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Abelian Manna model in three dimensions and below
Hoai Nguyen Huynh1, Gunnar Pruessner
1Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore. n.huynh10@imperial.ac.uk
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 26, 2012
Summary
The Abelian Manna model exhibits universal behavior across different lattice dimensions, revealing a consistent relationship between critical exponents and lattice properties for both regular and fractal structures.
Area of Science:
- Complex Systems
- Statistical Physics
- Dynamical Systems
Background:
- Self-organized criticality (SOC) describes systems that naturally evolve to a critical state.
- The Abelian Manna model is a fundamental model for studying SOC phenomena.
- Understanding exponent behavior across different lattice dimensions is crucial for SOC theory.
Purpose of the Study:
- To investigate the Abelian Manna model's behavior on 3D and fractal lattices.
- To accurately determine critical exponents related to avalanche distributions.
- To explore universality and establish relationships between exponents and lattice dimensionality.
Main Methods:
- High-accuracy moment analysis was employed to calculate critical exponents.
- The study extended previous analyses from lower-dimensional lattices.
- Rescaling of critical exponents incorporated lattice and random walker dimensions.
Main Results:
- Exponents for avalanche size, duration, and area distributions were precisely determined.
- Results confirmed universality below the upper critical dimension.
- Coefficients for an epsilon (ε) expansion were successfully calculated.
Conclusions:
- The findings reinforce the concept of universality in self-organized criticality.
- A remarkable, unified relation was discovered for critical exponents across regular and fractal lattices.
- This relation highlights the interconnectedness of lattice dimension and critical behavior in SOC models.
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