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Symmetries and fixed point stability of stochastic differential equations modeling self-organized criticality
Corral, Álvaro; Díaz Guilera, Albert
Universitat de Barcelona
A stochastic nonlinear partial differential equation is constructed for two different models exhibiting self-organized criticality: the Bak-Tang-Wiesenfeld (BTW) sandpile model [Phys. Rev. Lett. 59, 381 (1987); Phys. Rev. A 38, 364 (1988)] and the Zhang model [Phys. Rev. Lett. 63, 470 (1989)]. The dynamic renormalization group (DRG) enables one to compute the critical exponents. However, the nontrivial stable fixed point of the DRG transformation is unreachable for the original parameters of the models. We introduce an alternative regularization of the step function involved in the threshold condition, which breaks the symmetry of the BTW model. Although the symmetry properties of the two models are different, it is shown that they both belong to the same universality class. In this case the DRG procedure leads to a symmetric behavior for both models, restoring the broken symmetry, and makes accessible the nontrivial fixed point. This technique could also be applied to other problems with threshold dynamics.
Física estadística
Sistemes no lineals
Propietats magnètiques
Equacions d'estat
Regla de les fases i equilibri
Transformacions de fase (Física estadística)
Equacions diferencials estocàstiques
Statistical physics
Nonlinear systems
Magnetic properties
Equations of state
Phase rule and equilibrium
Phase transformations (Statistical physics)
Stochastic differential equations
(c) American Physical Society, 1997
The American Physical Society

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