Some Remarks on the Sandpile Model Thomas Worsch IAKS Vollmar, Fakult¨ at f¨ ur Informatik, Universit¨ at Karlsruhe
TSUNAMI
Earthquakes The US Geological Survey National Earthquake Information Center locates about 20000 earthquakes each year. Relative frequencies: magnitude number 6-6.9 120 7-7.9 18 8-9.9 1 Gutenberg-Richter law The frequency of earthquakes of a given size is an inverse power of the energy released. log( N ) ≈ a − b log( E ) where b ≈ 1
Overview • Sandpile model Part 1: some nice theoretical results • Chip firing games • Random addition of sand • (spectral) graph theory Part 2: some experiments • What is the size of an avalanche? • Are there aftershocks? • an invitation to open problem
The sandpile model (Bak, Tang, Wiesenfeld 1987) two-dimensional CA with von Neumann neighborhood think of a finite set of cells, e.g. a square of size n × n set of states: { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 } Local rule: Denote by x the current state of a cell and by v the number of (at most 4) neighbors which are in a state ≥ 4 : • If x ≤ 3 , then x ← x + v . • If x ≥ 4 , then x ← x + v − 4 .
Example 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 4 0 1 0 1 1 1 1 1 1 1 1 1 1 0 3 0 0 4 0 1 0 1 1 1 1 1 1 1 0 3 0 0 3 0 0 4 0 1 0 1 1 1 1 0 3 0 0 3 0 0 3 0 0 4 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0
Example 5 4 4 4 4 4 4 5 4 3 3 3 3 3 3 4 4 3 3 3 3 3 3 4 4 3 3 3 3 3 3 4 4 3 3 3 3 3 3 4 4 3 3 3 3 3 3 4 4 3 3 3 3 3 3 4 5 4 4 4 4 4 4 5
Example 3 2 2 2 2 2 2 3 2 5 4 4 4 4 5 2 2 4 3 3 3 3 4 2 2 4 3 3 3 3 4 2 2 4 3 3 3 3 4 2 2 4 3 3 3 3 4 2 2 5 4 4 4 4 5 2 3 2 2 2 2 2 2 3
Example 3 3 3 3 3 3 3 3 3 3 2 2 2 2 3 3 3 2 5 4 4 5 2 3 3 2 4 3 3 4 2 3 3 2 4 3 3 4 2 3 3 2 5 4 4 5 2 3 3 3 2 2 2 2 3 3 3 3 3 3 3 3 3 3
Example 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 3 3 3 3 3 2 5 5 2 3 3 3 3 2 5 5 2 3 3 3 3 3 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
Example 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
Example 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 Each cell fired exactly once.
Part 1: some nice theoretical results
Chip Firing Games Given: a finite connected undirected graph G = ( V, E ) without loops • Place a finite number of chips (grains of sand) at the nodes of a graph. • Whenever the number of chips at a node is at least as large as its degree, the node fires : it moves one chip to each of the neighbors. In each global step all nodes/cells/. . . are handled.
Chip Firing Games 2: boundary cells with Dirichlet condition The connected graph G ′ = ( V ′ , E ′ ) consists of • an “inner” graph G as before and • a non-empty set V ′ � V of boundary cells • modified rule : chips on boundary cells are immediately removed.
Chip Firing Games 2: boundary cells with Dirichlet condition The connected graph G ′ = ( V ′ , E ′ ) consists of • an “inner” graph G as before and • a non-empty set V ′ � V of boundary cells • modified rule : chips on boundary cells are immediately removed. The sandpile model is a such CFG: • cells as the nodes • edges between nodes which represent neighboring cells • 4 n boundary cells around the inner n × n square
A cell is • stable iff its number of chips is strictly less than its degree, • unstable otherwise. A global configuration is • stable iff all cells are stable, • unstable iff at least one cell is unstable.
Chip Firing Games: running it Fact In a CFG with boundary cells each unstable configuration leads to a stable one after a finite number of steps. Call this a relaxation of the initial configuration.
Chip Firing Games: running it Fact In a CFG with boundary cells each unstable configuration leads to a stable one after a finite number of steps. Call this a relaxation of the initial configuration. Theorem (Chung/Ellis) Let f ( x ) denote how often node x fires during a relaxation. Let D = diam ( G ) , N = | V | , and K the total number of chips in the initial configuration. Then � � N 3 f ( x ) ≤ D · K · x ∈ V
Adding some randomization Given a stable configuration • choose with equal probability one of the cells, • add one grain of sand to it, and • relax to the next stable configuration. This is a Markov chain .
Adding some randomization Given a stable configuration • choose with equal probability one of the cells, • add one grain of sand to it, and • relax to the next stable configuration. This is a Markov chain . Interest Observe the avalanches triggered by the randomly added sand. The distribution of their sizes is “the same” as for earthquakes.
Recurrent configurations Call a configuration c recurrent iff no matter where you add one grain of sand, by addition of more sand and relaxations you can reach c again.
Recurrent configurations Call a configuration c recurrent iff no matter where you add one grain of sand, by addition of more sand and relaxations you can reach c again. Questions • Are there non-/recurrent configurations? How many? • Is there any additional structure? • How do you check whether a configuration is recurrent?
Recurrent configurations Answers (Homework, Dhar, Creutz, . . . ) • All cells in state 3 : recurrent configuration. • Configurations with two neighboring 0 are non-recurrent. • The set of recurrent configurations is a group (for pointwise addition with subsequent relaxation).
Recurrent configurations Theorem Add to each cell as many chips as there are neighboring boundary cells and relax. The inner configuration is recurrent iff each cell fires exactly once .
Recurrent configurations Theorem Add to each cell as many chips as there are neighboring boundary cells and relax. The inner configuration is recurrent iff each cell fires exactly once . From this you can get: For each recurrent configuration there is a spanning forest G ′ with the outer nodes V ′ � V as the set of roots.
Recurrent configurations Theorem Add to each cell as many chips as there are neighboring boundary cells and relax. The inner configuration is recurrent iff each cell fires exactly once . From this you can get: For each recurrent configuration there is a spanning forest G ′ with the outer nodes V ′ � V as the set of roots. Theorem There is a bijection between the set of recurrent configurations and the set of rooted spanning forests .
Some graph theory Definition Denote by A the adjacency matrix . d 1 0 · · · 0 . . 0 d 2 . The degree matrix is D = . . ... . . 0 0 · · · 0 d n The Laplace matrix is L = D − A . Denote by L G the restriction of L to the rows and columns corresponding to the nodes of the inner graph G .
Some graph theory Definition Denote by A the adjacency matrix . d 1 0 · · · 0 . . 0 d 2 . The degree matrix is D = . . ... . . 0 0 · · · 0 d n The Laplace matrix is L = D − A . Denote by L G the restriction of L to the rows and columns corresponding to the nodes of the inner graph G . Theorem (Chung/Ellis) The number of rooted spanning forests of G ′ is det L G .
Part 2: some experiments
What is the “size” of an avalanche? You can count • the number of single cell firings • the number of global CA steps • the number of cells “covered” by the avalanche (i.e. which fired at least once) Differences? Relations?
Avalanches: cell firings versus CA steps 60000 "ava42" using 2:3 50000 40000 30000 20000 10000 0 0 5e+06 1e+07 1.5e+07 2e+07 2.5e+07
Avalanches: cell firings versus CA steps 100000 "ava42" using 2:3 2*x**0.6 10000 1000 100 10 1000 10000 100000 1e+06 1e+07 1e+08
Avalanches: cell firings versus area "ava42" using 2:4 x 4*x**0.7 1000000 1e+06 100000 10000 1000 1000 10000 100000 1e+06 1e+07 1e+08
Avalanches: steps versus area 1e+06 "ava42" using 3:4 x**1.6 x**1.45 x**1.3 100000 10000 1000 10 100 1000 10000 100000
Aftershocks in the sandpile model?
Aftershocks in the sandpile model? a preliminary observation — still too few experimental data
Aftershocks in the sandpile model? a preliminary observation — still too few experimental data Experiment • wait for a large avalanche • put the next 100 grains to cells in the vicinity of the one triggering that avalanche • check whether next large avalanches occur earlier than with completely random placement
Aftershocks in the sandpile model? Result • 1000 × 1000 square • 1152 avalanches covering at least 10000 cells • average number of grains you have to wait until you observe an avalanche covering at least 50% of the area of the original avalanche: – localized placement: 17 . 9 – random placement: 33 . 6
An invitation to open problems • “aftershocks” need much more work • proofs • or at least a better understanding of avalanches • non-squares • higher dimensions
Thank you very much for your attention! Slides : http://liinwww.ira.uka.de/~worsch/talks/gdansk.pdf
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