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Time-Inhomogeneous Fokker-Planck Equation for Wave Distributions in the Abelian Sandpile Model
1Institute for Theoretical Physics, University of Stellenbosch, Private Bag X1, 7602 Matieland, South Africa and and Institute of Atomic Physics, INFLPR, Lab 22, P.O. Box MG-36 R76900, Bucharest, Romania.
This study links Abelian sandpile model dynamics to Fokker-Planck equations, estimating critical exponents and dimensions for loop-erased random walks. Findings reveal avalanche size distributions as differences between power laws.
Area of Science:
- Statistical Mechanics
- Complex Systems
- Mathematical Physics
Background:
- The Abelian sandpile model exhibits critical phenomena relevant to self-organized criticality.
- Understanding avalanche dynamics and critical exponents is crucial for complex systems research.
Purpose of the Study:
- To connect Abelian sandpile model topplings to Fokker-Planck equations.
- To estimate critical exponents and dimensions using a novel theoretical approach.
- To analyze avalanche size distributions in different dimensions.
Main Methods:
- Formulating wave topplings as first arrival distributions for a scale-invariant, time-inhomogeneous Fokker-Planck equation.
- Applying a linear conjecture linking time inhomogeneity to loop-erased random walk (LERW) critical exponents.
- Utilizing numerical results to support theoretical estimations.
Main Results:
- The study provides a framework for estimating the lower critical dimension of the Abelian sandpile model.
- Exact values for the LERW critical exponent in three dimensions are derived.
- The avalanche size distribution in two dimensions is characterized as the difference between two closed power laws.
Conclusions:
- The research offers a new perspective on Abelian sandpile dynamics through Fokker-Planck equations.
- The findings contribute to a deeper understanding of critical phenomena and scaling laws in complex systems.
- The derived critical exponents and dimensions have implications for various fields, including physics and computer science.
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