slides

Kadanoff Sand Pile Model
Avalanches and Fixed Point.
Kévin Perrot and Eric Rémila
Université de Lyon
Laboratoire de l’Informatique du Paralléllisme
umr 5668 CNRS - ENS de Lyon - Université Lyon 1
DISCO - Valparaiso,
2011.11
KSPM - Definition
Suggested by L. Kadanoff et al (1989).
. A parameter D
. An initial configuration
(N, 0, 0, ...)
. One transition rule
≥D
KSPM - Definition
Suggested by L. Kadanoff et al (1989).
. A parameter D
. An initial configuration
(N, 0, 0, ...)
. One transition rule
≥D
D=3
N = 24
KSPM - Definition
Suggested by L. Kadanoff et al (1989).
. A parameter D
. An initial configuration
(N, 0, 0, ...)
. One transition rule
≥D
D=3
N = 24
KSPM - Definition
Suggested by L. Kadanoff et al (1989).
. A parameter D
. An initial configuration
(N, 0, 0, ...)
. One transition rule
≥D
D=3
N = 24
KSPM - Definition
Suggested by L. Kadanoff et al (1989).
. A parameter D
. An initial configuration
(N, 0, 0, ...)
. One transition rule
≥D
D=3
N = 24
KSPM - Definition
Suggested by L. Kadanoff et al (1989).
. A parameter D
. An initial configuration
(N, 0, 0, ...)
. One transition rule
≥D
D=3
N = 24
KSPM - Definition
Suggested by L. Kadanoff et al (1989).
. A parameter D
. An initial configuration
(N, 0, 0, ...)
. One transition rule
≥D
D=3
N = 24
KSPM - Definition
Suggested by L. Kadanoff et al (1989).
. A parameter D
. An initial configuration
(N, 0, 0, ...)
. One transition rule
≥D
D=3
N = 24
KSPM - Definition
Suggested by L. Kadanoff et al (1989).
. A parameter D
. An initial configuration
(N, 0, 0, ...)
. One transition rule
≥D
D=3
N = 24
KSPM - Definition
Suggested by L. Kadanoff et al (1989).
. A parameter D
. An initial configuration
(N, 0, 0, ...)
. One transition rule
≥D
D=3
N = 24
KSPM - Definition
Suggested by L. Kadanoff et al (1989).
. A parameter D
. An initial configuration
(N, 0, 0, ...)
. One transition rule
≥D
D=3
N = 24
KSPM - Definition
Suggested by L. Kadanoff et al (1989).
. A parameter D
. An initial configuration
(N, 0, 0, ...)
. One transition rule
≥D
D=3
N = 24
KSPM - Definition
Suggested by L. Kadanoff et al (1989).
. A parameter D
. An initial configuration
(N, 0, 0, ...)
. One transition rule
≥D
D=3
N = 24
KSPM - Definition
Suggested by L. Kadanoff et al (1989).
. A parameter D
. An initial configuration
(N, 0, 0, ...)
. One transition rule
≥D
D=3
N = 24
KSPM - Definition
Suggested by L. Kadanoff et al (1989).
. A parameter D
. An initial configuration
(N, 0, 0, ...)
. One transition rule
≥D
D=3
N = 24
fixed point
KSPM - Definition
Suggested by L. Kadanoff et al (1989).
. A parameter D
. An initial configuration
(N, 0, 0, ...)
. One transition rule
≥D
D=3
N = 24
Remark : KSPM(2)=SPM
fixed point
Motivations
Why studying sand piles ?
Motivations
Why studying sand piles ?
. From local rules to global behavior : Emergence.
Motivations
Why studying sand piles ?
. From local rules to global behavior : Emergence.
. Sand piles are at the edge between discrete and continuous
phenomena.
Motivations
Why studying sand piles ?
. From local rules to global behavior : Emergence.
. Sand piles are at the edge between discrete and continuous
phenomena.
. They emphasize Self-Organized Criticality.
Bak, Tang, Wiesenfeld (1987)
. Property of dynamical systems which have critical attractors.
. A small disturbance can involve a deep reorganization.
Motivations
Why studying sand piles ?
. From local rules to global behavior : Emergence.
. Sand piles are at the edge between discrete and continuous
phenomena.
. They emphasize Self-Organized Criticality.
Bak, Tang, Wiesenfeld (1987)
. Property of dynamical systems which have critical attractors.
. A small disturbance can involve a deep reorganization.
Adding a single grain can create an unbounded avalanche !
KSPM - Representation
. sequence of height differences
( 4 , 2 , 5 , 0 , 1 , 0ω )
KSPM - Representation
. sequence of height differences
. rule application ⇒ a column gives some height difference
( 4 , 2 , 5 , 0 , 1 , 0ω )
KSPM - Representation
. sequence of height differences
. rule application ⇒ a column gives some height difference
( 4 , 2 , 5 , 0 , 1 , 0ω )
. KSPM is a Linear Chip Firing Game.
s
4
2
5
0
1
0
KSPM - Known Results
Goles, Morvan, Phan (2002)
D = 3 and N = 24
(24, 0ω )
0
(21, 0, 1, 0ω )
0
(18, 0, 2, 0ω )
0
(15, 0, 3, 0ω )
0
(12, 0, 4, 0ω )
0
(9, 0, 5, 0ω )
0
(6, 0, 6, 0ω )
0
(3, 0, 7, 0ω )
(0, 0, 8, 0ω )
ω
(0, 2, 5, 0, 1, 0 )
ω
2 (15, 2, 0, 0, 1, 0 )
(12, 2, 1, 0, 1, 0ω )
0
(9, 2, 2, 0, 1, 0ω )
0
ω
2 (6, 2, 3, 0, 1, 0 ) 2
(3, 2, 4, 0, 1, 0ω ) 0 (6, 4, 0, 0, 2, 0ω )
2
(3,04, 1, 0, 2, 1
0ω )
(8, 1, 0, 1, 2, 0ω )
(0, 4, 2, 0, 2, 1
0ω )
(5,01, 1, 1, 2, 0ω )
(2, 1, 2, 1, 2, 0ω )
KSPM - Known Results
Goles, Morvan, Phan (2002)
D = 3 and N = 24
(24, 0ω )
0
(21, 0, 1, 0ω )
0
(18, 0, 2, 0ω )
0
(15, 0, 3, 0ω )
0
(12, 0, 4, 0ω )
0
(9, 0, 5, 0ω )
0
(6, 0, 6, 0ω )
0
(3, 0, 7, 0ω )
(0, 0, 8, 0ω )
ω
(0, 2, 5, 0, 1, 0 )
. (KSPM(D, N), →) is a lattice.
ω
2 (15, 2, 0, 0, 1, 0 )
(12, 2, 1, 0, 1, 0ω )
0
(9, 2, 2, 0, 1, 0ω )
0
ω
2 (6, 2, 3, 0, 1, 0 ) 2
(3, 2, 4, 0, 1, 0ω ) 0 (6, 4, 0, 0, 2, 0ω )
2
(3,04, 1, 0, 2, 1
0ω )
(8, 1, 0, 1, 2, 0ω )
(0, 4, 2, 0, 2, 1
0ω )
(5,01, 1, 1, 2, 0ω )
(2, 1, 2, 1, 2, 0ω )
KSPM - Known Results
Goles, Morvan, Phan (2002)
D = 3 and N = 24
(24, 0ω )
0
(21, 0, 1, 0ω )
0
(18, 0, 2, 0ω )
0
(15, 0, 3, 0ω )
0
(12, 0, 4, 0ω )
0
(9, 0, 5, 0ω )
0
(6, 0, 6, 0ω )
0
(3, 0, 7, 0ω )
(0, 0, 8, 0ω )
ω
(0, 2, 5, 0, 1, 0 )
. (KSPM(D, N), →) is a lattice.
. A unique fixed point π(n).
ω
2 (15, 2, 0, 0, 1, 0 )
(12, 2, 1, 0, 1, 0ω )
0
(9, 2, 2, 0, 1, 0ω )
0
ω
2 (6, 2, 3, 0, 1, 0 ) 2
(3, 2, 4, 0, 1, 0ω ) 0 (6, 4, 0, 0, 2, 0ω )
2
(3,04, 1, 0, 2, 1
0ω )
(8, 1, 0, 1, 2, 0ω )
(0, 4, 2, 0, 2, 1
0ω )
(5,01, 1, 1, 2, 0ω )
(2, 1, 2, 1, 2, 0ω )
KSPM - Known Results
Goles, Morvan, Phan (2002)
D = 3 and N = 24
(24, 0ω )
0
(21, 0, 1, 0ω )
0
(18, 0, 2, 0ω )
0
(15, 0, 3, 0ω )
0
(12, 0, 4, 0ω )
0
(9, 0, 5, 0ω )
0
(6, 0, 6, 0ω )
0
(3, 0, 7, 0ω )
(0, 0, 8, 0ω )
ω
(0, 2, 5, 0, 1, 0 )
. (KSPM(D, N), →) is a lattice.
. A unique fixed point π(n).
ω
2 (15, 2, 0, 0, 1, 0 )
(12, 2, 1, 0, 1, 0ω )
0
(9, 2, 2, 0, 1, 0ω )
0
ω
2 (6, 2, 3, 0, 1, 0 ) 2
. All paths are equivalent : same
fired columns, with same
occurrences.
(3, 2, 4, 0, 1, 0ω ) 0 (6, 4, 0, 0, 2, 0ω )
2
(3,04, 1, 0, 2, 1
0ω )
(8, 1, 0, 1, 2, 0ω )
(0, 4, 2, 0, 2, 1
0ω )
(5,01, 1, 1, 2, 0ω )
(2, 1, 2, 1, 2, 0ω )
KSPM - Known Results
Goles, Morvan, Phan (2002)
D = 3 and N = 24
(24, 0ω )
0
(21, 0, 1, 0ω )
0
(18, 0, 2, 0ω )
0
(15, 0, 3, 0ω )
0
(12, 0, 4, 0ω )
0
(9, 0, 5, 0ω )
0
(6, 0, 6, 0ω )
0
(3, 0, 7, 0ω )
(0, 0, 8, 0ω )
ω
(0, 2, 5, 0, 1, 0 )
. (KSPM(D, N), →) is a lattice.
. A unique fixed point π(n).
ω
2 (15, 2, 0, 0, 1, 0 )
(12, 2, 1, 0, 1, 0ω )
0
(9, 2, 2, 0, 1, 0ω )
0
ω
2 (6, 2, 3, 0, 1, 0 ) 2
. All paths are equivalent : same
fired columns, with same
occurrences.
. Fixed point shape ?
(3, 2, 4, 0, 1, 0ω ) 0 (6, 4, 0, 0, 2, 0ω )
2
(3,04, 1, 0, 2, 1
0ω )
(8, 1, 0, 1, 2, 0ω )
(0, 4, 2, 0, 2, 1
0ω )
(5,01, 1, 1, 2, 0ω )
(2, 1, 2, 1, 2, 0ω )
KSPM - Known Results
Goles, Morvan, Phan (2002)
D = 3 and N = 24
(24, 0ω )
0
(21, 0, 1, 0ω )
0
(18, 0, 2, 0ω )
0
(15, 0, 3, 0ω )
0
(12, 0, 4, 0ω )
0
(9, 0, 5, 0ω )
0
(6, 0, 6, 0ω )
0
(3, 0, 7, 0ω )
(0, 0, 8, 0ω )
(0, 2, 5, 0, 1, 0ω )
. (KSPM(D, N), →) is a lattice.
. A unique fixed point π(n).
ω
2 (15, 2, 0, 0, 1, 0 )
(12, 2, 1, 0, 1, 0ω )
0
(9, 2, 2, 0, 1, 0ω )
0
ω
2 (6, 2, 3, 0, 1, 0 ) 2
. All paths are equivalent : same
fired columns, with same
occurrences.
. Fixed point shape ?
. Small perturbation effect ?
(3, 2, 4, 0, 1, 0ω ) 0 (6, 4, 0, 0, 2, 0ω )
2
(3,04, 1, 0, 2, 1
0ω )
(8, 1, 0, 1, 2, 0ω )
(0, 4, 2, 0, 2, 1
0ω )
(5,01, 1, 1, 2, 0ω )
(2, 1, 2, 1, 2, 0ω )
Inductive Approach - Avalanches
1. Start from the fixed point : π(k − 1)
Inductive Approach - Avalanches
1. Start from the fixed point : π(k − 1)
2. Add a single grain of column 0 : π(k − 1)↓0
Inductive Approach - Avalanches
1. Start from the fixed point : π(k − 1)
2. Add a single grain of column 0 : π(k − 1)↓0
3. Fire, until reaching a fixed point : π(π(k − 1)↓0 )
Inductive Approach - Avalanches
1. Start from the fixed point : π(k − 1)
2. Add a single grain of column 0 : π(k − 1)↓0
3. Fire, until reaching a fixed point : π(π(k − 1)↓0 )
π(k) = π(π(k − 1)↓0 )
This allows an inductive way for computing π(k).
Inductive Approach - Avalanches
1. Start from the fixed point : π(k − 1)
2. Add a single grain of column 0 : π(k − 1)↓0
3. Fire, until reaching a fixed point : π(π(k − 1)↓0 )
π(k) = π(π(k − 1)↓0 )
This allows an inductive way for computing π(k).
The k th avalanche : the set of fired columns in step 3.
Inductive Approach - Avalanches
1. Start from the fixed point : π(k − 1)
2. Add a single grain of column 0 : π(k − 1)↓0
3. Fire, until reaching a fixed point : π(π(k − 1)↓0 )
π(k) = π(π(k − 1)↓0 )
This allows an inductive way for computing π(k).
The k th avalanche : the set of fired columns in step 3.
1 Each column is fired at most once in an avalanche
Illustration on examples
∗
∗
∗
∗
π(103)↓0 → π(104)↓0 → π(105)↓0 → π(106)↓0 → π(107)
D=3
Illustration on examples
∗
∗
∗
∗
π(103)↓0 → π(104)↓0 → π(105)↓0 → π(106)↓0 → π(107)
D=3
Illustration on examples
∗
∗
∗
∗
π(103)↓0 → π(104)↓0 → π(105)↓0 → π(106)↓0 → π(107)
D=3
Illustration on examples
∗
∗
∗
∗
π(103)↓0 → π(104)↓0 → π(105)↓0 → π(106)↓0 → π(107)
D=3
Illustration on examples
∗
∗
∗
∗
π(103)↓0 → π(104)↓0 → π(105)↓0 → π(106)↓0 → π(107)
D=3
Illustration on examples
∗
∗
∗
∗
π(103)↓0 → π(104)↓0 → π(105)↓0 → π(106)↓0 → π(107)
D=3
Illustration on examples
∗
∗
∗
∗
π(103)↓0 → π(104)↓0 → π(105)↓0 → π(106)↓0 → π(107)
D=3
Illustration on examples
∗
∗
∗
∗
π(103)↓0 → π(104)↓0 → π(105)↓0 → π(106)↓0 → π(107)
D=3
Illustration on examples
∗
∗
∗
∗
π(103)↓0 → π(104)↓0 → π(105)↓0 → π(106)↓0 → π(107)
D=3
Illustration on examples
∗
∗
∗
∗
π(103)↓0 → π(104)↓0 → π(105)↓0 → π(106)↓0 → π(107)
D=3
Illustration on examples
∗
∗
∗
∗
π(103)↓0 → π(104)↓0 → π(105)↓0 → π(106)↓0 → π(107)
D=3
Illustration on examples
∗
∗
∗
∗
π(103)↓0 → π(104)↓0 → π(105)↓0 → π(106)↓0 → π(107)
D=3
Illustration on examples
∗
∗
∗
∗
π(103)↓0 → π(104)↓0 → π(105)↓0 → π(106)↓0 → π(107)
D=3
Illustration on examples
∗
∗
∗
∗
π(103)↓0 → π(104)↓0 → π(105)↓0 → π(106)↓0 → π(107)
D=3
Illustration on examples
∗
∗
∗
∗
π(103)↓0 → π(104)↓0 → π(105)↓0 → π(106)↓0 → π(107)
D=3
Illustration on examples
∗
∗
∗
∗
π(103)↓0 → π(104)↓0 → π(105)↓0 → π(106)↓0 → π(107)
D=3
Illustration on examples
∗
∗
∗
∗
π(103)↓0 → π(104)↓0 → π(105)↓0 → π(106)↓0 → π(107)
D=3
Illustration on examples
∗
∗
∗
∗
π(103)↓0 → π(104)↓0 → π(105)↓0 → π(106)↓0 → π(107)
D=3
Illustration on examples
∗
∗
∗
∗
π(103)↓0 → π(104)↓0 → π(105)↓0 → π(106)↓0 → π(107)
D=3
Illustration on examples
∗
∗
∗
∗
π(103)↓0 → π(104)↓0 → π(105)↓0 → π(106)↓0 → π(107)
D=3
Illustration on examples
∗
∗
∗
∗
π(103)↓0 → π(104)↓0 → π(105)↓0 → π(106)↓0 → π(107)
D=3
Illustration on examples
∗
∗
∗
∗
π(103)↓0 → π(104)↓0 → π(105)↓0 → π(106)↓0 → π(107)
D=3
Illustration on examples
∗
∗
∗
∗
π(103)↓0 → π(104)↓0 → π(105)↓0 → π(106)↓0 → π(107)
D=3
Illustration on examples
∗
∗
∗
∗
π(103)↓0 → π(104)↓0 → π(105)↓0 → π(106)↓0 → π(107)
D=3
Illustration on examples
∗
∗
∗
∗
π(103)↓0 → π(104)↓0 → π(105)↓0 → π(106)↓0 → π(107)
D=3
Illustration on examples
∗
∗
∗
∗
π(103)↓0 → π(104)↓0 → π(105)↓0 → π(106)↓0 → π(107)
D=3
Illustration on examples
∗
∗
∗
∗
π(103)↓0 → π(104)↓0 → π(105)↓0 → π(106)↓0 → π(107)
D=3
Avalanches ←→ SOC
First Avalanches
N=98
N=97
N=96
N=93
N=90
N=87
N=86
N=85
N=82
N=79
N=76
N=75
N=74
N=71
N=68
N=65
N=64
N=63
N=60
N=57
N=54
N=53
N=52
N=49
N=46
N=43
N=42
N=39
N=36
N=35
N=32
N=29
N=26
N=25
N=22
N=19
N=18
N=15
N=12
N=9
N=6
N=3
N=192
N=189
N=186
N=183
N=180
N=179
N=178
N=175
N=172
N=169
N=168
N=165
N=162
N=161
N=158
N=157
N=156
N=153
N=152
N=151
N=148
N=145
N=142
N=141
N=140
N=137
N=134
N=131
N=130
N=129
N=126
N=123
N=120
N=117
N=114
N=113
N=112
N=109
N=108
N=107
N=104
N=101
N=278
N=275
N=272
N=271
N=270
N=267
N=266
N=265
N=262
N=261
N=260
N=257
N=254
N=251
N=250
N=249
N=246
N=245
N=242
N=239
N=238
N=235
N=232
N=229
N=228
N=227
N=224
N=221
N=218
N=217
N=216
N=213
N=210
N=207
N=206
N=205
N=202
N=201
N=200
N=197
N=196
N=195
Holes Positions on Avalanches,
. Hole : an unfired column, such that there exists a fired
column with largest rank.
Holes Positions on Avalanches,
. Hole : an unfired column, such that there exists a fired
column with largest rank.
. The BIG conjecture : Let L(D, N) be the largest hole of the
N th avalanche
2 L(D, N) is in O(log N) .
Holes Positions on Avalanches,
. Hole : an unfired column, such that there exists a fired
column with largest rank.
. The BIG conjecture : Let L(D, N) be the largest hole of the
N th avalanche
2 L(D, N) is in O(log N) .
. Only proved for D = 3, unfortunately.
Holes Positions on Avalanches,
. Hole : an unfired column, such that there exists a fired
column with largest rank.
. The BIG conjecture : Let L(D, N) be the largest hole of the
N th avalanche
2 L(D, N) is in O(log N) .
. Only proved for D = 3, unfortunately.
. Consequence : except for leftmost columns,
we do not care about holes
√
(since the support of the sand pile π(N) is in Θ( N)).
Beyond the Holes : Long Avalanches
. Peak : column whose height difference is D −1.
. Peaks are masters of long avalanches.
≥D
D
π(N)
peak (D −1)
Beyond the Holes : Long Avalanches
. Peak : column whose height difference is D −1.
. Peaks are masters of long avalanches.
≥D
D
π(N)
peak (D −1)
Beyond the Holes : Long Avalanches
. Peak : column whose height difference is D −1.
. Peaks are masters of long avalanches.
≥D
D
π(N)
peak (D −1)
Beyond the Holes : Long Avalanches
. Peak : column whose height difference is D −1.
. Peaks are masters of long avalanches.
≥D
D
π(N)
peak (D −1)
Beyond the Holes : Long Avalanches
. Peak : column whose height difference is D −1.
. Peaks are masters of long avalanches.
≥D
D
π(N)
peak (D −1)
≥D
D
π(N)
peak (D −1)
Beyond the Holes : Long Avalanches
. Peak : column whose height difference is D −1.
. Peaks are masters of long avalanches.
≥D
D
π(N)
peak (D −1)
≥D
D
π(N)
0 +1+1+1+1+1
peak (D −1)
Avalanche effect :
. the last reached peak := 0,
. the D − 1 columns just after the last reached peak : + 1,
. other columns : unchanged.
3 The avalanche effect is easily computable beyond the holes
Transduction
Governing peak : rightmost peak before the interval
5 4 2 5 1
Transduction
Governing peak : rightmost peak before the interval
...
5 4 2 0 2
...
5 4 2 5 1
Transduction
Governing peak : rightmost peak before the interval
...
0 5 3 1 3
...
...
5 4 2 0 2
...
5 4 2 5 1
Transduction
Governing peak : rightmost peak before the interval
...
0 0 4 2 4
...
...
0 5 3 1 3
...
...
5 4 2 0 2
...
5 4 2 5 1
Transduction
Governing peak : rightmost peak before the interval
...
1 1 5 2 4
...
...
0 0 4 2 4
...
...
0 5 3 1 3
...
...
5 4 2 0 2
...
5 4 2 5 1
The governing peak really governs the evolution of the interval
Transduction
Governing peak : rightmost peak before the interval
...
...
...
...
1 1 5 2 4
...
0 0 4 2 4
...
1
0 5 3 1 3
...
0
5 4 2 0 2
...
3
5 4 2 5 1
output
2|301
54251
11524
Transduction
input
The governing peak really governs the evolution of the interval
Wave Pattern - Example for D = 3
1|01
21
11
0|∅
1|∅
10
00
1|∅
0|∅
0|∅
0|∅
1|1
20
0|0
0|10
1|∅
1|∅
0|∅
12
22
1|10
t(0100001) = 01001
Wave Pattern - Example for D = 3
1|01
21
11
0|∅
1|∅
10
00
1|∅
0|∅
0|∅
0|∅
1|1
20
0|0
0|10
1|∅
1|∅
0|∅
12
22
1|10
t(0100001) = 01001
∀u ∈ {0, 1}∗ , ∃n in 0(log |u|) such that t n (u) is a prefix of (01)ω
Wave Pattern - Example for D = 3
1|01
21
11
0|∅
1|∅
10
00
1|∅
0|∅
0|∅
0|∅
1|1
20
0|0
0|10
1|∅
1|∅
0|∅
12
22
1|10
t(0100001) = 01001
∀u ∈ {0, 1}∗ , ∃n in 0(log |u|) such that t n (u) is a prefix of (01)ω
Starting from a prefix of (01)ω , the sand pile becomes a wave
Wave Pattern - Example for D = 3
1|01
21
11
0|∅
1|∅
10
00
1|∅
0|∅
0|∅
0|∅
20
0|0
0|10
1|∅
1|∅
D −1
0|∅
12
22
1|10
t(0100001) = 01001
...
1|1
2
1
Wave (D − 1, ..., 1)k
∀u ∈ {0, 1}∗ , ∃n in 0(log |u|) such that t n (u) is a prefix of (01)ω
Starting from a prefix of (01)ω , the sand pile becomes a wave
Conclusion
N
O(log N)
1 2 3
π(N)
√
Θ( N)
O(log N)
4
columns
D=3
1
2
3
4
emergent density on avalanches
density used to built a transducer
transducer iteration involves a balanced evolution
balanced evolution involves the wave pattern
general D
Thank you for Sandpiling, Mister Goles