Sandpil model under rotational constraint: Scaling universality and crossover
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2012
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Abstract
Sandpile model introduced by Bak, Tang and Wiesenfeld (BTW) was taken as paradigm to study Self Organized Criticality (SOC), a phenomena of non-existence of any characteristic size or time of events in the non-equilibrium steady state of a class of slowly driven system. The effect of rotational constraint on the toppling dynamics of a sandpile model is studied here constructing a rotational sandpile model (RSM). RSM has deterministic toppling rule except the first toppling and develops internal stochasticity during time evolution. There exists a well defined prescription to study critical behaviour in equilibrium critical phenomena. However, nonequilibrium models generally lack such a well defined method. The critical properties of sandpile models are usually studied by calculating the probability distributions of different avalanche properties and performing moment analysis of them. In this thesis, the critical avalanche properties of RSM are studied not only by applying usual numerical techniques but also inventing several new techniques. RSM is found to belong to a new universality class different from that of a stochastic sandpile model (SSM) as well as that of the deterministic BTW model. It was known that the BTW model follows a complicated multi-scaling whereas the SSM follows finite size scaling (FSS). The finite size effect on the critical exponents of the RSM is verified performing moment analysis of the distribution functions. It is found that they follow usual FSS ansatz. Not only the RSM but different variants of the RSM constructed with changes in the toppling rule of RSM under the same rotational symmetry are found to belong to the same universality class of RSM. The RSM universality class seems to be a robust universality class which remains unaffected due to the change in the details of the toppling rule. The existence of such a robust universality class has been confirmed and analyzed further applying the newer techniques invented in this thesis. These newer techniques are based on a DmicroscopicD avalanche parameter namely the toppling number Si, the number of times a site i toppled in an avalanche. The multifractal aspect of an avalanche is explored defining a multifractal measure, the toppling number density Di = Si/ P Si. Toppling number density is found to...
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Supervisor: S. B. Santra
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PHYSICS