Bak–Tang–Wiesenfeld sandpile

Bak–Tang–Wiesenfeld sandpile

In physics, the Bak–Tang–Wiesenfeld sandpile model is the first discovered example of a dynamical system displaying self-organized criticality and is named after Per Bak, Chao Tang and Kurt Wiesenfeld.

The model is a cellular automaton. At each site on the lattice there is a value that corresponds to the slope of the pile. This slope builds up as grains of sand are randomly placed onto the pile, until the slope exceeds a specific threshold value at which time that site collapses transferring sand into the adjacent sites, increasing their slope. This random placement of sand at a particular site may have no effect and it may cause a cascading reaction that will affect every site on the lattice. These "avalanches" are an example of the Eden growth model.

The iteration rules for the 2D model are as follows:

Starting with a flat surface z(x,y) = 0 for all x and y:

Add a grain of sand::z(x,y) ightarrow z(x,y) + 1.And avalanche if z(x,y) > z_c::z(x,y) ightarrow z(x,y) - 4:z( x pm 1, y) ightarrow z( x pm 1, y) + 1:z(x, y pm 1) ightarrow z( x, y pm 1 ) + 1.

This system is interesting in that it is attracted to its critical state, at which point the correlation length of the system and the correlation time of the system go to infinity, without any fine tuning of a system parameter. This contrasts with earlier examples of critical phenomena, such as the phase transitions between solid and liquid, or liquid and gas, where the critical point can only be reached by precise tuning (usually of temperature). Hence, in the sandpile model we can say that the criticality is self-organized.

Once the sandpile model reaches its critical state there is no correlation between the system's response to a and the details of a perturbation. Generally this means that dropping another grain of sand onto the pile may cause nothing to happen, or it may cause the entire pile to collapse in a massive slide. The model also displays 1/f noise, a feature common to many complex systems in nature.

This model only displays critical behaviour in 2 or more dimensions. The sandpile model can be expressed in 1D; however, instead of evolving to its critical state, the 1D sandpile model instead reaches a minimally stable state where every lattice site goes toward the critical slope.

The iteration rules for the 1D model are:

Adding a grain of sand at x::z(x) ightarrow z(x) + 1.And an avalanche at x if z > z_c::z(x) ightarrow z(x) - 2:z( x pm 1 ) ightarrow z( x pm 1) +1.

References

* [http://xstructure.inr.ac.ru/x-bin/theme3.py?level=1&index1=-15547 Sandpile model on arxiv.org]

Further reading

* cite book
author = Per Bak
date = 1996
title = How Nature Works: The Science of Self-Organized Criticality
publisher = Copernicus
location = New York
id = ISBN 0-387-94791-4

* cite journal
author = Per Bak, Chao Tang and Kurt Wiesenfeld
date = 1987
title = Self-organized criticality: an explanation of 1/ƒ noise
journal = Physical Review Letters
volume = 59
pages = 381–384
doi = 10.1103/PhysRevLett.59.381

* cite journal
author = Per Bak, Chao Tang and Kurt Wiesenfeld
date = 1988
title = Self-organized criticality
journal = Physical Review A
volume = 38
pages = 364–374
doi = 10.1103/PhysRevA.38.364


Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Per Bak — Infobox Scientist box width = 300px name = Per Bak image width = 200px caption = Per Bak at the ITCP 1992. birth date = December 8 1948 birth place = Brønderslev, Denmark death date = October 16 2002 death place = Copenhagen, Denmark residence =… …   Wikipedia

  • Kurt Wiesenfeld — is an American physicist working primarily on non linear dynamics. His works primarily concern stochastic resonance, spontaneous synchronization of coupled oscillators, and non linear laser dynamics. Since 1987, he has been professor of physics… …   Wikipedia

  • Chao Tang — is a Chinese physicist and professor at the University of California at San Francisco. In 1987, as a post doctoral research scientist in the Solid State Theory Group of Brookhaven National Laboratory, he and another fellow post doctoral scientist …   Wikipedia

  • Self-organized criticality — In physics, self organized criticality (SOC) is a property of (classes of) dynamical systems which have a critical point as an attractor. Their macroscopic behaviour thus displays the spatial and/or temporal scale invariance characteristic of the …   Wikipedia

  • Chaos theory — This article is about chaos theory in Mathematics. For other uses of Chaos theory, see Chaos Theory (disambiguation). For other uses of Chaos, see Chaos (disambiguation). A plot of the Lorenz attractor for values r = 28, σ = 10, b = 8/3 …   Wikipedia

  • List of mathematics articles (B) — NOTOC B B spline B* algebra B* search algorithm B,C,K,W system BA model Ba space Babuška Lax Milgram theorem Baby Monster group Baby step giant step Babylonian mathematics Babylonian numerals Bach tensor Bach s algorithm Bachmann–Howard ordinal… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”