Simple models of earthquake faults:Examples of driven dissipative

systemsA symposium in honor of Gene F. Mazenko

University of Chicago, May 18, 2015

Harvey Gould, Clark University and BostonUniversity

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Collaborators

Bill Klein, Boston UniversityJohn Rundle, University of California, DavisKristy Tiampo, University of Western OntarioJan Tobochnik, Kalamazoo College

Former graduate studentsRachele Dominguez, Boston University →

Randolph-Macon CollegeChris Serino, Boston University → Lincoln LabsJunchao Xia, Clark → BioMaPS Institute, Rutgers

Current graduate studentsTyler Xuan Gu, Boston UniversityJames Silva, Boston University

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Motivation: Gutenberg-Richter scaling

Total number of events with moment greater thanor equal to moment M (energy)

NM ∝ M−β (β ≈ 2/3)

Is Gutenberg-Richter scaling related to criticalphenomena and self-organized criticality?

Will discuss three models in order of decreasingcomplexity that yield power law scaling:

I Burridge-Knopoff, blocks and springs; need tosolve Newton’s equations.

I Rundle-Jackson-Brown, stress and energy,cellular automaton.

I Olami-Feder-Christensen (OFC), stress, cellularautomaton. 3 / 16

Burridge-Knopoff slider-block model (1967)

moving plate

V

frictional surface fixed plate

2D Nearest -Neighbor Burridge- Knopoff Model

KCKC

KL

Blocks connected by springs and coupled by springsto loader plate and pulled over a rough surfacedescribed by a velocity-weakening friction law.Need to solve Newton’s equations numerically.Carlson and Langer (1989); Xia, Gould, Klein, andRundle (2005).

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Rundle, Jackson, Brown (RJB) Cellular Automaton Model

Cellular automaton version of Burridge-Knopoffmodel. Ignore inertia effects. Each site has stress σiand energy εi . Rundle and Jackson (1977), Rundleand Brown (1991).

The evolution of the stress is same as Olami, Feder,and Christensen model (1992) so will emphasize thelatter.

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Olami, Feder, and Christensen (OFC) model

Stress σi on each site i with failure threshold σFand mean residual stress σR . Dissipation coefficientα. Initially distribute stress at random.

1. If σi ≥ σF , reduce σi to σR + η and distributestress (1− α)(σi − σR − η) to neighbors; ηrepresents noise.

2. Check neighbor sites and go to step 1.

3. Continue until σi < σF for all i . Number of sitesthat fail constitutes an earthquake of size s.

4. Reload system: Find site with maximum stressand bring it to failure by adding σF − σmax to allsites.

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What have we learned?

• ns , number of events of size s, does not scale ifstress distributed only to nearest neighbors (R = 1).• Give stress equally to all sites within distanceR 1. Long-range mimics elastic force.

ns ∼ s−3/2 (s 1)

-8

-7

-6

-5

-4

-3

-2

-1

0

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

log s

log

ns

R = 30, α = 0.01, σF = 1.0, σR = 0.25± 0.25, L = 256,3× 107 events, periodic boundary conditions.

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More results for R 1 (near-mean field)

• Form of Langevin equation same as for Isingmodel. =⇒ OFC model in equilibrium for R 1.

lnP(E ) ∝ E , E =1

2

N∑i=1

σ2i (OFC model).

• Scaling of events similar to scaling of clusters inmean-field Ising model near spinodal.

ns ∼ s−τe−∆h sσ (τ = 5/2, σ = 1)

(τ = 7/3 for Ising mean-field critical point.)Ising clusters generated by tossing random bondsbetween parallel spins. Earthquakes grow from asingle site.

s ns(random bond) = ns(seed)

ns ∼ s−3/2 (number of events)

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Spread of ns s 1 not due to poor statistics

-8

-7

-6

-5

-4

-3

-2

-1

0

1.5 2.5 3.5 4.5log s

τfc = 1.48 ± 0.04

τfn = 1.51 ± 0.01

τan = 2.01 ± 0.01

2.01.0 3.0 4.0 5.0

log

ns

τ =

3/2 clusters σmin ∈ [0.99, 1.0)

3/2 failed nucleation σmin ∈ [0.90, 0.99)

2 arrested nucleation σmin ∈ [0.0, 0.90)

Time between given type of event ∝ e−t/τ . 9 / 16

Punctuated equilibrium

6100

6120

6140

6160

6180

6200

6220

1.841 105 1.842 105 1.843 105 1.844 105 1.845 105 1.846 105 1.847 105

initi

al s

tress

event0 0.5 1 1.5 2 2.5 3 3.5 4

x 105

0

500

1000

1500

2000

2500

3000

time

Ωσ(0)/Ωσ(t)

σi(t) =1

t

∫ t

0

σi(t′)dt ′ <σ(t)> =

1

N

N∑i=1

σi(t).

Ω(t) =1

N

N∑i=1

[σi(t)−<σ(t)>

]2(Thirumalai-Mountain)

If system is effectively ergodic, Ω(t) ∝ 1/t. 10 / 16

Gutenberg-Richter scaling [Serino et al. (2011)]

• OFC model describes individual faults rather thanfault systems. The latter satisfy Gutenberg-Richterscaling, single faults may or may not.• s ∝ M in mean-field limit.β = τ − 1 = (3/2)− 1 = 1/2 [empirical β ≈ 2/3]• Randomly eliminate fraction q of sites. Dead sitesdue to different levels of fracture or gouge. Whenstress is transferred to a dead site, it is dissipated.• Model fault system as collection of noninteractingOFC models or “faults” with different values of q.

ns ∼∫ 1

0

dq D(q)ns(q) ∼∫ 1

0

dq1

qx

[1

(1− q)

e−q2s

sτ

]∼ 1

s τ.

τ = 2− x/2 =⇒ x = 2/3.11 / 16

What about foreshocks, aftershocks, and quasiperiodiclarge events?

• In addition to three types of scaling events, thereexist rare large “breakout” events that do not scale.• Precursors and aftershocks exist in the OFCmodel for R = 1, but not for R 1.• Add asperity sites with larger σF . Find foreshocksprior to a main shock followed by aftershocks.• Spatial and temporal patterns observed in naturalseismicity are strongly influenced by the underlyingphysical properties and are not solely the result of asimple cascade mechanism [Kazemian et al. (2015)].

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Burridge-Knopoff model

• Larger αBK =⇒ friction force decreases morerapidly with increasing velocity.• R = 1 (no theory):

P(M) ∼ M−2 (αBK & 1).

M ∝∑

i ui ; ui = displacement of block i ; sum overblocks in event. Larger events have characteristicsize. No power law behavior for ns .• R 1. One block connected to 2R neighborswith rescaled spring constant k/R .

ns ∼

s−2 (αBK < 1) no theory

s−3/2 (αBK > 1) same as OFC model

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Summary

I Olami-Feder-Christensen and Rundle-Jackson-Brown models effectively ergodic for R 1.

I Gutenberg-Richter scaling associated with apseudospinodal.

I Behavior of R 1 Burridge-Knopoff modeldepends on how quickly friction force decreaseswith increasing velocity.

I Real earthquake faults can have differentbehavior depending on friction force, range ofstress transfer, and presence of asperities.

I Does R = 1 OFC have a critical point at α = 0?I Do other driven dissipative systems become

effective ergodic in mean-field limit?14 / 16

References• B. Gutenberg and C. F. Richter, Seismicity of the Earth and AssociatedPhenomenon, 2nd ed. (Princeton University Press, 1954).• R. Burridge and L. K. Knopoff, “Model and theoretical seismicity,”Bull. Seismol. Soc. Am. 57, 341 (1967).• P. Bak, C. Tang, and K. Wiesenfeld, “Self-organized criticality,” Phys.Rev. Lett. 59, 381 (1987).• J. M. Carlson and J. S. Langer, “Properties of earthquakes generatedby fault dynamics,” Phys. Rev. Lett. 62, 2632 (1989); ibid., “Mechanicalmodel of an earthquake fault,” Phys. Res. A 40, 6470 (1989).• J. B. Rundle and D. D. Jackson, “Numerical simulation of earthquakesequences,” Bull. Seismol. Soc. Am. 67, 1363 (1977); J. B. Rundle andS. R. Brown, “Origin of rate dependence in frictional sliding,” J. Stat.Phys. 65, 403 (1991).• Z. Olami, J. S. Feder, and K. Christensen, “Self-organized criticality ina continuum, nonconservative cellular automaton modeling earthquakes,”Phys. Rev. Lett. 68, 1244 (1992).• J. B. Rundle, W. Klein, S. Gross, and D. L. Turcotte, “Boltzmannfluctuations in numerical simulations of nonequilibrium lattice thresholdsystems, Phys. Rev. Lett. 75, 1658 (1995).

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• W. Klein, J. B. Rundle, and C. D. Ferguson, “Scaling and nucleation inmodels of earthquake faults,” Phys. Rev. Lett. 78, 3793 (1997).• W. Klein, M. Anghel, C. D. Ferguson, J. B. Rundle, and J. S. SaMartins, “Statistical analysis of a model for earthquake faults withlong-range stress transfer,” in Geocomplexity and the Physics ofEarthquakes, edited by J. B. Rundle, D. L. Turcotte, and W. Klein(American Geophysical Union, 2000).• J. Xia, H. Gould, W. Klein, and J. B. Rundle, “Simulation of theBurridge-Knopoff model of earthquakes with variable range stresstransfer,” Phys. Rev. Lett. 95, 248501 (2005); ibid., “Near-mean- fieldbehavior in the generalized Burridge-Knopoff earthquake model withvariable-range stress transfer,” Phys. Rev. E 77, 031132 (2008).• C. A. Serino, K. F. Tiampo, and W. Klein, “New approach toGutenberg-Richter scaling,” Phys. Rev. Lett. 106, 108501 (2011).• J. Kazemian, K. F. Tiampo, W. Klein, and R. Dominguez, “Foreshockand aftershocks in simple earthquake models,” Phys. Rev. Lett. 114,088501 (2015).• J. Xia, C. A. Serino, M. Anghel, J. Tobochnik, H. Gould, and W. Klein,“Scaling behavior of the distribution of failed sites in the long-rangeOlami-Feder-Christensen model,” unpublished.

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