong-coupling limit, " i.e., the hop-ping term (1) will be treated as a perturbation and foreach hopping the interaction term (2) will be applied anynumber of times until further application of it no longerproduces an eAect. The boundary conditions are suchthat particles can leave the system from the left edge butnot from the right edge. Thus if a particle tries to hopbeyond site L, it lands on site L [5]. In general, one mayconsider the hopping term1479VOLUME 67, NUMBER 12 PHYSICAL REVIEW LETTERS 16 SEPTEMBER 1991but we shall choose pp=1 in the following, so that parti-cles are injected one by one from the left edge corre-sponding to sand being dropped one by one at site i =0.We call each particle injection a time step. Starting fromany given configuration, the system will reach a "limit cy-cle" after a finite number of time steps. The period of thelimit cycle is N . The states in the limit cycle are charac-terized by two conditions: (1) For any N consecutivesites, there is at least one site i such that a(i) =N. (2)Let o(L) =k; then k ~ 1 and there exists at least one sitei such that cr(i) =N for L —k ~ i (L. Thus the systemorganizes into intervals of N consecutive sites. For N=2,we assume I.=2K for simplicity so that there are K inter-vals. By condition (1) above, there are five allowed statesfor each interval, namely, u) =(02), u2=(12), w =(22),v) =(21), and v2=(20) where the integers in the brack-ets give the particle number of the two consecutive sites.A state in the limit cycle is described by a sequence[z~z2 zk.], where z; C [u~, u2, w, v), v2], with the con-straints that (1) z; =u) or u2 implies z; —) =u), u2, or w,(2) z; =v~ or v2 implies z;+~ =v), v2, or w, and (3)z~&v2. We shall consider sequences of lengths otherthan K. An "allowed sequence" is one such that condi-tions (1) and (2) are satisfied. In particular, u;, w, and v;may be regarded as allowed sequences of length one. It iseasy to show that the number of allowed sequences[z~z2 zk] of length k is nk —~if zk =u ~, u2 or z~ =v), v2and is mk —] if zk =w, v], v2 or z [ =u ~, uq, w, where(22k+)+1) ~ (22k+2 1) (4)To describe time evolution of the system, we introduceinteger-valued "time functions" tj defined on the allowedsequences. For sequences of length one, we havet, (u )=)0, t~(u2) =n~, t~(w) =2n~,r, (v)) =n, +), r)(v2) =n, +)+m, .In general,krp([z)z2 . zk ] ) = g tz )(uz'), —(s)ktI([z)z2 zk]) —g t&+i )(u~) . —It can be shown that p(r[ z)'z'2' z)r]) provides time or-dering for the states in the limit cycle. For a given al-lowed sequence [z)z2 zk], we denote its "next se-quence" by [z)z2 zk]+1 which is defined by the rela-tionrp([z)z2 ' ' ' zk]+1) =rp([z)z2 ' ' ' zk])+1 . (7)Then we have[u; ]+1=[u;+1 ],[ww ]+1=[v)w ],[wv( ]+ I =[v) v; ],rp([z }z2 ' ' ' ZI] ) =rp([z )z2 ' ' zk] )+rk ([zk+) ' ' ' zl] ),which in fact describes the "scaling" property of the timeevolution. In plain words, the time evolution of a state atthe kth interval is given as follows. Each state of u [ or u2will last nk —~times steps and that of v[, v2, or w wi11 lastmk —[ times steps. If we ignore this repetition of a givenstate, the time evolution of the kth interval is describedby the time evolution of an allowed sequence of lengthK —k+1. Consider the Kth interval. The four states u [,uq, w, and v~appear in this order and each repeats n~ —[,n~ —[, m~ —[, and m~ —[ time steps, respectively. At the(K —1)th interval, there are sixteen time intervals corre-sponding to the sixteen allowed sequences [zk )zk] withz~&vq. The time ordering of the sixteen intervals isdetermined by tp([zk )zeal). T—hus we see that, in gen-eral, the state [z ~z2. . . zk] in the limit cycle correspondsto a path in a tree. It is now possible to identify a mul-tifractal along the time axis. We rescale the time vari-able so that each time step is of length 6'=2 and theduration for a limit cycle fits into the time interval [0,1].Consider the states [z)z2 . zk. ] in the limit cycle whichcontain N~ u~'s, N2 up's, Np w's, N~ & ]'s, and N2 v2's.The corresponding time steps for such states form a frac-tal in the interval [0,1] with the fractal dimension[wu; ]+1=[u)u;+1. ],[v) ]+1=[v2+1 ][v2 v2[ ]]+1= [u) u) [ ]+1].In the last relation of Eq. (8), the number of consecutivev2's is the same as that of consecutive u ['s. It is instruc-tive to note the relation1 42+gof(41 42 40 41 (2) = 4) I» + + 42 ln2 ln2where g; =N;/K, g; =N;/K, go =Np/K, and satisfy P;(g;Denoting the time average by angular brackets ( . ),(cr(2i —1))= —', —1/2 ', (a(2i)) = —', —1/2 '+'.1+ +(oin 1+ +g;42 4o+g;)+go=i.we find(10)For the two-point function, we get, for example,1 1 3 1 1&a(2i —I )o(2j —1))—(o(2i —I))&o(2j —1))=— 2(. , ) —— 2,. — 2) . (12)VOLUME 67, NUMBER 12 PHYSICAL REVIEW LETTERS 16 SEPTEMBER 1991Neglecting the boundary effect, we see that the two-pointfunction decreases exponentially with a correlation lengthg =a/ln2, where a is the lattice spacing.For each injection of a particle from the left edge, anumber of consecutive local states will be aAected. Wedefine the corresponding affected number of sites to bethe cluster size. Then one can show that the distributionof the cluster size decreases exponentially with thecharacteristic length g. A similar result holds when weconsider the distribution of the durations of avalanches ifwe define the duration of an avalanche to be the numberof times the interaction term in Eq. (2) is applied [6l.We conclude that for one dimension, the deterministicsandpile model does not show the "self-organized critical-ity" in the sense of a power-law distribution for, say, thecluster size [7]. The conclusion may very well be other-wise when a random perturbation is applied [2]. We doobserve an interesting tree structure in the time evolutionof the system. This "ordering, " however, is not reflectedeither in the two-point correlation function or in the dis-tribution of the cluster size.We wish to thank Dr. C. K. Chan for many useful dis-cussions. This work is supported in part by a grant fromthe National Science Council, Taiwan, Republic of Chi-na, under Contract No. NSC80-0208-M001-17.[1] Per Bak, Chao Tang, and Kurt Wiesenfeld, Phys. Rev.Lett. 59, 381 (1987); Phys. Rev. A 38, 364 (1988); ChaoTang and Per Bak, Phys. Rev. Lett. 60, 2347 (1988).[2] Leo P. Kadanoff, Sidney R. Nagel, Lei Wu, and Su-minZhou, Phys. Rev. A 39, 6524 (1989).[3] Kurt Wiesenfeld, James Theiler, and Bruce McNamara,Phys. Rev. Lett. 65, 949 (1990).[4] Deepak Dhar, Phys. Rev. Lett. 64, 1613 (1990).[5] Recall that here the "particle" refers to slope variable andshould not be confused with sand grain.[6] The expert may wonder how this can happen since if sandis dropped only at the left edge, the average number ofavalanches should be of order L. In fact, the distributionfor the number of avalanches T decreases exponentiallyfor 0~ T~ —,' L and increases exponentially for —,' L ~ T~ L. The precise statement should be the distribution ofT decreases exponentially toward its minimum valuewhich is attained when T= —, L.[7] However, the power spectrum for the distribution of sandgrains dropping out of the right edge as time developsmay very well show a I/f behavior.

Physical Review Letters

Exact solution of a deterministic sandpile model in one dimension

http://tkuir.lib.tku.edu.tw:8080/dspace/retrieve/81654/PhysRevLett.67.1479.pdf

Exact solution of a deterministic sandpile model in one dimension

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