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/