Hello!

These are rolling set of notes primarily for archiving and personal use. They are updated as and when I get time.

These are lecture notes made out of course on Self Organised Criticality at IISER, Pune.

Fractals

"The fractal geometry of nature"-Benoit Mandelbrot

Definition of fractal

A fractal is by definition a set for which the Hausdorff Besicovitch dimension strictly exceeds the topological dimension} \begin{equation*} D>D_{T} \end{equation*}

Effective Dimension

Effective dimension has a subjective basis (dimensions of veil thread ball as studied by physicists). It is a matter of approximation and, therefore of degree of resolution.

Certain ill-defined transitions between zones of well-defined dimension are reinterpreted as being fractal zones within which \( D>D_{T} \)

Spatial Homogeneity, scaling and self similarity

Homogeneous distribution on a line, plane or space has two very desirable properties. It is invariant under displacement, and it is invariant under change of scale.

Fractals

"The fractal geometry of nature"-Benoit Mandelbrot

Definition of fractal

A fractal is by definition a set for which the Hausdorff Besicovitch dimension strictly exceeds the topological dimension} \begin{equation*} D>D_{T} \end{equation*}

Effective Dimension

Effective dimension has a subjective basis (dimensions of veil thread ball as studied by physicists). It is a matter of approximation and, therefore of degree of resolution.

Certain ill-defined transitions between zones of well-defined dimension are reinterpreted as being fractal zones within which \( D>D_{T} \)

Spatial Homogeneity, scaling and self similarity

Homogeneous distribution on a line, plane or space has two very desirable properties. It is invariant under displacement, and it is invariant under change of scale.

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.

• 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..