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Published on: August 5, 2016
Renormalization-group approach to the Manna sandpile
1Department of Physics, National Chung Cheng University, Chia-Yi 62102, Taiwan and National Center for Theoretical Sciences at Taipei, Physics Division, National Taiwan University, Taipei 10617, Taiwan.
Summary
A new renormalization group (RG) scheme for the q-state Manna model was developed. This RG scheme accurately predicts height probabilities and critical exponents, closely matching numerical simulations for various q values.
Area of Science:
- Statistical Physics
- Complex Systems
Background:
- The q-state Manna model is a crucial model in statistical physics for understanding self-organized criticality.
- Existing methods for analyzing the Manna model face challenges in accurately predicting critical exponents.
Purpose of the Study:
- To propose a novel renormalization group (RG) scheme for the q-state Manna model.
- To validate the RG scheme by comparing its predictions with numerical simulation results.
Main Methods:
- Development of an RG scheme based on stochastic characteristics and scale similarity.
- Enumeration of RG evolution events within a 2x2 RG cell for q=2, 3, and 4.
- Fixed point analysis to determine height probabilities and critical exponents (τ and z).
Main Results:
- The proposed RG scheme yields height probabilities highly consistent with numerical simulations.
- Calculated values for the toppling number exponent (τ) and dynamical exponent (z) are presented.
- RG-derived values for τ and z for q=4 show strong agreement with simulation data.
Conclusions:
- The developed RG scheme provides an effective analytical tool for studying the q-state Manna model.
- The RG approach offers a reliable alternative to numerical simulations for determining critical exponents.
- The findings highlight the applicability of RG methods to complex systems exhibiting self-organized criticality.
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