Briefly, a cellular automaton consist of a lattice of discrete identical sites, each site taking on a finite set of values. The values of the sites evolve in discrete time steps according to a set of rules that specify the value of each site in terms of the values of neighboring sites. The main components of a CA are:

-- the grid

-- the neighboors

-- the set of rules

-- the initial conditions.

In our examples we use a rectangular grid and two type of neighboor vonNewman (a) and Moore (b), see figure

Here we illustrate a number of 3 discrete systems: forest fire, sand piles, and game of life.

Bibliography:

Stephen Wolfram,

Jane Molofsky and James D. Bever,

The forest fire model is a probabilistc cellular automaton defined on d-dimensional lattice, in our exemaple a 2D square.

Each site can be in one of three states: empty, green tree or burning tree.

The update rules:

(a) an empty site become a green tree with a*p * probability,

(b) a burning site become an emty site and

(c) a green tree become a burning tree if at least one of its neighbors is a burning tree.

Runnig examples:

*p*=0.03;0.04;0.05;0.08

Bibliography:

P. Bak, K. Chen, and C. Tang,* A forest fire model and some thoughts on turbulence*,
Phys. Lett. A. 147(5,6), (1990)

B. Drossel and F. Schwabl,* Self Organized Critical Forest-Fire Model*, Phys. Rev. Lett. 69(11),
(1992)

Each site can be in one of three states: empty, green tree or burning tree.

The update rules:

(a) an empty site become a green tree with a

(b) a burning site become an emty site and

(c) a green tree become a burning tree if at least one of its neighbors is a burning tree.

Runnig examples:

Bibliography:

P. Bak, K. Chen, and C. Tang,

B. Drossel and F. Schwabl,

Sand pile models is an example of dynamical system with extend spatial degrees of freedom.
This kind of systems evolve into * self-organized critical state* [Bak].

The rules of evolution:

vonNewmann $$ \begin{array}{rl} z(x,y)\rightarrow&z(x,y)-4\\ z(x\pm 1,y)\rightarrow&z(x\pm 1,y)+1\\ z(x,y\pm 1)\rightarrow&z(x,y\pm 1)+1\\ &{\rm for\,\,} z(x,y)\leq 5 \end{array} $$ Moore $$ \begin{array}{rl} z(x,y)\rightarrow&z(x,y)-8\\ z(x\pm 1,y)\rightarrow&z(x\pm 1,y)+1\\ z(x,y\pm 1)\rightarrow&z(x,y\pm 1)+1\\ z(x\pm 1,y\pm 1)\rightarrow&z(x\pm ,y\pm 1)+1\\ &{\rm for\,\,} z(x,y)\leq 9 \end{array} $$ Running examples:

Source grain numbers= 100, 30 000, 60 000;

Bibliography:

P. Bak, C. Tang and K. Wiesefeld* Self organized criticality*,
Phys. Lett. A. 38(1), (1988)

Deepak Dhar,* Theoretical studies of self-organized criticality*, Physica A, 369, pp. 29-70, 2006.

The rules of evolution:

vonNewmann $$ \begin{array}{rl} z(x,y)\rightarrow&z(x,y)-4\\ z(x\pm 1,y)\rightarrow&z(x\pm 1,y)+1\\ z(x,y\pm 1)\rightarrow&z(x,y\pm 1)+1\\ &{\rm for\,\,} z(x,y)\leq 5 \end{array} $$ Moore $$ \begin{array}{rl} z(x,y)\rightarrow&z(x,y)-8\\ z(x\pm 1,y)\rightarrow&z(x\pm 1,y)+1\\ z(x,y\pm 1)\rightarrow&z(x,y\pm 1)+1\\ z(x\pm 1,y\pm 1)\rightarrow&z(x\pm ,y\pm 1)+1\\ &{\rm for\,\,} z(x,y)\leq 9 \end{array} $$ Running examples:

Source grain numbers= 100, 30 000, 60 000;

Bibliography:

P. Bak, C. Tang and K. Wiesefeld

Deepak Dhar,

A cell can be in two states: alive cell or dead cell.

The rule of evolution:

Dead cell: Become an alive cell if it has exactly 3 alive neigbors, otherwise reamin dead

Alive cell: Become a dead cell if it has more of 4 alive cell or less than 1 alive cell, otherwise remain alive.

Runnig examples:

Bibliography:

Conway's Game of Life - Stanford University

The rule of evolution:

Dead cell: Become an alive cell if it has exactly 3 alive neigbors, otherwise reamin dead

Alive cell: Become a dead cell if it has more of 4 alive cell or less than 1 alive cell, otherwise remain alive.

Runnig examples:

Bibliography:

Conway's Game of Life - Stanford University

Rule of random walk.