Math 2.0

Geophysics Definition

Olami–Feder–Christensen model

Builds on Cellular automatonAbelian sandpile model

Definition

Fix an integer L≥2L \ge 2, a threshold Fth>0F_{\mathrm{th}} > 0 and a transfer parameter α\alpha with 0<α≤1/40 < \alpha \le 1/4. A state is a function FF from the square lattice {1,…,L}2\set{1, \dots, L}^2 to [0,Fth)[0, F_{\mathrm{th}}), where F(i)F(i) is the force on site ii. The Olami–Feder–Christensen model evolves a state by the following rules.

  1. Driving: all values F(i)F(i) increase by the same amount until the largest value equals FthF_{\mathrm{th}}.
  2. Toppling: every site ii with F(i)≥FthF(i) \ge F_{\mathrm{th}} is reset to 00. Each nearest neighbour jj of ii inside the lattice gets F(j)↦F(j)+αF(i)F(j) \mapsto F(j) + \alpha F(i), with F(i)F(i) taken before the reset.
  3. Relaxation: rule 2 is repeated until all sites are below FthF_{\mathrm{th}}; then rule 1 starts again.

The set of topplings caused by one driving step is an avalanche, and the number of topplings is its size. Force passed across the boundary of the lattice is lost.

Remarks

Example: with α=0.2\alpha = 0.2, a site with F(i)=FthF(i) = F_{\mathrm{th}} in the interior topples, its value becomes 00, and its four neighbours gain 0.2Fth0.2 F_{\mathrm{th}} each. The total force drops by (1−4α)Fth=0.2Fth(1 - 4\alpha) F_{\mathrm{th}} = 0.2 F_{\mathrm{th}}, so the bulk dynamics is dissipative. For α=1/4\alpha = 1/4 the bulk dynamics conserves the total force, and loss happens only at the boundary. Non-example: the abelian sandpile model of Bak, Tang and Wiesenfeld adds integer grains at random single sites. It moves a fixed number of grains in each toppling, so it fails rule 1 and the proportional transfer of rule 2. The model is a deterministic cellular automaton with real values, apart from the random initial state.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

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Abelian group, concept, no evidence levelAbelian groupconceptCellular automaton, concept, no evidence levelCellular automatonconceptCofactor, concept, no evidence levelCofactorconceptDeterminant, concept, no evidence levelDeterminantconceptGraph, concept, no evidence levelGraphconceptLaplacian matrix, concept, no evidence levelLaplacian matrixconceptMultigraph, concept, no evidence levelMultigraphconceptSpanning tree, concept, no evidence levelSpanning treeconceptKirchhoff's matrix tree theorem, statement, no evidence levelKirchhoff's matrix tr…statementAbelian sandpile model, example, no evidence levelAbelian sandpile modelexampleOlami–Feder–Christensen model, definition, no evidence levelOlami–Feder–Christens…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Abelian group concept
  2. Cellular automaton concept
  3. Cofactor concept
  4. Determinant concept
  5. Graph concept
  6. Laplacian matrix concept
  7. Multigraph concept
  8. Spanning tree concept
  9. Kirchhoff's matrix tree theorem statement
  10. Abelian sandpile model example
  11. Olami–Feder–Christensen model definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Olami%E2%80%93Feder%E2%80%93Christensen_model
About this page
Tag7DZV6MYEAC
URIm2://7DZV6MYEAC
Kinddefinition
WritingW0 stub
MSC86A15, 82C22, 37B15
DefinesOlami–Feder–Christensen model
UsesCellular automaton, Abelian sandpile model
Changed3 November 2026

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