Self-organized criticality - mit.edu

1 Self-organized criticalityVladyslav A. GolykMassachusetts Institute of Technology, Department of Physics, Cambridge, Massachusetts 02139, USAWe study the concept of the Self-organized criticality (SOC) and its application to a wide rangeof scientific problems with very different backgrounds. In particular, we discuss the Bak-Tang-Wiesenfeld sandpile model which displays SOC behavior and by computing the critical exponentfor the two-dimensional model we find the agreement with the known result. Finally, we providea new example of Zipf s law and discuss the connection of power-laws found in nature to the numbers:I.

2 INTRODUCTIONThe discovery of the Self-organized criticality (SOC)is one of ground-breaking achievements of statisticalphysics in the last couple of decades. Self-organized crit-icality is a very rich phenomenon as it combines self-organization and criticality to describe complexity. Thisconcept was first introduced by P. Bak and the collabora-tors in the seminal paper in 1987[1]. SOC is a property ofdynamical systems to organize its microscopic behaviorto be spatial (and/or temporal) scale independent.

3 Thatresembles of the critical behavior of the critical pointof phase transitions. However, in contrast to the usualphase transitions, the systems displaying SOC do not re-quire external tuning of the control parameters, evolutions, organizes itselfinto the , the dynamical system organizes itself into a statewith complex, but rather general structure. Complexityarises in the sense that no single characteristic event sizeexists, no scale present to guide the system s evo-lution. Despite the complexity, system exhibits simplestatistical properties governed by power laws.

4 For exam-ple, the number of eventsDas a function of its sizes(where a big event is less likely to happen compared to asmaller one) is given by,D(s) =As ,(1)whereAis some constant and is the exponent describ-ing statistical features of a SOC state. Remarkably, someof the exponents can be same for systems with very dif-ferent microscopic is typically observed in slowly-driven non-equilibrium systems with extended degrees of freedomand a high level of non-linearity. Many individual ex-amples have been identified since Bak s original paper,but to date there is no known set of general characteris-tics that guarantee a system to display SOC.

5 Phenomenaof strikingly different backgrounds were claimed to ex-hibit SOC behavior: sandpiles, earthquakes, forest fires,rivers, mountains, cities, literary texts, electric break-down, motion of magnetic flux lines in superconductors,water droplets on surfaces, dynamics of magnetic do-mains, growing surfaces, human brains, etc.[2]. The com-mon feature for all mentioned systems is that the tem-poral and/or spatial power-law correlations extend overseveral decades where intuitively one may anticipate thatthe physical laws would vary observations of power-law distributions ofspatially-extended objects have triggered a need fora theoretical explanation.

6 Unfortunately, no unifyingmathematical formalism has been elaborated so far andit appears unclear how to identify whether a given sys-tem displays SOC behavior or it is something worse, there exists no generally accepted definitionof , there exist a few mathematical mod-els which seem to display SOC behavior: Bak-Tang-Wiesenfeld sandpile, Olami-Feder-Christensen earth-quake model, Lattice Gas model, Critical Forest Firemodel, etc. In the present work we describe the originalBTW sandpile model which is the first discovered exam-ple of a dynamical system displaying SOC[1].

7 We per-form computations for the two-dimensional model usingparameters different from the original one. We obtain anexponent which is in a good agreement with the knownone, thus again indicating that SOC state is achievedwithout any need of parameters , we find another system exhibiting power-lawstatistical properties. It appears, that distribution ofskyscrapers heights obeys power-law. Finally, we discussthe open question which systems exhibiting power-lawcharacteristics can be considered to exhibit SOC BAK-TANG-WIESENFELD SANDPILEMODELC onsider a flat table of finite size with one grain ofsand added per unit time interval so that the system hasenough time to equilibrate before the next grain dropsdown.

8 The grains can be added either randomly or atsome fixed position of the table. As a result of frictionbetween the grains the system does not automaticallyequilibrate to a ground state of flat height profile. Ini-tially, the grains are most likely to stay at the same placeswhere they landed, however as we carry on adding moresand, the height profile becomes steeper and small sand2slides or avalanches can occur. If the grain lands on topof other grain it may topple to a lower level overcomingfriction due to gravity.

9 This toppling causes local dis-turbance which does not affect the large-scale picture, there are no correlations between distant parts of thesandpile. However, as the slope increases, a single grainis more likely to cause other grains to topple and eventu-ally the slope reaches a certain maximal value when theamounts of sand being added and falling off the edgesare balanced. Clearly, now the local dynamics no longergoverns the process and the avalanches span the entiresystem leading to complexity.

10 This is the SOC state withits own complex emergent dynamics which cannot be de-scribed by local dynamics laws. That is why it is naturalto expect that SOC state is robust to the modificationsof the systems, which is the crucial requirement for SOCto describe real world. For example, by changing the sizeof our system (as long as it still stays large), by addingdifferent barriers on the table, by adding some amountof wet sand, the critical state dynamics stays exactly thesame. That has been demonstrated on the example ofthe Bak-Tang-Wiesenfeld (BTW) sandpile show how the BTW sandpile model works, we con-sider a 2D flat surface defined byz(x,y) = 0 for allxandy(again, this initial condition does not affect finalself-organized critical behavior) and start adding a grainof sand at a random position (x,y):z(x,y) z(x,y) + 1.