Kim Christensen
Blackett Laboratory, Imperial College London 05.12.2006
Complexity and Criticality
• Criticality
–
Ising model
–
Phase transition at (Tc , 0)
–
Scale invariance and £xed points
• Complexity
–
Earthquakes and rainfall
–
Rice-pile experiment
–
Oslo rice-pile model
–
Self-organised criticality
m(T, 0)
1
↑
↓
↑
↓
↑
↑
↑
↑
↑
↓
↓
↑
↓
↓
↑
↓
↓
↑
↑
↑
↓
↓
↓
↑
↑
0
−1
0
1
T /Tc
De£nition
Criticality: Ising Model
The simplest model of a ferromagnet consists of N spins s i = ±1 = ↑ or ↓,
i = 1, . . . , N with constant nearest-neighbour interaction J > 0 placed in a
uniform external £eld H . The energy of microstate {si }
= {s1 , s2 , . . . , sN }
↑
↓
↑
↓
↑
E{si } = spin-spin interaction + spin-external £eld interaction
= −J
X
hiji
si sj − H
N
X
si .
i=1
The partition function
Z(T, H) =
X
{si }
The free energy per spin
¡
The magnetisation per spin
m(T, H) = −
¢
exp −βE{si } .
µ
↑
↑
↑
↑
↓
∂f
∂H
¶
T
.
1
f (T, H) = − kB T ln Z.
N
The susceptibility per spin
χ(T, H) =
µ
∂m
∂H
¶
T
.
↓
↑
↓
↓
↑
↓
↓
↑
↑
↑
↓
↓
↓
↑
↑
Phase transition at (T, H)
f (T, H)
= (Tc , 0)
Criticality: Ising Model
Free energy per spin in mean-£eld model.
H
Critical point (T, H)
= (Tc , 0)
m(T, H)
T
H
T
Magnetisation per spin in mean-£eld model.
Phase transition at (T, H)
Assume H
= (Tc , 0)
Criticality: Ising Model
= 0. In equilibrium, the free energy is minimised
F = hEi − T S.
T = 0: Energy minimised: spins are aligned: m(0, 0) = ±1.
Con£gurations are self-similar. The correlation length ξ(0, 0) = 0.
T = ∞: Entropy maximised: spins are randomly orientated: m(∞, 0) = 0.
Con£gurations are self-similar. The correlation length ξ(∞, 0) = 0.
T = Tc : hEi and T S balanced. Spins are “undetermined”: m(Tc , 0) = 0.
Con£gurations are self-similar. The correlation length ξ(Tc , 0) = ∞.
T →0
T = Tc
T →∞
Scale invariance and £xed points
m(T, 0)
1
0
0
Criticality: Ising Model
m(T, 0) ∝ ±(Tc − T )β for T → Tc− .
1
T /Tc
−1
ξ(T, 0)
ξ
Scaling factor b
ξ/b
ξ(T, 0) ∝ |Tc − T |−ν for T → Tc .
ξ/b2
0
> 1.
1
T /Tc
Scale invariance and £xed points
T < Tc
Criticality: Ising Model
T = Tc
T > Tc
ξ → ξ/b
ξ/b → ξ/b2
Fixed points ξ
= ξ/b
ξ=0
ξ=∞
ξ=0
Warning
Equilibrium vs. Non-equilibrium
• No truly isolated natural systems exist.
• Most systems have a ¤ux of mass or energy passing though them.
• Most systems are in a non-equilibrium steady state.
• Take great care not to apply results from equilibrium systems outside their
range of validity.
Plate tectonics
Complexity: Earthquakes
Palace Hotel, San Francisco, U.S.A.
5:12AM – 18 April, 1906.
World-wide occurrence of earthquakes.
Outline plate boundaries.
1.1.1997 − 30.6.1997, M > 4
Earthquake catalogue
Scale invariance: Gutenberg-Richter Law
Complexity: Earthquakes
At a fault, strain builds up slowly.
Energy released through earthquakes.
Gutenberg-Richter Law:
N (S > s) ∝ s−b .
Large truck passing by
Small atom bomb
100 hydrogen bombs
N (S
> s) earthquakes/year
N(S>s) [earthquakes/year]
10
10
10
10
10
10
5
4
3
b = 0.95
2
1
0
−1
10
−2
10
0
1
2
3
4
5
6
7
Magnitude
=loglog
Magnitudem
m=
10(S)
10 s
8
Rain event
Complexity: Rainfall
Europe, August 2002.
Level of River Elbe =
9.39m.
Rain event over Grand Canyon
dissipates energy in the
atmosphere.
Rain gauges
Standard rain gauge: Tipping bucket.
Resolution of rain rate qmin = 0.25 mm/h.
Temporal resolution ∆t = ? min.
Radar: Resolution of rain rate
qmin = 0.005 mm/h.
Temporal resolution ∆t = 1 min.
Complexity: Rainfall
A rain event is a sequence of successive
non-zero rain rates. The event size
M = q(t + 1) + · · · + q(t + T ).
with event duration T .
Complexity: Rainfall
Rain rate q(t) [mm/min]
Rain event de£nition
1.50
1.25
1.00
0.75
0.50
0.25
0
0
50000
100000 150000 200000 250000
Rain-equivalent of Gutenberg-Richter Law:
N (M ) ∝ M −τ M .
N (M ) [event/year/mm]
Time t [min]
10
10
5
4
10
10
2
10
10
10
τM = 1.4
3
1
0
-1
-2
10 -4
10
10
-3
10
-2
10
-1
10
0
10
Event size M [mm]
1
10
2
Conclusion
Complexity: Earthquakes and Rainfall
• Rain is a complicated spatio-temporal phenomenon.
• Trickles, drizzle, bursts, showers, downpours, and torrents.
• Identifying rain events as the basic entities reveals that
– The frequency-event size distribution is scale free.
This is the rain-equivalent of the Gutenberg-Richter law for earthquakes.
– The frequency-drought duration distribution is scale free.
This is the rain-equivalent of the Omori law for earthquakes.
Rain is “Earthquake in the Sky”.
Experimental setup
Complexity: Rice-pile experiment
L
• Reaches statistically stationary state where hin¤uxi = hout¤uxi.
• Avalanches dissipate energy.
Statistically stationary states
Complexity: Rice-pile experiment
Avalanche size E
E = 294
L = 33
Complexity: Rice-pile experiment
E = 2868
L = 33
Energy dissipated by avalanche, E , measured in unit of mgδ = 1.54µJ .
Focus on avalanche-size probability density, P (E; L) dE , in system of size L.
Avalanche-size probability
Complexity: Rice-pile experiment
0.3
1
0.8
0.6
Rescaled avalanche sizes, E/Emax
as function of number of additions.
E/Emax
0.4
0.2
0.2
0
0
2000
4000
t 6000
10
1000
t
1200
L = 16
L = 33
L = 66
L = 105
P (E; L) ∝ E
for E
À 1.
P (E; L)
10
10
-4
τE ≈ 2
-5
10
10
1400
-2
-3
−τE
10000
0.1
0
800
Avalanche-size probability densities:
8000
-6
-7
10
0
10
1
10
2
10
3
10
E [1.54µJ]
4
10
10
5
Conclusion
Complexity: Earthquakes, rainfall, and avalanches
System
Crust of Earth
Atmosphere
Granular pile
Energy source
Convection
Sun
Adding grains
Energy storage
Tension
Vapour
Potential
Threshold
Friction
Saturation
Friction
Relaxation
Earthquake
Rain event
Avalanche
Common basis:
• Slowly driven non-equilibrium systems.
• Threshold dynamics.
• Relaxation event dissipates energy.
• Reaches statistically stationary state where hin¤uxi = hout¤uxi.
• Relaxation events of all sizes up to a system dependent cutoff.
De£nition
Complexity: Oslo rice-pile model
Lattice with L sites, vertical wall at left boundary and open at right boundary.
• The height, hi , is the number of grains at column i, with hL+1 = 0.
• The local slopes zi = hi − hi+1 , i = 1, . . . , L.
1
2
L
1
2
L
1
V
Train model of earthquakes.
2
L
De£nition
Complexity: Oslo rice-pile model
The algorithm for the Oslo rice-pile model:
1. Initialise the critical slopes zic
∈ {1, 2} and place the system in an
arbitrary metastable state with zi ≤ zic for all i.
2. Add a grain at site i
3. If zi
= 1: z1 → z1 + 1.
> zic , the site relaxes and
zi → z i − 2
zi±1 → zi±1 + 1.
The critical slope zic is chosen randomly zic
∈ {1, 2}
A new metastable state is reached when zi ≤ zic for all i.
4. Proceed to 2 and reiterate.
Avalanche-size probability
Complexity: Oslo rice-pile model
1.0
Rescaled avalanche sizes, s/smax
as function of number of additions.
s/smax
0.8
0.6
0.4
0.2
0
0
2000
4000
6000
t
8000
10000
0
10
L = 64
L = 256
L = 1024
L = 4096
-2
10
P (s; L) ∝ s
−τs
D
G(s/L ).
P (s; L)
Avalanche-size probability densities:
-4
10
-6
10
-8
10
-10
10
-12
10
-14
10
0
10
1
10
2
10
3
10
4
10
s
5
10
6
10
7
10
8
10
Data collapse reveals scaling function G
Complexity: Oslo rice-pile model
0
Transformed avalanche-size
probability.
sτs P (s; L)
10
-1
10
10
-2
-3
10
0
10
1
10
2
10
3
10
4
10
s
5
10
6
10
7
10
8
10
9
10
0
sτs P (s; L) ∝ G(s/LD ).
Data collapse obtained using
exponents τs = 1.55, D = 2.25.
sτs P (s; L)
10
-1
10
10
-2
-3
10
-8
10
-7
10
-6
10
-5
10
-4
10
-3
10
s/LD
-2
10
-1
10
0
10
10
1
Conclusion
Complexity: Oslo rice-pile model
• The susceptibility is the average avalanche size: hsi = L.
• The slowly driven pile organises itself, without any external £ne-tuning of
control parameters, into a highly susceptible state where the susceptibility
diverges with system size.
• The Oslo rice-pile model displays self-organised criticality.
• The Oslo model is the “Ising Model” for self-organised criticality.
Collaborators
Kim Christensen
Per Bak.
Leon Danon, University of Barcelona, Spain.
Tim Scanlon, Imperial.
Vidar Frette, Haugesund Højskole, Norway.
Anders Malthe-Sørenssen, University of Oslo, Norway.
Jens Feder, PGP, University of Oslo, Norway.
Torstein Jøssang, PGP, University of Oslo, Norway.
Paul Meakin, PGP, University of Oslo, Norway.
Alvaro Corral, Universitat Autonoma de Barcelona, Spain.
Gunnar Pruessner, Imperial.
Matthew Stapleton, Imperial.
Nicholas Moloney, Elte University, Budapest, Hungary.
Ole Peters, Santa Fe Institute and Los Alamos Natinal Laboratory, U.S.A.
Christopher Hertlein, Germany.
Papers/animations available via: www.cmth.ph.ic.ac.uk/people/k.christensen/