Ref: SOC lecture notes; Section-8, Theoretical studies of self-organized criticality(Physica A-2006)

The Directed Abelian Sandpile Model

  • Variant of BTW mdel and much more tractable analytically.
  • Existence of a preffered direction, particles fall in only one direction(down).
  • There can be two knds of pile a conical or tent shaped pile.

Model Description

  • Taking square lattice the problem can be described as convention that occupation is only possible at sites such that \( X+Y = even \) or a square lattice oriented in \( (1,1) \) direction.
  • Can also be thought of as child playing with wooden blocks on staircase.

Lattice

  • Sand Grains are added anywhere on top edge with equal probability.
  • On each toppling, one grain of sand is transferred to each of the two downward neighbors.
  • Particles leave the system if there is a toppling at the bottom layer.
  • For any site the stable heights are either \( 0 \) or \( 1 \)

We can give the time evolution for the heights as
If \( Z(X,Y) \geq 2 \) then \begin{cases} Z(X,Y) \rightarrow Z(X,Y) = 2 \\ Z(X \pm 1, Y \pm 1) \rightarrow Z(X \pm 1, Y \pm 1) + 1 \end{cases}

We want to ask questions such as, What is...

  • Steady State
  • Avalnche Distribution
  • Avalanche Shapes and so on..