Ricepiles: Experiment and Models

Size: px
Start display at page:

Download "Ricepiles: Experiment and Models"

Transcription

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.

On self-organised criticality in one dimension

On self-organised criticality in one dimension On self-organised criticality in one dimension Kim Christensen Imperial College ondon Department of Physics Prince Consort Road SW7 2BW ondon United Kingdom Abstract In critical phenomena, many of the

More information

arxiv:cond-mat/ v1 [cond-mat.stat-mech] 17 Jul 2003

arxiv:cond-mat/ v1 [cond-mat.stat-mech] 17 Jul 2003 Anisotropy and universality: the Oslo model, the rice pile experiment and the quenched Edwards-Wilkinson equation. arxiv:cond-mat/0307443v1 [cond-mat.stat-mech] 17 Jul 2003 Gunnar Pruessner and Henrik

More information

On the avalanche size distribution in the BTW model. Abstract

On the avalanche size distribution in the BTW model. Abstract On the avalanche size distribution in the BTW model Peter L. Dorn, David S. Hughes, and Kim Christensen Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 2BW, United Kingdom (July

More information

arxiv:cond-mat/ v2 [cond-mat.stat-mech] 15 Jul 2004

arxiv:cond-mat/ v2 [cond-mat.stat-mech] 15 Jul 2004 Avalanche Behavior in an Absorbing State Oslo Model Kim Christensen, Nicholas R. Moloney, and Ole Peters Physics of Geological Processes, University of Oslo, PO Box 148, Blindern, N-316 Oslo, Norway Permanent:

More information

Generalized Manna Sandpile Model with Height Restrictions

Generalized Manna Sandpile Model with Height Restrictions 75 Brazilian Journal of Physics, vol. 36, no. 3A, September, 26 Generalized Manna Sandpile Model with Height Restrictions Wellington Gomes Dantas and Jürgen F. Stilck Instituto de Física, Universidade

More information

Nonconservative Abelian sandpile model with the Bak-Tang-Wiesenfeld toppling rule

Nonconservative Abelian sandpile model with the Bak-Tang-Wiesenfeld toppling rule PHYSICAL REVIEW E VOLUME 62, NUMBER 6 DECEMBER 2000 Nonconservative Abelian sandpile model with the Bak-Tang-Wiesenfeld toppling rule Alexei Vázquez 1,2 1 Abdus Salam International Center for Theoretical

More information

Avalanches in Fractional Cascading

Avalanches in Fractional Cascading Avalanches in Fractional Cascading Angela Dai Advisor: Prof. Bernard Chazelle May 8, 2012 Abstract This paper studies the distribution of avalanches in fractional cascading, linking the behavior to studies

More information

The Bak-Tang-Wiesenfeld sandpile model around the upper critical dimension

The Bak-Tang-Wiesenfeld sandpile model around the upper critical dimension Phys. Rev. E 56, 518 (1997. 518 The Bak-Tang-Wiesenfeld sandpile model around the upper critical dimension S. Lübeck and K. D. Usadel Theoretische Tieftemperaturphysik, Gerhard-Mercator-Universität Duisburg,

More information

Complex Systems Methods 10. Self-Organized Criticality (SOC)

Complex Systems Methods 10. Self-Organized Criticality (SOC) Complex Systems Methods 10. Self-Organized Criticality (SOC) Eckehard Olbrich e.olbrich@gmx.de http://personal-homepages.mis.mpg.de/olbrich/complex systems.html Potsdam WS 2007/08 Olbrich (Leipzig) 18.01.2007

More information

arxiv:cond-mat/ v1 [cond-mat.soft] 12 Sep 1999

arxiv:cond-mat/ v1 [cond-mat.soft] 12 Sep 1999 Intermittent Granular Flow and Clogging with Internal Avalanches S. S. Manna 1,2 and H. J. Herrmann 1,3 1 P. M. M. H., École Supérieure de Physique et Chimie Industrielles, 10, rue Vauquelin, 75231 Paris

More information

Sandpile models and random walkers on finite lattices. Abstract

Sandpile models and random walkers on finite lattices. Abstract Sandpile models and random walkers on finite lattices Yehiel Shilo and Ofer Biham Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel Abstract Abelian sandpile models, both deterministic,

More information

arxiv: v2 [cond-mat.stat-mech] 6 Jun 2010

arxiv: v2 [cond-mat.stat-mech] 6 Jun 2010 Chaos in Sandpile Models Saman Moghimi-Araghi and Ali Mollabashi Physics department, Sharif University of Technology, P.O. Box 55-96, Tehran, Iran We have investigated the weak chaos exponent to see if

More information

Self-organized Criticality in a Modified Evolution Model on Generalized Barabási Albert Scale-Free Networks

Self-organized Criticality in a Modified Evolution Model on Generalized Barabási Albert Scale-Free Networks Commun. Theor. Phys. (Beijing, China) 47 (2007) pp. 512 516 c International Academic Publishers Vol. 47, No. 3, March 15, 2007 Self-organized Criticality in a Modified Evolution Model on Generalized Barabási

More information

Anastasios Anastasiadis Institute for Space Applications & Remote Sensing National Observatory of Athens GR Penteli, Greece

Anastasios Anastasiadis Institute for Space Applications & Remote Sensing National Observatory of Athens GR Penteli, Greece CELLULAR AUTOMATA MODELS: A SANDPILE MODEL APPLIED IN FUSION Anastasios Anastasiadis Institute for Space Applications & Remote Sensing National Observatory of Athens GR-15236 Penteli, Greece SUMMARY We

More information

Distribution of Residence Times of Grains in Sandpile Models.

Distribution of Residence Times of Grains in Sandpile Models. Distribution of Residence Times of Grains in Sandpile Models. A Thesis Submitted to the Tata Institute of Fundamental Research, Mumbai For the degree of Doctor of Philosophy in Physics by Punyabrata Pradhan

More information

How self-organized criticality works: A unified mean-field picture

How self-organized criticality works: A unified mean-field picture PHYSICAL REVIEW E VOLUME 57, NUMBER 6 JUNE 1998 How self-organized criticality works: A unified mean-field picture Alessandro Vespignani International Centre for Theoretical Physics (ICTP), P.O. Box 586,

More information

arxiv: v1 [cond-mat.stat-mech] 6 Mar 2008

arxiv: v1 [cond-mat.stat-mech] 6 Mar 2008 CD2dBS-v2 Convergence dynamics of 2-dimensional isotropic and anisotropic Bak-Sneppen models Burhan Bakar and Ugur Tirnakli Department of Physics, Faculty of Science, Ege University, 35100 Izmir, Turkey

More information

Criticality in Earthquakes. Good or bad for prediction?

Criticality in Earthquakes. Good or bad for prediction? http://www.pmmh.espci.fr/~oramos/ Osvanny Ramos. Main projects & collaborators Slow crack propagation Cracks patterns L. Vanel, S. Ciliberto, S. Santucci, J-C. Géminard, J. Mathiesen IPG Strasbourg, Nov.

More information

A Simple Model of Evolution with Variable System Size

A Simple Model of Evolution with Variable System Size A Simple Model of Evolution with Variable System Size Claus Wilke and Thomas Martinetz Institut für Neuroinformatik Ruhr-Universität Bochum (Submitted: ; Printed: September 28, 2001) A simple model of

More information

Time correlations in self-organized criticality (SOC)

Time correlations in self-organized criticality (SOC) SMR.1676-8 8th Workshop on Non-Linear Dynamics and Earthquake Prediction 3-15 October, 2005 ------------------------------------------------------------------------------------------------------------------------

More information

Avalanches, transport, and local equilibrium in self-organized criticality

Avalanches, transport, and local equilibrium in self-organized criticality PHYSICAL REVIEW E VOLUME 58, NUMBER 5 NOVEMBER 998 Avalanches, transport, and local equilibrium in self-organized criticality Afshin Montakhab and J. M. Carlson Department of Physics, University of California,

More information

Self-organized criticality and the self-organizing map

Self-organized criticality and the self-organizing map PHYSICAL REVIEW E, VOLUME 63, 036130 Self-organized criticality and the self-organizing map John A. Flanagan Neural Networks Research Center, Helsinki University of Technology, P.O. Box 5400, FIN-02015

More information

arxiv:cond-mat/ v1 [cond-mat.stat-mech] 16 Jan 2004

arxiv:cond-mat/ v1 [cond-mat.stat-mech] 16 Jan 2004 arxiv:cond-mat/0401302v1 [cond-mat.stat-mech] 16 Jan 2004 Abstract Playing with sandpiles Michael Creutz Brookhaven National Laboratory, Upton, NY 11973, USA The Bak-Tang-Wiesenfeld sandpile model provdes

More information

The Sandpile Model on Random Apollonian Networks

The Sandpile Model on Random Apollonian Networks 1 The Sandpile Model on Random Apollonian Networks Massimo Stella Bak, Teng and Wiesenfel originally proposed a simple model of a system whose dynamics spontaneously drives, and then maintains it, at the

More information

Effects of Interactive Function Forms and Refractoryperiod in a Self-Organized Critical Model Based on Neural Networks

Effects of Interactive Function Forms and Refractoryperiod in a Self-Organized Critical Model Based on Neural Networks Commun. Theor. Phys. (Beijing, China) 42 (2004) pp. 121 125 c International Academic Publishers Vol. 42, No. 1, July 15, 2004 Effects of Interactive Function Forms and Refractoryperiod in a Self-Organized

More information

arxiv:cond-mat/ v1 17 Aug 1994

arxiv:cond-mat/ v1 17 Aug 1994 Universality in the One-Dimensional Self-Organized Critical Forest-Fire Model Barbara Drossel, Siegfried Clar, and Franz Schwabl Institut für Theoretische Physik, arxiv:cond-mat/9408046v1 17 Aug 1994 Physik-Department

More information

Extra! Extra! Critical Update on Life. by Rik Blok. for PWIAS Crisis Points

Extra! Extra! Critical Update on Life. by Rik Blok. for PWIAS Crisis Points Extra! Extra! Critical Update on Life by Rik Blok for PWIAS Crisis Points March 18, 1998 40 min. Self-organized Criticality (SOC) critical point: under variation of a control parameter, an order parameter

More information

Criticality on Rainfall: Statistical Observational Constraints for the Onset of Strong Convection Modelling

Criticality on Rainfall: Statistical Observational Constraints for the Onset of Strong Convection Modelling Criticality on Rainfall: Statistical Observational Constraints for the Onset of Strong Convection Modelling Anna Deluca, Álvaro Corral, and Nicholas R. Moloney 1 Introduction A better understanding of

More information

arxiv:cond-mat/ v1 [cond-mat.stat-mech] 16 Dec 1997

arxiv:cond-mat/ v1 [cond-mat.stat-mech] 16 Dec 1997 arxiv:cond-mat/9712183v1 [cond-mat.stat-mech] 16 Dec 1997 Sandpiles on a Sierpinski gasket Frank Daerden, Carlo Vanderzande Departement Wiskunde Natuurkunde Informatica Limburgs Universitair Centrum 3590

More information

Supplementary Information: The origin of bursts and heavy tails in human dynamics

Supplementary Information: The origin of bursts and heavy tails in human dynamics Supplementary Information: The origin of bursts and heavy tails in human dynamics Albert-László Barabási Department of Physics, University of Notre Dame, IN 46556, USA (Dated: February 7, 2005) 1 Contents

More information

Self-Organization in Models of Sandpiles, Earthquakes, and Flashing Fireflies.

Self-Organization in Models of Sandpiles, Earthquakes, and Flashing Fireflies. Self-Organization in Models of Sandpiles, Earthquakes, and Flashing Fireflies. Kim Christensen Institute of Physics and Astronomy University of Aarhus DK - 8000 Aarhus C Denmark Present address: Department

More information

L. E. Aragón et al. f pin

L. E. Aragón et al. f pin epl draft Avalanches in Tip-Driven Interfaces in Random Media arxiv:1510.06795v1 [cond-mat.dis-nn] 23 Oct 2015 L.E. Aragón, A.B. Kolton 1, P. Le Doussal, K.J. Wiese 2 and E.A. Jagla 1 1 CONICET - Centro

More information

arxiv:cond-mat/ v1 [cond-mat.stat-mech] 3 May 2000

arxiv:cond-mat/ v1 [cond-mat.stat-mech] 3 May 2000 Different hierarchy of avalanches observed in Bak-Sneppen evolution model arxiv:cond-mat/0005067v1 [cond-mat.stat-mech] 3 May 2000 W. Li 1, and X. Cai 1,2,3, 1 Institute of Particle Physics, Hua-zhong

More information

Spatial and Temporal Behaviors in a Modified Evolution Model Based on Small World Network

Spatial and Temporal Behaviors in a Modified Evolution Model Based on Small World Network Commun. Theor. Phys. (Beijing, China) 42 (2004) pp. 242 246 c International Academic Publishers Vol. 42, No. 2, August 15, 2004 Spatial and Temporal Behaviors in a Modified Evolution Model Based on Small

More information

CHARACTERISING THE FAILURE AND REPOSE ANGLES OF IRREGULARLY SHAPED THREE-DIMENSIONAL PARTICLES USING DEM

CHARACTERISING THE FAILURE AND REPOSE ANGLES OF IRREGULARLY SHAPED THREE-DIMENSIONAL PARTICLES USING DEM Ninth International Conference on CFD in the Minerals and Process Industries CSIRO, Melbourne, Australia 10-12 December 2012 CHARACTERISING THE FAILURE AND REPOSE ANGLES OF IRREGULARLY SHAPED THREE-DIMENSIONAL

More information

arxiv:cond-mat/ v1 [cond-mat.stat-mech] 27 May 1999

arxiv:cond-mat/ v1 [cond-mat.stat-mech] 27 May 1999 Bubbling and Large-Scale Structures in Avalanche Dynamics arxiv:cond-mat/9905404v1 [cond-mat.stat-mech] 27 May 1999 Supriya Krishnamurthy a, Vittorio Loreto a and Stéphane Roux b a) Laboratoire Physique

More information

Self-organized criticality as an absorbing-state phase transition

Self-organized criticality as an absorbing-state phase transition PHYSICAL REVIEW E VOLUME 57, NUMBER 5 MAY 1998 Self-organized criticality as an absorbing-state phase transition Ronald Dickman, 1, * Alessandro Vespignani, 2, and Stefano Zapperi 3, 1 Department of Physics

More information

arxiv:cond-mat/ v1 [cond-mat.stat-mech] 27 Mar 2002

arxiv:cond-mat/ v1 [cond-mat.stat-mech] 27 Mar 2002 Sandpiles with height restrictions arxiv:cond-mat/0203565v1 [cond-mat.stat-mech] 27 Mar 2002 Ronald Dickman 1,, Tânia Tomé 1, and Mário J. de Oliveira 2 1 Departamento de Física, ICEx, Universidade Federal

More information

Absorbing-state phase transitions in fixed-energy sandpiles

Absorbing-state phase transitions in fixed-energy sandpiles PHYSICAL REVIEW E VOLUME 62, NUMBER 4 OCTOBER 2000 Absorbing-state phase transitions in fixed-energy sandpiles Alessandro Vespignani, 1 Ronald Dickman, 2 Miguel A. Muñoz, 3 and Stefano Zapperi 4 1 The

More information

Granular Micro-Structure and Avalanche Precursors

Granular Micro-Structure and Avalanche Precursors Granular Micro-Structure and Avalanche Precursors L. Staron, F. Radjai & J.-P. Vilotte Department of Applied Mathematics and Theoretical Physics, Cambridge CB3 0WA, UK. Laboratoire de Mécanique et Génie

More information

Simple models for complex systems toys or tools? Katarzyna Sznajd-Weron Institute of Theoretical Physics University of Wrocław

Simple models for complex systems toys or tools? Katarzyna Sznajd-Weron Institute of Theoretical Physics University of Wrocław Simple models for complex systems toys or tools? Katarzyna Sznajd-Weron Institute of Theoretical Physics University of Wrocław Agenda: Population dynamics Lessons from simple models Mass Extinction and

More information

Dynamical Synapses Give Rise to a Power-Law Distribution of Neuronal Avalanches

Dynamical Synapses Give Rise to a Power-Law Distribution of Neuronal Avalanches Dynamical Synapses Give Rise to a Power-Law Distribution of Neuronal Avalanches Anna Levina 3,4, J. Michael Herrmann 1,2, Theo Geisel 1,2,4 1 Bernstein Center for Computational Neuroscience Göttingen 2

More information

Applying a cellular automaton model to describing the anomalous relaxation of the metastable states in the disordered porous media

Applying a cellular automaton model to describing the anomalous relaxation of the metastable states in the disordered porous media Journal of Physics: Conference Series PAPER OPEN ACCESS Applying a cellular automaton model to describing the anomalous relaxation of the metastable states in the disordered porous media o cite this article:

More information

Avalanche dynamics, surface roughening, and self-organized criticality: Experiments on a three-dimensional pile of rice

Avalanche dynamics, surface roughening, and self-organized criticality: Experiments on a three-dimensional pile of rice Avalanche dynamics, surface roughening, and self-organized criticality: Experiments on a three-dimensional pile of rice C. M. Aegerter, R. Günther, and R. J. Wijngaarden Division of Physics and Astronomy,

More information

Interface depinning versus absorbing-state phase transitions

Interface depinning versus absorbing-state phase transitions Powered by TCPDF (www.tcpdf.org) This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Author(s): Alava, Mikko J. & Muñoz,

More information

Renormalization approach to the self-organized critical behavior of sandpile models

Renormalization approach to the self-organized critical behavior of sandpile models PHYSICAL REVIEW E VOLUME 51, NUMBER 3 MARCH 1995 Renormalization approach to the self-organized critical behavior of sandpile models Alessandro Vespignani,I,2 Stefano Zapperi,I,3 and Luciano Pietronero

More information

Self-organized Criticality and Synchronization in a Pulse-coupled Integrate-and-Fire Neuron Model Based on Small World Networks

Self-organized Criticality and Synchronization in a Pulse-coupled Integrate-and-Fire Neuron Model Based on Small World Networks Commun. Theor. Phys. (Beijing, China) 43 (2005) pp. 466 470 c International Academic Publishers Vol. 43, No. 3, March 15, 2005 Self-organized Criticality and Synchronization in a Pulse-coupled Integrate-and-Fire

More information

A Modified Earthquake Model Based on Generalized Barabási Albert Scale-Free

A Modified Earthquake Model Based on Generalized Barabási Albert Scale-Free Commun. Theor. Phys. (Beijing, China) 46 (2006) pp. 1011 1016 c International Academic Publishers Vol. 46, No. 6, December 15, 2006 A Modified Earthquake Model Based on Generalized Barabási Albert Scale-Free

More information

Scale-invariant behavior in a spatial game of prisoners dilemma

Scale-invariant behavior in a spatial game of prisoners dilemma PHYSICAL REVIEW E, VOLUME 65, 026134 Scale-invariant behavior in a spatial game of prisoners dilemma Y. F. Lim and Kan Chen Department of Computational Science, National University of Singapore, Singapore

More information

ON SELF-ORGANIZED CRITICALITY AND SYNCHRONIZATION IN LATTICE MODELS OF COUPLED DYNAMICAL SYSTEMS

ON SELF-ORGANIZED CRITICALITY AND SYNCHRONIZATION IN LATTICE MODELS OF COUPLED DYNAMICAL SYSTEMS International Journal of Modern Physics B, c World Scientific Publishing Company ON SELF-ORGANIZED CRITICALITY AND SYNCHRONIZATION IN LATTICE MODELS OF COUPLED DYNAMICAL SYSTEMS CONRAD J. PÉREZ, ÁLVARO

More information

Effects of Interactive Function Forms in a Self-Organized Critical Model Based on Neural Networks

Effects of Interactive Function Forms in a Self-Organized Critical Model Based on Neural Networks Commun. Theor. Phys. (Beijing, China) 40 (2003) pp. 607 613 c International Academic Publishers Vol. 40, No. 5, November 15, 2003 Effects of Interactive Function Forms in a Self-Organized Critical Model

More information

Quasi-Stationary Simulation: the Subcritical Contact Process

Quasi-Stationary Simulation: the Subcritical Contact Process Brazilian Journal of Physics, vol. 36, no. 3A, September, 6 685 Quasi-Stationary Simulation: the Subcritical Contact Process Marcelo Martins de Oliveira and Ronald Dickman Departamento de Física, ICEx,

More information

arxiv:cond-mat/ v1 10 Jul 1996

arxiv:cond-mat/ v1 10 Jul 1996 Self - organized - criticality and synchronization in pulse coupled relaxation oscillator systems; the Olami, Feder and Christensen and the Feder and Feder model arxiv:cond-mat/9607069v1 10 Jul 1996 Samuele

More information

1/ f noise and self-organized criticality

1/ f noise and self-organized criticality 1/ f noise and self-organized criticality Lecture by: P. H. Diamond, Notes by: Y. Zhang June 11, 2016 1 Introduction Until now we have explored the intermittent problem, from which the multiplicative process

More information

Asynchronous updating of threshold-coupled chaotic neurons

Asynchronous updating of threshold-coupled chaotic neurons PRAMANA c Indian Academy of Sciences Vol. 70, No. 6 journal of June 2008 physics pp. 1127 1134 Asynchronous updating of threshold-coupled chaotic neurons MANISH DEV SHRIMALI 1,2,3,, SUDESHNA SINHA 4 and

More information

Self-organized Criticality and its implication to brain dynamics. Jaeseung Jeong, Ph.D Department of Bio and Brain Engineering, KAIST

Self-organized Criticality and its implication to brain dynamics. Jaeseung Jeong, Ph.D Department of Bio and Brain Engineering, KAIST Self-organized Criticality and its implication to brain dynamics Jaeseung Jeong, Ph.D Department of Bio and Brain Engineering, KAIST Criticality or critical points Criticality indicates the behavior of

More information

Computational Mechanics of the Two Dimensional BTW Model

Computational Mechanics of the Two Dimensional BTW Model Computational Mechanics of the Two Dimensional BTW Model Rajesh Kommu kommu@physics.ucdavis.edu June 8, 2010 Abstract Some aspects of computational mechanics in two dimensions are investigated in this

More information

WHAT IS a sandpile? Lionel Levine and James Propp

WHAT IS a sandpile? Lionel Levine and James Propp WHAT IS a sandpile? Lionel Levine and James Propp An abelian sandpile is a collection of indistinguishable chips distributed among the vertices of a graph. More precisely, it is a function from the vertices

More information

arxiv:cond-mat/ v1 [cond-mat.stat-mech] 29 Apr 2002

arxiv:cond-mat/ v1 [cond-mat.stat-mech] 29 Apr 2002 N-Site approximations and CAM analysis for a stochastic sandpile arxiv:cond-mat/0204608v1 [cond-mat.stat-mech] 29 Apr 2002 Ronald Dickman Departamento de Fí sica, ICEx, Universidade Federal de Minas Gerais,

More information

AVALANCHES IN A NUMERICALLY SIMULATED SAND DUNE DYNAMICS

AVALANCHES IN A NUMERICALLY SIMULATED SAND DUNE DYNAMICS Fractals, Vol. 11, No. 2 (2003) 183 193 c World Scientific Publishing Company AVALANCHES IN A NUMERICALLY SIMULATED SAND DUNE DYNAMICS B. S. DAYA SAGAR,,, M. B. R. MURTHY and P. RADHAKRISHNAN Faculty of

More information

Building blocks of self-organized criticality, part II: transition from very low drive to high drive

Building blocks of self-organized criticality, part II: transition from very low drive to high drive Building blocks of self-organized criticality, part II: transition from very low to high Ryan Woodard and David E. Newman University of Alaska Fairbanks Fairbanks, Alaska 99775-5920, USA Raúl Sánchez Universidad

More information

On the Damage-Spreading in the Bak-Sneppen Model. Max-Planck-Institut fur Physik komplexer Systeme, Nothnitzer Str. 38, D Dresden.

On the Damage-Spreading in the Bak-Sneppen Model. Max-Planck-Institut fur Physik komplexer Systeme, Nothnitzer Str. 38, D Dresden. On the Damage-Spreading in the Bak-Sneppen Model Angelo Valleriani a and Jose Luis Vega b Max-Planck-Institut fur Physik komplexer Systeme, Nothnitzer Str. 38, D-01187 Dresden. We explain the results recently

More information

Granular Matter and the Marginal Rigidity State.

Granular Matter and the Marginal Rigidity State. 1 Granular Matter and the Marginal Rigidity State. R Blumenfeld*, SF Edwards and RC Ball*, *Department of Physics, University of Warwick, CV4 7AL Coventry, UK. Polymers and Colloids, Cavendish Laboratory,

More information

The nuclear fragmentation problem and Bormio's contribution to its solution

The nuclear fragmentation problem and Bormio's contribution to its solution The nuclear fragmentation problem and Bormio's contribution to its solution 1 Department of Physics and Astronomy, Michigan State University and National Superconducting Cyclotron Laboratory 567 Wilson

More information

Dynamics of a shallow fluidized bed

Dynamics of a shallow fluidized bed PHYSICAL REVIEW E VOLUME 60, NUMBER 6 DECEMBER 1999 Dynamics of a shallow fluidized bed Lev S. Tsimring, 1 Ramakrishna Ramaswamy, 2 and Philip Sherman 1 1 Institute for Nonlinear Science, University of

More information

Invariant measures and limiting shapes in sandpile models. Haiyan Liu

Invariant measures and limiting shapes in sandpile models. Haiyan Liu Invariant measures and limiting shapes in sandpile models Haiyan Liu c H. Liu, Amsterdam 2011 ISBN: 9789086595259 Printed by Ipskamp Drukkers, Enschede, The Netherlands VRIJE UNIVERSITEIT Invariant measures

More information

arxiv:cond-mat/ v1 [cond-mat.stat-mech] 23 Feb 2004

arxiv:cond-mat/ v1 [cond-mat.stat-mech] 23 Feb 2004 arxiv:cond-mat/42564v1 [cond-mat.stat-mech] 23 Feb 24 Exact solution of the totally asymmetric Oslo model Gunnar Pruessner Department of Mathematics, Imperial College London, 18 Queen s Gate, London SW7

More information

Self-Organized Criticality (SOC) Tino Duong Biological Computation

Self-Organized Criticality (SOC) Tino Duong Biological Computation Self-Organized Criticality (SOC) Tino Duong Biological Computation Agenda Introduction Background material Self-Organized Criticality Defined Examples in Nature Experiments Conclusion SOC in a Nutshell

More information

arxiv:cond-mat/ v2 [cond-mat.stat-mech] 9 Mar 1998

arxiv:cond-mat/ v2 [cond-mat.stat-mech] 9 Mar 1998 The Anisotropic Bak Sneppen Model D A Head Institute of Physical and Environmental Sciences, Brunel University, Uxbridge, Middlesex, UB8 3PH, United Kingdom arxiv:cond-mat/9802028v2 [cond-mat.stat-mech]

More information

Flux noise resulting from vortex avalanches using a simple kinetic model

Flux noise resulting from vortex avalanches using a simple kinetic model PHYSICAL REVIEW B VOLUME 60, NUMBER 13 1 OCTOBER 1999-I Flux noise resulting from vortex avalanches using a simple kinetic model G. Mohler and D. Stroud Department of Physics, The Ohio State University,

More information

Sandpiles. Chapter Model definition

Sandpiles. Chapter Model definition Chapter 5 Sandpiles {chap:sandpile The sky is blue, the sun is high, and you are sitting idle on a beach, a cold beer in one hand and a handful of dry sand in the other. Sand is slowly trickling through

More information

arxiv:chao-dyn/ v1 5 Mar 1996

arxiv:chao-dyn/ v1 5 Mar 1996 Turbulence in Globally Coupled Maps M. G. Cosenza and A. Parravano Centro de Astrofísica Teórica, Facultad de Ciencias, Universidad de Los Andes, A. Postal 26 La Hechicera, Mérida 5251, Venezuela (To appear,

More information

A model for the transmission of contact forces in granular piles

A model for the transmission of contact forces in granular piles Author: Javier Cristín. Facultat de Física, Universitat de Barcelona, Diagonal 645, 08028 Barcelona, Spain Advisor: Carmen Miguel. Abstract:Granular matter is fundamentally different from other, more conventional,

More information

Self-organized scale-free networks

Self-organized scale-free networks Self-organized scale-free networks Kwangho Park and Ying-Cheng Lai Departments of Electrical Engineering, Arizona State University, Tempe, Arizona 85287, USA Nong Ye Department of Industrial Engineering,

More information

A damped pendulum forced with a constant torque

A damped pendulum forced with a constant torque A damped pendulum forced with a constant torque P. Coullet, J.M. Gilli, M. Monticelli, and N. Vandenberghe Institut Non Linéaire de Nice, UMR 128 CNRS - UNSA, 1361 Route des Lucioles, 06560, Valbonne,

More information

arxiv: v3 [cond-mat.stat-mech] 16 Dec 2007

arxiv: v3 [cond-mat.stat-mech] 16 Dec 2007 arxiv:0711.2732v3 [cond-mat.stat-mech] 16 Dec 2007 The angle of repose of spherical grains in granular Hele-Shaw cells: A molecular dynamics study Hamed Maleki Physics department, University of Birjand,

More information

Dry granular flows: gas, liquid or solid?

Dry granular flows: gas, liquid or solid? Dry granular flows: gas, liquid or solid? Figure 1: Forterre & Pouliquen, Annu. Rev. Fluid Mechanics, 2008 1 Characterizing size and size distribution Grains are not uniform (size, shape, ) Statistical

More information

A Sandpile to Model the Brain

A Sandpile to Model the Brain A Sandpile to Model the Brain Kristina Rydbeck Kandidatuppsats i matematisk statistik Bachelor Thesis in Mathematical Statistics Kandidatuppsats 2017:17 Matematisk statistik Juni 2017 www.math.su.se Matematisk

More information

Self-organized criticality and observable features of avalanching systems. Michal Bregman

Self-organized criticality and observable features of avalanching systems. Michal Bregman Self-organized criticality and observable features of avalanching systems Michal Bregman October 7, 2005 Contents 1 Introduction 6 1.1 Characterization of the SOC state......................... 7 1.2 Numerical

More information

Minimal Model Study for ELM Control by Supersonic Molecular Beam Injection and Pellet Injection

Minimal Model Study for ELM Control by Supersonic Molecular Beam Injection and Pellet Injection 25 th Fusion Energy Conference, Saint Petersburg, Russia, 2014 TH/P2-9 Minimal Model Study for ELM Control by Supersonic Molecular Beam Injection and Pellet Injection Tongnyeol Rhee 1,2, J.M. Kwon 1, P.H.

More information

On a self-organized critical forest-fire model

On a self-organized critical forest-fire model J. Phys. A Math. Gen. 26 (1993) 2081-2089. Printed in the UK On a self-organized critical forest-fire model Peter Grassberger Physics Departmen4 University of Wuppertal. D-5~600 Wuppertsl 1, Federal Republic

More information

A sandpile experiment and its implications for self-organized criticality and characteristic earthquake

A sandpile experiment and its implications for self-organized criticality and characteristic earthquake Earth Planets Space, 55, 283 289, 2003 A sandpile experiment and its implications for self-organized criticality and characteristic earthquake Naoto Yoshioka Graduate School of Integrated Science, Yokohama

More information

arxiv: v1 [cond-mat.stat-mech] 16 Nov 2016

arxiv: v1 [cond-mat.stat-mech] 16 Nov 2016 A traffic model with an absorbing-state phase transition M. L. L. Iannini, and Ronald Dickman, arxiv:1611.05307v1 [cond-mat.stat-mech] 16 Nov 2016 Departamento de Física and National Institute of Science

More information

Theoretical studies of self-organized criticality

Theoretical studies of self-organized criticality Physica A 369 (2006) 29 70 www.elsevier.com/locate/physa Theoretical studies of self-organized criticality Deepak Dhar Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha

More information

arxiv: v2 [cond-mat.stat-mech] 24 Aug 2014

arxiv: v2 [cond-mat.stat-mech] 24 Aug 2014 Hyperuniformity of critical absorbing states Daniel Hexner and Dov Levine, Department of Physics, Technion, Haifa, Israel Initiative for the Theoretical Sciences - CUNY Graduate Center 65 Fifth Avenue,

More information

Tectonophysics 485 (2010) Contents lists available at ScienceDirect. Tectonophysics. journal homepage:

Tectonophysics 485 (2010) Contents lists available at ScienceDirect. Tectonophysics. journal homepage: Tectonophysics 485 (2010) 321 326 Contents lists available at ScienceDirect Tectonophysics journal homepage: www.elsevier.com/locate/tecto Short Communication Criticality in earthquakes. Good or bad for

More information

Nonlinear Dynamical Behavior in BS Evolution Model Based on Small-World Network Added with Nonlinear Preference

Nonlinear Dynamical Behavior in BS Evolution Model Based on Small-World Network Added with Nonlinear Preference Commun. Theor. Phys. (Beijing, China) 48 (2007) pp. 137 142 c International Academic Publishers Vol. 48, No. 1, July 15, 2007 Nonlinear Dynamical Behavior in BS Evolution Model Based on Small-World Network

More information

Lecture 6: Flow regimes fluid-like

Lecture 6: Flow regimes fluid-like Granular Flows 1 Lecture 6: Flow regimes fluid-like Quasi-static granular flows have plasticity laws, gaseous granular flows have kinetic theory -- how to model fluid-like flows? Intermediate, dense regime:

More information

Potential and Kinetic Energy: Roller Coasters Student Version

Potential and Kinetic Energy: Roller Coasters Student Version Potential and Kinetic Energy: Roller Coasters Student Version Key Concepts: Energy is the ability of a system or object to perform work. It exists in various forms. Potential energy is the energy an object

More information

A self-organized criticality model for the magnetic field in toroidal confinement devices

A self-organized criticality model for the magnetic field in toroidal confinement devices A self-organied criticality model for the magnetic field in toroidal confinement devices Hein Isliker Dept. of Physics University of Thessaloniki In collaboration with Loukas Vlahos Sani Beach, Chalkidiki,

More information

arxiv:cond-mat/ v1 [cond-mat.stat-mech] 20 Dec 1996

arxiv:cond-mat/ v1 [cond-mat.stat-mech] 20 Dec 1996 Sensitivity to initial conditions and nonextensivity in biological arxiv:cond-mat/9612196v1 [cond-mat.stat-mech] 20 Dec 1996 evolution Francisco A. Tamarit, Sergio A. Cannas Facultad de Matemática, Astronomía

More information

Any live cell with less than 2 live neighbours dies. Any live cell with 2 or 3 live neighbours lives on to the next step.

Any live cell with less than 2 live neighbours dies. Any live cell with 2 or 3 live neighbours lives on to the next step. 2. Cellular automata, and the SIRS model In this Section we consider an important set of models used in computer simulations, which are called cellular automata (these are very similar to the so-called

More information

Branching Process Approach to Avalanche Dynamics on Complex Networks

Branching Process Approach to Avalanche Dynamics on Complex Networks Journal of the Korean Physical Society, Vol. 44, No. 3, March 2004, pp. 633 637 Branching Process Approach to Avalanche Dynamics on Complex Networks D.-S. Lee, K.-I. Goh, B. Kahng and D. Kim School of

More information

Coarsening process in the 2d voter model

Coarsening process in the 2d voter model Alessandro Tartaglia (LPTHE) Coarsening in the 2d voter model May 8, 2015 1 / 34 Coarsening process in the 2d voter model Alessandro Tartaglia LPTHE, Université Pierre et Marie Curie alessandro.tartaglia91@gmail.com

More information

Controlling chaos in random Boolean networks

Controlling chaos in random Boolean networks EUROPHYSICS LETTERS 20 March 1997 Europhys. Lett., 37 (9), pp. 597-602 (1997) Controlling chaos in random Boolean networks B. Luque and R. V. Solé Complex Systems Research Group, Departament de Fisica

More information

Myllys, M.; Maunuksela, J.; Alava, Mikko; Ala-Nissilä, Tapio; Timonen, J. Scaling and noise in slow combustion of paper

Myllys, M.; Maunuksela, J.; Alava, Mikko; Ala-Nissilä, Tapio; Timonen, J. Scaling and noise in slow combustion of paper Powered by TCPDF (www.tcpdf.org) This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Myllys, M.; Maunuksela, J.; Alava,

More information

Theory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute Of Technology, Madras

Theory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute Of Technology, Madras Theory of Computation Prof. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute Of Technology, Madras Lecture No. # 25 Problems and Solutions (Refer Slide Time: 00:16) Today,

More information

Epidemic dynamics and endemic states in complex networks

Epidemic dynamics and endemic states in complex networks PHYSICAL REVIEW E, VOLUME 63, 066117 Epidemic dynamics and endemic states in complex networks Romualdo Pastor-Satorras 1 and Alessandro Vespignani 2 1 Departmento de Física i Enginyeria Nuclear, Universitat

More information

arxiv:cond-mat/ v2 [cond-mat.stat-mech] 3 Oct 2005

arxiv:cond-mat/ v2 [cond-mat.stat-mech] 3 Oct 2005 Growing Directed Networks: Organization and Dynamics arxiv:cond-mat/0408391v2 [cond-mat.stat-mech] 3 Oct 2005 Baosheng Yuan, 1 Kan Chen, 1 and Bing-Hong Wang 1,2 1 Department of Computational cience, Faculty

More information

Modeling and Visualization of Emergent Behavior in Complex Geophysical Systems for Research and Education

Modeling and Visualization of Emergent Behavior in Complex Geophysical Systems for Research and Education Modeling and Visualization of Emergent Behavior in Complex Geophysical Systems for Research and Education NATALIA A. SMIRNOVA, VADIM M. URITSKY Earth Physics Department St.Petersburg State University Ulyanovskaya

More information