Exact solution of a stochastic directed sandpile model.

Area of Science:

• Statistical Mechanics
• Complex Systems
• Dynamical Processes

Background:

• Directed sandpile models are used to study self-organized criticality.
• Previous work by Dhar and Ramaswamy analyzed deterministic toppling rules.

Purpose of the Study:

• To introduce and analytically solve a directed sandpile model with stochastic toppling rules.
• To determine the universality class and critical exponents of this new model.
• To compare its behavior with the deterministic counterpart.

Main Methods:

• Analytical solution of the stochastic directed sandpile model.
• Calculation of critical exponents D(//) and tau in different dimensions.
• Identification of the upper critical dimension.

Main Results:

• The stochastic model belongs to a different universality class than the deterministic model.
• In two dimensions, critical exponents are D(//)=7/4 and tau=10/7.
• In one dimension, critical exponents are D(//)=3/2 and tau=4/3.
• The upper critical dimension is three, with mean-field exponents D(//)=2 and tau=3/2.

Conclusions:

• The introduction of stochasticity significantly alters the behavior of directed sandpile models.
• The calculated critical exponents provide key insights into the model's statistical properties.
• The upper critical dimension marks a transition to mean-field behavior.