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Two-component Abelian sandpile models.

F C Alcaraz1, P Pyatov, V Rittenberg

  • 1Instituto de Física de São Carlos, Universidade de São Paulo, Caixa Postal 369, 13560-590 São Carlos, São Paulo, Brazil. alcaraz@if.sc.usp.br

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Summary
This summary is machine-generated.

Multicomponent Abelian sandpile models reveal nonlinear toppling probability relations due to associativity. Studies show two-component models with conservation laws exhibit only trivial avalanches.

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Area of Science:

  • Complex systems
  • Statistical mechanics
  • Algebraic structures

Background:

  • One-component Abelian sandpile models feature independent toppling probabilities.
  • Multicomponent models introduce dependencies and nonlinear relations.
  • Associativity of Abelian algebras is key to understanding these relations.

Purpose of the Study:

  • To derive nonlinear relations among toppling probabilities in multicomponent Abelian sandpile models.
  • To investigate the impact of associativity on toppling dynamics.
  • To analyze Abelian sandpile models with two conservation laws.

Main Methods:

  • Derivation of nonlinear relations based on the associativity condition.
  • Focus on two-component quadratic Abelian algebras.
  • Analysis of avalanche behavior in models with two conservation laws.

Main Results:

  • Nonlinear relations among toppling probabilities are established for two-component quadratic Abelian algebras.
  • The associativity condition is shown to impose these constraints.
  • Abelian sandpile models with two conservation laws result in trivial avalanches.

Conclusions:

  • The associativity of Abelian algebras fundamentally alters toppling probabilities in multicomponent sandpile models.
  • Two-component models with conservation laws are constrained to exhibit only simple, trivial avalanches.
  • This research clarifies the dynamics of complex sandpile systems.