Finite-size scaling of critical avalanches

Area of Science:

• Complex systems
• Statistical physics
• Non-linear dynamics

Background:

• Self-organized criticality (SOC) describes systems that naturally evolve to a critical state.
• Avalanche size distributions in SOC systems typically follow power laws with finite-size cutoffs.
• Previous studies often assumed a universal avalanche size exponent.

Purpose of the Study:

• To investigate the behavior of avalanche size distributions in self-organized critical systems.
• To systematically examine the influence of system size on avalanche size distributions.
• To accurately estimate critical exponents using scaling methods.

Main Methods:

• Implementation of the scaling method to identify scaling functions.
• Analysis of data collapse to ensure accurate estimation of critical exponents.
• Simulation of sandpile models (e.g., prototype sandpile, train/Oslo sandpile).

Main Results:

• Demonstrated that avalanche size distributions can decrease with increasing system size.
• Identified distinct scaling functions and critical exponents for avalanche and system sizes.
• Estimated the avalanche size exponent for the prototype sandpile on a square lattice to be 1.
• Observed a logarithmic system size dependence in the distribution, consistent with normalization.
• Found the avalanche size exponent for the train/Oslo sandpile model to be slightly less than 1, differing from prior estimates.

Conclusions:

• The typical power-law distribution with a cutoff is not the only behavior observed in SOC systems.
• Accurate estimation of critical exponents requires careful consideration of system size effects.
• The findings challenge previous estimates and provide a more nuanced understanding of avalanche dynamics in SOC models.