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Related Experiment Video

Updated: May 18, 2026

Three-Dimensional Shape Modeling and Analysis of Brain Structures
05:33

Three-Dimensional Shape Modeling and Analysis of Brain Structures

Published on: November 14, 2019

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
PubMed
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.

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Last Updated: May 18, 2026

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05:33

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Published on: November 14, 2019

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.