Grain Sorting in the One Dimensional Sand Pile Model. March 20, labri, ura cnrs 1 304, F Talence Cedex,

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1 aoratoire Bordelais de Recherche en Informatique, ura cnrs 304, Universit Bordeaux I, 35, cours de la i ration, Talence Cedex, France. Raort de Recherche Num ro 5797 Grain Sorting in the One Dimensional Sand Pile Model J r me Olivier Durandose y March 20, 997 lari, ura cnrs 304, Universit Bordeaux I, 35, cours de la i ration, F Talence Cedex, France. jdurand@lari.uordeaux.fr Astract We study the evolution of a dimensional ile, emty at rst, which receives a grain in its rst stack at each iteration. The nal osition of grains is singular: grains are sorted according to their arity. They are sorted on traezoidal areas alternating on oth side of a diagonal line of sloe 2. This is exlained and roved y means of a local study. Each generated ile, encoded in height dierences, is the concatenation of four atterns: 22, 33, 0202 and. The relative length of the rst two atterns and the last two atterns converges to 2. We make asymtotic exansions and rove that all the lengths of the ile are increasing roortionally to the square root of the numer of iterations. Introduction We consider an innite sequence of stacks. Each stack can hold any nite numer of grains. This numer is also called the height of the stack. Sand Pile Model (sm) and Chi Firing Games (cfg) are ased on local interactions. Both model conserve the total numer of grains. In sm, a grain goes to a neighor stack if the height dierence etween stacks is more than a given threshold ; whereas in cfg a stack gives a chi to each of its neighors if its numer of chis is aove a certain value. In the dimensional case, oth games are dimensional cellular automata and they are equivalent via some simle encoding. Both sm and cfg, like Petri Nets [], are used to model in arallel comuting the ows of information in a system. sm is used to model gradiandriven dynamic loadalancing. Grains model data or tasks, and stacks, a rocessor network [2]. The aim is to nd a simle, fast and relatively inexensive local rearrangement which ensures that all rocessors have almost the same amount of work. sm is imortant for granular ows in hysics. It admits invariants, entroylike functions and veries the so called `SelfOrganized Criticality' and is related to the =f henomenon [2, 9, 0, 3]. This work was artially suorted y ECOS and the French Cooeration in Chile. y This research was done while the author was in the Deartamento de Ingenier a Matem tica, Facultad de Ciencias F sicas y Matem ticas, Universidad de Chile, Santiago, Chile.

2 Goles and Kiwi study the sequential dimensional sm and the related cfg [6, 7, 8]. They have roved the uniqueness of the nal ile whatever the order of the iterations as well as descried the dynamics in various sequential cases. The rolem we study in this article is the arallel evolution of a dimensional sm, emty at rst, which receives a grain onto its rst stack at each iteration. It can also e seen as sand driing in a thin ut long hourglass. In Sect. 2, we dene the sm and cfg models and recall that they are equivalent in dimension. We also dene the driing rocess we study. In Sect. 3, we rove that the generated iles, in height dierences, are made of four atterns: 22, 33, 0202 and. The frontiers etween atterns act like signals. The silhouettes of each ile is made of two arts of dierent sloes: 2 then. In Sect. 4, grains are marked deending on their arity. Even and odd grains are arranged in a very secial way: they are located in traezoidal areas alternating on oth side of a diagonal line of sloe 2. We exlain this y looking locally at the interactions etween moving grains and signals. In Sect. 5, we give asymtotic aroximations of the dierent arameters. We do this y making a continuous aroximation of the ile and use a dierential resolution like Anderson et al. in []. We rove that the length of the art of sloe is 2 times the length of the art of sloe 2 and that all the length of the ile are increasing roortionally to the square root of the numer of iterations. 2 Denitions The dimensional sand ile is an innite sequence of stacks. Each stack can hold any nite numer of grains. We use the notation of Goles and Kiwi [8]. The only dierence is that our model is arallel. A ile is encoded y the sequence of the numer of grains, or height, of the stacks. It is then denoted with square rackets: = [[ 0 : : : k ]]. We call sloe the dierence of height, i i, etween two consecutive stacks. If more stacks are considered, the sloe is the average sloe. If a stack is higher y at last 2 grains than the next stack, then one grains tumles down. This is deicted y the movement of the grains a to f in Fig.. The starting ile is emty. At each iteration, a grain falls onto stack numer. Grains c to f in Fig. are the newly arrived grains. The numer of grains is nite. Excet for the grain added to the ile at each iteration, the numer of grains is constant. The numer of grains is then equal to the iteration numer. c a [[ ]] hh ii d c a [[ ]] hh ii e d c a [[ ]] hh 3 0 ii f e c d a [[ ]] hh 3 ii Figure : Iterations 4 to 7. Since grains are only moved to smaller stacks, a direct induction roves that only decreasing sequences are generated from the initial ile. A ile is now an element of NN decreasing to zero. This ensures that height dierences etween any two consecutive stacks is always ositive. Denition et ( n ) e the following threshold function: 8n 2 Z, ( n ) = if 0 n, otherwise 0. et e a ile. The dynamics of sm with driing is driven y the following transition function F : F () 0 = 0 ( 0 2 ) + ; 0 < i; F ()i = i ( i i+ 2 ) + ( i i 2 ) : 2

3 The negative terms corresond to the ossiility of giving a grain to the next stack, while the ositive term corresonds to the ossiility of getting a grain from the revious stack. All the stacks are udated at the same time. This is a arallel rocess. Denition 2 A ile can e encoded y the list of the height dierences etween stacks: for any ile ' () = hh ( 0 ) ( 2 ) ( 2 3 ) : : :ii. With this encoding, the transition function ecomes: (x) 0 = x 0 2 ( x 0 2 ) + ( x i+ 2 ) + ; 8i; 0 < i; (x)i = x i + ( x i 2 ) 2 ( x i 2 ) + ( x i+ 2 ) : We call chi the dierence of height etween a stack and the next one. Denition 2 is equivalent to: if a stack has more than two chis, it `res' one to oth its neighors. This is the chi ring game (cfg). In a dimensional lattice, sm and cfg are equivalent with this encoding Iterations Height Piles 0 Figure 2: Iterations 0 to 50. Figure 2 illustrates the rst 50 stes of this dynamic. The lengths and heights, as well as the sloes, exhiit some regularity. After some iterations, there are two stras of triangles drawn on the surface as deicted in Fig. 3 for iteration 00 to Triangles and Signals Piles are encoded in height dierence in Fig. 4 (stes to 20). Triangles aear with atterns 22, 33, 0202 and. Those atterns are stale. It should e noted that for the second and third atterns, digits are alternating, like in a chessoard and the frontier etween them is either 2 or 30. et " e the emty word. The Kleene oerator is denoted *; i.e., (3) is " or 3 or 33 or 333 : : : We use the theory of languages in the next roosition in order to get a synthetic exression. Proosition The ile, encoded in height dierences, is a word of the following language: 2 ( " j 3 ) ( 3 ) ( " j 2 ) ( 0 2 ) ( " j 0 ) : 3

4 22 50 Height Iterations Piles 00 Figure 3: Iterations 00 to 50. Proof. We rove the Proosition y induction. It is true for the rst 20 iterations as it can e seen on Figure 4. Interaction, as exressed in denitions and 2, only deends on the two closest neighors. It is enough to look locally at the interactions of the frontiers on Fig. 4. Suose that the n th ile is the concatenation of four arts with the atterns 22, 33, 0202 and resectively. We call frontier the limit etween two atterns and order the limits of the ile. We denote (left), M (middle) and R (right) the ositions of the frontiers etween resectively rst and second, second and third, third and fourth atterns. They are reresented on Fig. 4 where and R ehave like signals moving on oth sides of M. Geometric denitions are given in Fig. 5. et us rst investigate each signal alone, from left to right: is going left (right) if it is equal to 2j (2j3) (lines 07 to 7) ; M is not moving (lines 96 to 02) ; R is going left (right) if it is equal to 0j (2j) (lines 94 to 04). While the roosition is true, only the following encounters are ossile, from left to right: on the left order, ounces (lines 59 to 65) ; when meets M, it ounces and M is moved one ste to the right (lines 8 to 87) ; when R meets M, it ounces and M is moved one ste to the left (lines 50 to 57). The order of the signals is ket, and the only ossile encounter with more than two frontiers is MR. The meeting can e exactly synchronous (lines 40 to 44) or not (lines 62 to 67 and 03 to 09). In all cases the order is ket and no other case arises. The dynamics is very simle excet when signal or R reaches one of its limits; the rest is only linear dislacement. When reaches the left order, it only ounces ack. When R reaches the right order, it ounces ack and the total length is increased y one. When R comes ack to the center, the total length has een increased y one. In height dierences, iles are the concatenation of four arts of atterns 22, 3, 02 and resectively. The two rst arts have a sloe of 2 while the two last arts have a sloe of. This exlains the shae of iles (in heights) as deicted in Fig. 5. 4

5 # M R M R M R Figure 4: Reresentation with height dierences :G +" 2 6 D +" 2 G D ", " 2 2 {, 0, } M R 4 aeling Grains Figure 5: Geometric denitions of G, D, and M. Grains are laeled according to the iteration during which they enter the ile. On Fig. 6, at the th iteration, all odd grains are sotted in lack. Their localization is singular. The odd grains, like the even ones, are located on traezoidal areas delimited y the axis, lines of sloe and 2, and a diagonal line. These areas are alternating like in a chessoard. The diagonal searation seems to e a straight line. There is also some kind of relation etween the intersections of the line of sloe and 2 with the axis and the edges in the middle as deicted in Fig. 6. We do not have any exlanation nor roof for this henomenon yet. Nevertheless, if the diagonal searation is a line, ecause of such coincidences, its sloe would e: 5

6 2a a Figure 6: Position of the odd grains (in lack). a= = ( + 2a)=(a + ) which leads to =a = 2. We rove in Sec. 5 that indeed it is a straight line with sloe 2. Theorem Odd and even grains are always sorted in traezoid areas delimited y a diagonal, lines of sloe and 2, and the axes. With gures 7 to, we rove that the grains are always on either side of the frontier, deending on their arity. In all these gures, grains are either lack or white deending on their arity. Grains for which arity is unknown are drawn with a little circle. The grains which do not move any more are reresented y their silhouette. et us rst consider that signal is away from the left order. Even and odd grains come alternatively and go down the ile one after the other as deicted in Fig. 7. Grains ehave like a wave of marles on stairs Figure 7: Arrival of new grains From this, a direct induction roves that the attern 22 corresonds to an evenodd wave of grains. et us consider that signal is going right. As deicted in Fig. 8, the wave is just going down with scarce grains running in front of it. 6

7 2 2 j j j3 Figure 8: Signal goes right. Going right, the signal encounters the middle order M as deicted in Fig. 9. The rst grain crosses the order and ecause of the height dierence, the second gets locked. The third asses over the second and restrains the fourth from assing, and so on. M 2j3 j2 0 M 2 2j3j j j 2 M 2 2j 3j0 Figure 9: Signal reaches the middle order alone. M 2j 3 j2 The henomena of one grain getting locked and the next assing over it, one layer u, is the way the signal goes right as deicted in Fig. 0. When reaches the left order, it ends uilding a layer and go ack to the middle on the new layer as deicted in Fig. 0. In comarison to Fig. 7, we know now that the grains that are running in front of the wave are all of the same arity. The rst grain of the wave is of the same arity as the rst grain of the revious wave and as the grains the scarce grains, so that the henomena starts again and loos j j j 3 3 j j j Figure 0: Signal goes left and reaches the left order. In the eginning of this roof, we never take signal R into account. When R is away from the middle M, it has no actions whatsoever since the selection of grains is made efore. Signal R only orders the grains on layers in the right art. When or R meets M it only moves it and that does not change the dynamic of Fig. 9. But, when all three signals, M and R meet, things are dierent as deicted in Fig.. This time, the fate of odd and even grains are switch. The changes in the destination of odd and even grains in Fig. 6 are directly linked to the synchronous encounter of and R detailed in Section 3. In Fig. 6, the searation lines reresent the silhouettes of iles at some iterations and the diagonal 7

8 M R 2j3 j2 0 j MR 2 2j3j0j 2 2 2j MR 2 2jj2j M R 2j 3j0 2 j Figure : Signals and R exactly synchronized. searation is the trace of the middle order M. Since there are as much even grains as odd grains, the 2 symmetric areas in Fig. 6 have the same surface, i.e., they corresond to the same numer of grains. 5 Asymtotic Behavior All the results in this section can also e found in [5]. The roof of [5] is too long to t here, we give here a shorter one which we feel is more like an a osteriori verication. Theorem 2 The diagonal searation is a line of sloe 2. The value of G increases (decreases) y one for each round tri of (R). The value of D is decreases (increases) y one (two) for each loo of (R). The round tri delay for a signal is twice the length of its art it evolves in, u to a constant. Since every quantities go to innity, when G and D are very ig, the equations can e extended to continuity as in []: 8 >< >: dg = dt 2:G dt 2:D ; dd = dt 2:G + 2 dt This equations can e solved with the hyothesis D = It leads to: G:dG = With this hyothesis, the ossile solutions are: 2:D : 2 G which comes from the oservations of Sec dt : G = D = s G = t + c ; r 2 t + 2c : Where t is the time (or numer of fallen grains or numer of iterations) and c is a constant. From Fig. 5, the numer of fallen grains n is also the total area of Fig. 5, i.e., of the two triangles and of the rectangle. We get the following aroximations: n D2 2 + G:D + G2 (2 + r n G ; D G r 2) G 2 ; n 2 + : () 8

9 It should e noted that oth triangles of Fig. 5 have almost the same area, G 2. This is coherent with the surface oservations of Section 4. The rectangle is equally arted y the diagonal and even and odd grains are equally arted on oth sides of the diagonal. 6 Conclusion A more random distriution of odd and even grains might have een exected, on the contrary grains are sorted. This is imortant, ecause if even and odd grains/tasks are very dierent, in a dimensional rocessor array sequentially fed using sm load alancing technique, disarities arise. When taken modulo 3, 5 or more, there is no such segregated location as efore, grains are more uniformly sread. The way grains sread as a wave and xed in the silhouette is very interesting. It gives a hysical meaning to the signals. When goes right it sread grains. When it goes left, it makes a one over two selection. Signal is going right and left while the grains are always running to the right. The signal R is acting similarly. When it goes right it is sreading the grains on a new layer, oening it. When it goes left it xes them. When grains and signals are going in oosite directions, since they have seed one, signals only meet every other grain. These signals, on the hysical oint of view, are very interesting ecause they corresond to the waves roagation on a ile of sand when you dig at the ottom. We have roved that the ile is exending in the square root of the numer of fallen grains (or iterations). This is asolutely normal when one consider that the grains (linear) are lling a surface (quadratic). The relative length of the two arts is 2. To comare with the work of Anderson et al. [], on the one hand, they found a quadratic relaxation time for the cfg starting with the ile ::: 0 0 n 0 0 ::: and nal ile ::: 0 0 ::: 0 ::: But when considered as stacks of grains, they corresond to ::: n n n 0 0 ::: and ::: n n (n) (n2) ::: 2 0 ::: resectively. This is a very dierent case ecause of the inuence of the left order which is high and feeds grains to the right art. On the other hand, they also oserved geometric atterns and signals roagations. References [] R. Anderson,. ov sz, P. Shor, J. Sencer, E. Tardos, and S. Winograd. Disks, alls and walls: Analysis of a cominatorial game. American Mathematical Monthly, 96:48493, 989. [2] P. Bak, T. Tang, and K. Wiesenfeld. Selforganized criticality: An exlanation of =f noise. Physical Review etters, 59(4):38384, 987. [3] J. Bitar and E. Goles. Parallel chi ring games on grahs. Theoretical Comuter Science, 92:29300, 992. [4] A. Bj rner,. ov sz, and P. W. Shor. Chiring games on grahs. Euroean Journal of Cominatorics, 2:28329, 99. [5] J. O. Durandose. Automates Cellulaires, Automates Partitions et Tas de Sale. PhD thesis, lari, 996. In French. [6] E. Goles and M. Kiwi. Sandile dynamics in onedimensional ounded lattice. In Boccara, Goles, Martinez, and Picco, editors, Cellular Automata and Cooerative Systems, ages Kluwer, 99. [7] E. Goles and M. Kiwi. Dynamics of sandiles games on grahs. In latin'92, numer 583 in ecture Notes in Comuter Science, ages SringerVerlag, 992. [8] E. Goles and M. Kiwi. Games on line grahs and sand iles. Theoretical Comuter Science, 5:32349, 993. [9] P. Grasserger and S. Manna. Some more sandiles

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