Ricepiles: Experiment and Models
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1 Progress of Theoretical Physics Supplement No. 139, Ricepiles: Experiment and Models Mária Markošová ) Department of Computer Science and Engineering Faculty of Electrical Engineering and Information Technology Slovak University of Technology, Ilkovičova 3, Bratislava, Slovakia The theory of self organized criticality (SOC) describes the avalanche dynamics of the spatially extended systems, consisting of many small elements (such as granular material). Numerically and analytically, SOC is studied on the models called sandpile cellular automata. Several numerical and analytical results concerning the avalanche statistics at the critical state have been achieved and the question of how well these results describe the real avalanche dynamics of a granular material has arised. To answer this question the ricepile experiment has been done and the avalanche statistics has been carefully measured. This further stimulated an interest to study in more detail, which physical properties of granular material are important for SOC state to be established. Several ricepile models have been created and numerically analysed. This paper exhibits a short review of the experimental results and the ricepile models. 1. Introduction To understand the dynamics of the granular material is important from the scientific as well as from the practical point of view. One of the interesting theoretical approaches to the problem exhibits the theory of self organized criticality (SOC). 1) SOC theory describes, how a pile of sand organizes itself to the critical state under the action of a slow drive and a quick relaxation process, an avalanche. The SOC state is a stationary state, in which critical, power law scaling of the avalanche statistics is present. SOC theory is based on the careful numerical 1)- 3) and analytical 4), 5) studies of the model systems sandpile cellular automata. A natural step from the sandpile models, leads to the investigation of the real piles of granular material, and to the comparison of the theoretical results with practical measurements. In 1996 an experiment has been done at the Oslo University, in which the dynamical behaviour of the driven quasi-one-dimensional pile of rice has been investigated. The avalanche sizes in the steady state were measured, for the two types of grains (elongated and round), in terms of the dissipated potential energy. In the case of the ricepile, consisting of the elongated grains, SOC state has been established and the avalanche statistics, together with the critical exponents has been measured. 6) Because none of the known sandpile cellular automata exhibits the same statistics of avalanches and the same critical exponents as the experimental pile of rice, the model ricepiles were suggested and numerically studied. 7) - 9), 11), 13), 14) The model ricepile is, in principle, a cellular automaton defined on the onedimensional lattice, with randomness incorporated into the toppling rules and with the deterministic drive. Changes in the toppling rules are often manifested by the ) address: mark@oslik.elf.stuba.sk
2 490 M. Markošová different dynamical behaviour of the model and different universality class into which the model belongs. 9), 17) 2. Ricepile experiment The experimental device consists of two vertical, parallel plates with a thin gap in between. 6) At the bottom and at the left-hand side the gap is closed, at the top and the right-hand side it remains open. The grains are added to the left closed boundary and in between the plates quasi-one-dimensional ricepile is created. The grains leave the system at the right open boundary. In the stationary state the global slope of the pile is constant, but the avalanches redistribute the mass along the surface and the profile of pile is changing. The size of an avalanche was defined as the energy dissipated between the two consecutive profiles. P (E, L)dE is the probability that an avalanche with the system size L and dissipated energy between E and E +de occurs. The probability density has, in the stationary state, a form: P (E, L)=L β f(e/l µ ). (2.1) The experimental measurement reveals that for the elongated ricegrains the scaling function f(x) = const for x 1 and f(x) x τ for x>1. Normalization of P (E, L) gives β = µ, and from the average dissipated energy, which is equal to the added potential energy one can establish that β = µ =1.0. 6) Finite size scaling plots of (2.1) show that really β = µ =1.0 and also, that τ = The distribution has a power law tail, indicating SOC. For the round ricegrains, the results were different and no self organized critical behaviour has been found. 3. Ricepile models One of the main questions, which arises from the experimental results is, which physical properties of the pile material are important for SOC state to occur. What role, for example, plays the friction, the grain shape, the gravity and the inertia of rolling grains? The ricepile experiment reveals, that certainly the shape of the grain, better said the grain aspect ratio α = width length, (3.1) is very important. With the shape such properties as the grain packing, friction, the shape of the pile profile, inertia effects, are directly connected. Therefore in each ricepile model all these aspects should be taken into account. In the two threshold ricepile model friction and gravity effects are included in a simple way, through the parameter p. 7) - 9) The value of the parameter p decides, whether the grain will stop on the site, or roll further down the slope. In consistence with the experimental set up, the two threshold model is defined on the onedimensional lattice of size L, with a wall at the zero position and an open boundary
3 Ricepiles: Experiment and Models 491 at the position L +1. At the open boundary, particles are free to flow out of the system. As in the experiment, the system is driven by adding particles to the position one, at the closed end. Every time unit one particle is dropped into the system. The local slope z i of the pile is given as the difference of heights on the two subsequent sites i and i +1 (z i = h i h i+1 ). There are two different critical thresholds, namely the avalanche threshold z c and the gravity threshold z g. The dynamics of the two threshold ricepile model 7), 8) is as follows: 1. Each avalanche starts at i =1. Ifz 1 >z c, the site one is activated and topples a particle to the next nearest position (two), with the probability p. If even z 1 >z g ; p = Every particle, sliding from the position i to i+1 activates three columns, namely i 1, i and i + 1. The position i 1 is activated, because it possibly can become supercritical, when removing a grain from the i-th column. Columns i and i + 1 are activated, because they are destabilized by sliding or stopping particles, respectively. In the next time step, all supercritical active sites topple a particle to the i + 1-st position with probability p (p =1.0, if z i >z g ). 3. Step two is repeated, until there are no active sites in the system, that means, until the avalanche is not over. This relatively simple model has been numerically simulated under the conditions p =0.6, z c =1.0 and z g =4.0, but the results do not, to a great extent, depend on the parameter values. Avlanache size distribution (in which the size of the avalanche is given in terms of the number of topplings) was measured. It scales as P (s, L) s τ f s ( s L ν The avalanche lifetime distribution scales as P (T,L) T y f T ( T L σ ) ). (3.2) (3.3) and the energy avalanches (e.g. the avalanche size is given in terms of dissipated potential energy) distribution is ( ) E P (E, L) E α f E L ν. (3.4) E Because one toppling in average dissipates constant amount of potential energy, exponent α = τ =1.53 ± 0.05 and the same way ν = ν E =2.2 ± Exponents y and σ have been measured to be 1.84 ± 0.05 and 1.4 ± 0.05, respectively. The distribution of the energy avalanches scales with L according to Eq. (2.1), with the exponents β = µ 1.0, as has been theoretically predicted. But, in spite of this, the two threshold model does not give the measured value of α The set of the above-mentioned critical exponents enlist the model into the Manna universality class, 10) which seems to be different from the universality class of the real pile of rice. It is known, that the different toppling rules (more nonlocal, for example) may change the critical exponents, and thus also the universality class. 17), 9) The two threshold model can be modified and the toppling rules are changed as follows:
4 492 M. Markošová a) The number of particles toppled from the position i is constant and independent of the supercritical local slope z i the model is called limited. b) The number of toppled particles is a function of the supercritical local slope z i, the model is called unlimited. c) If the particle (or more particles) topples from the site i and moves only to the next nearest position i + 1, the model is defined as local. d) The model is called nonlocal, if n toppled particles moving from the i-th site are added subsequently to the n nearest downslope positions (one particle per site) i +1, i +2,, i + n. Thus four different models are recognized: 1. local limited model (LLIM) 2. local unlimited model (LUNLIM) 3. nonlocal limited model (NLIM) 4. nonlocal unlimited model (NUNLIM) The distributions of the avalanche sizes (3.2) (3.4) were numerically studied for the above-mentioned models, and the critical exponents have been measured. The results are in Fig. 1. As is evident, the models belong to the three different universality classes, but still none of them is the universality class of the real ricepile. In order to enhance the number of small avalanches and suppress the number of the big ones (which should lead to a steeper power law part and thus higher critical exponent τ), one threshold ricepile model has been established. 11) The dynamics of this model is similar to the two threshold one. The only difference lies in a fact that there is no second, gravity threshold. It means that if TWO THRESHOLDS ONE THRESHOLD MODEL NLIM LLIM NUNLIM LUNLIM NLIM LLIM NUNLIM LUNLIM τ=1.55 ν=2.25 UNIVERSALITY CLASSES τ=1.35 ν=1.55 τ=1.63 ν=2.75 OSLO Fig. 1. Division of the ricepile models into the different universality classes.
5 Ricepiles: Experiment and Models 493 the local slope z i is supercritical and active, it topples a particle with the probability p (0 <p<1) to the nearest neighbour downslope position z i+1. There, therefore, persists small, but nonzero probability that also the extremely large local slopes are possible. Physically this seems to be quite plausible. It is not probable that in the real piles of the granular material there exists a strict gravity threshold. If one investigates the average material transport J(p) as a function of p, where J(p) is defined as J(p) = n out(p) n in (p), (3.5) ( n out, n in is the number of outgoing, ingoing particles in a certain time interval) three different dynamical regimes are recognized: i) isolating (for 0 <p p ), in which all particles are absorbed in the system and none of them reaches the open boundary; ii) partially conductive (for p <p p c ), in which the pile profile grows up as a bulk, because a certain fraction of the particles, depending on p, is absorbed in the system; and iii) totally conductive (for p c <p 1), when the number of ingoing and outgoing particles is balanced. Between these different dynamical regimes there are two phase transitions, the first one at p , the second one at p c Close to the first phase transition point p, the steepness of the pile is still high enough to say that the local slopes are almost everywhere higher than the critical threshold z c. This is the reason that the spreading of the active sites in time is practically determined by the probability p; the same way as it is in the percolation process. In the space-time coordinate system, we have therefore a picture of directed percolation with three descendants and an absorbing boundary. 12) p is thus simply the critical percolation threshold. Avalanche size distribution for the different parameter values p has been numerically measured in the partially and entirely conductive regimes. Scaling (3.4), indicating self organized criticality was found in both cases. The critical exponents indicate, that the one threshold model still belongs to the Manna universality class (see Fig. 1). All four versions of the one threshold model (LLIM, LUNLIM, NLIM, NUNLIM) were also numerically studied. 13) The models belong to the two different universality classes (see Fig. 1), defined by the critical exponents, not corresponding to that of experimentally measured set. Theorists, participating at the Oslo ricepile experiment, established the Oslo ricepile model. 14) Using this model, some experimental facts are easily explained. The Oslo model is one-dimensional cellular automaton, driven at the closed boundary; z 1 z (3.6) The other boundary remains open. If z i >z c, the pile topples
6 494 M. Markošová z i z i 2, (3.7) z i±1 z i±1 +1. (3.8) Critical slope z c in the Oslo model varies dynamically between one and two every time unit. Qualitatively, the dynamics of the Oslo model is the same as the dynamics of the two threshold ricepile model with p =0.5, z c =1.0, z g =2.0 and belongs to the same Manna universality class. It is possible to measure experimentally the distribution of the tracer times. This term denotes a time, during which the tracer particle (e.g. coloured ricegrain) persists in the ricepile. The distribution is given by ( P (T,L) L β T T f L µ T ) (3.9) with µ T = β T =1.5 ±0.2 and f(x) = const for x<1 and f(x) =x α ; α =2.4 ±0.2 for x>1. From (3.9) it is clear, that T L µ T. (3.10) The angle of repose is independent on L and thus the average velocity of the tracer particle is V L 1 µ T = L 0.5±0.2. (3.11) Feeding rate of the tracers is also L independent. If λ L denotes the active zone depth, for the number of tracers crossing any section in the time interval t one has and t V λ L = const, (3.12) V 1 λ L. (3.13) Active zone depth can be measured. It has been proven experimentally that (3.11) is fulfilled. This model is interesting not only because of successful explanation of the experimental facts, but also from another point of view. Namely, it has been shown by Paczuski and Boetcher 15) that the Oslo model is easily mapped to the known, and theoretically well elaborated, interface depinning model. Recently, some other progress has been made in the modelling of granular material dynamics. To make the model closer to the reality, inertia effects have been included through the aspect ratio α (3.1). Increasing α has the effect of approaching the crossover point of the non SOC and SOC behaviour of the pile. Such crossover point has been numerically studied by Head and Rodges. 16)
7 Ricepiles: Experiment and Models Conclusion In this short review I presented some ricepile cellular automata which model the avalanche dynamics of the ricepile, together with the comparison of the numerical results with the experiment. Further studies are necessary in order to understand the dynamics of the granular material, such as rice. The progress goes through the better theoretical understanding of the probabilistic models 18), 19) as well as through establishing models, which include physical properties of the material in a more appropriate way. 16) This work was supported by the VEGA Grant No 2/6018/99 and the VEGA Grant No 1/4289/99. References 1) P. Bak, Ch. Tang and K. Wiesenfeld, Phys. Rev. Lett. 59 (1987), ) P. Bak and K. Sneppen, Phys. Rev. Lett. 71 (1993), ) K. Chen, P. Bak and S. P. Obukhov, Phys. Rev. A43 (1991), ) D. Dhar, Phys. Rev. Lett. 64 (1990), ) D. Dhar, cond. mat ) V. Frette, K. Christensen, A. Malthe-Sorensen, J. Feder, T. Jossang and P. Meakin, Nature 379 (1996), 49. 7) L. A. N. Amaral and K. B. Lauritsen, Phys. Rev. E54 (1996), R ) L. A. N. Amaral and K. B. Lauritsen, Physica A231 (1996), ) L. A. N. Amaral and K. B. Lauritsen, Phys. Rev. E56 (1997), ) S. S. Manna, J. of Phys. A24 (1992), L ) M. Markošová, M. H. Jensen, K. B. Lauritsen and K. Sneppen, Phys. Rev. E55 (1997), R ) K. B. Lauritsen, K. Sneppen, M. Markošová and M. H. Jensen, Physica A247 (1997), 1. 13) M. Markošová, cond. mat , to appear in Phys. Rev. E. 14) K. Christensen, A. Corral, V. Frette, J. Feder and T. Jossang, Phys. Rev. Lett. 77 (1996), ) M. Paczuski and S. Boetcher, Phys. Rev. Lett. 77 (1996), ) D. A. Head and G. J. Rodges, Phys. Rev. E55 (1997), ) L. Kadanoff, S. R. Nagel, L. Wu and S. M. Zhou, Phys. Rev. A39 (1989), ) A. Vespignani and S. Zapperi, Phys. Rev. E57 (1998), ) B. Tadić and D. Dhar, Phys. Rev. Lett. 79 (1999), 1519.
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