Defining an Energy in the Olami-Feder-Christensen Model

Transcription

1 Defining an Energy in the Olami-Feder-Christensen Model James B. Silva Boston University Collaborators : William Klein, Harvey Gould Kang Liu, Nick Lubbers, Rashi Verma, Tyler Xuan Gu

2 WHY STUDY EARTHQUAKES? Earthquakes have caused a tremendous amount of damage in the form of death tolls and economic damage Our understanding of earthquakes is not sufficient to prevent human and economic loss.

3 SHORT HISTORY OF EARTHQUAKE MONITORING Notable seismologist Harry Oscar Wood proposes a network of seismic stations in the article The earthquake problem in the western United States (1916) National Earthquake Hazards Reduction (NEHR) Act of 1977 Early warning systems using detection of fast compression waves installed nationwide in Japan since Early warning systems also installed in Mexico, Romania, and Taiwan.

4 Introduction Modeling Earthquake Fault Systems Olami-Feder-Christensen Defining an OFC energy Conclusions / Further Work T HE CHALLENGES OF UNDERSTANDING EARTHQUAKE FAULT SYSTEMS I Earthquake fault systems appear to be unique: I I I Different levels of quasi - periodicity Different compositions of individual faults Different levels of plate movement

5 THE GOALS OF MODELING EARTHQUAKES How do the physical properties of earthquake fault systems lead to earthquake events, their frequency, and occasionally quasi-periodicity? Can physics help reach the end goal of prediction in a reasonable time frame? Does research in earthquakes overlap with research on other avalanche processes (neurons in the brain, financial crisis, etc)?

6 THE GOALS OF MODELING EARTHQUAKES Governments across the world are starting to collect large amount of data on earthquake activity. How do we analyze this data and how does physics help guide this analysis? Are some vital measurements missing that should be collected?

7 THE PHYSICAL PROPERTIES OF EARTHQUAKE FAULT SYSTEMS Gutenberg-Richter Law (1). The number of earthquakes of size greater than a value M are given by the following: log 10 N = a bm Omori s Law (2). The rate of aftershocks decays in time as the following: R(t) = K (t+c) p

8 BUILDING A MODEL FOR AN EARTHQUAKE FAULT SYSTEM Block spring model: Linear springs connecting media on a rough surface that is loaded with a moving plate. (3) Loader plate drags the blocks until a block slips (fails) initiating an avalanche event. K c : Spring constant between a block and another block. K l : Spring constant between the blocks and the loader plate.

9 BUILDING A MODEL FOR AN EARTHQUAKE FAULT SYSTEM Model has a well defined energy. Simplifications (Rundle-Jackson-Brown [1977] (4) ): If a block slips then the block moves before the other blocks can react. Zero-velocity limit. Block system is pulled by a moving plate slow enough to fail a single site.

10 HOW EFFECTIVE OF A MODEL IS RUNDLE-JACKSON-BROWN Gutenberg-Richter Law scaling is observed in the models event size frequency distribution but an exponential cutoff is present. Short range nature of the stress transfer dampens the creation of large events.

11 IMPROVING THE MODEL: LONG RANGE STRESS TRANSFER Gutenberg-Richter Law scaling is observed in the models event size frequency distribution. Block spring interactions lead to power law scaling in event size distribution.

12 OLAMI-FEDER CHRISTENSEN RJB model is computationally expensive. Simplifications will be introduced. Olami-Feder-Christensen [1992] (5) Focus on buildup and failure of stress site. Displacement is not necessary but rather the sum of displacement is focus.

13 OLAMI-FEDER CHRISTENSEN

14 OLAMI-FEDER CHRISTENSEN

15 OLAMI-FEDER CHRISTENSEN

16 OLAMI-FEDER CHRISTENSEN

17 OLAMI-FEDER CHRISTENSEN

18 CHALLENGES FOR THE OFC MODEL OFC model does not have a well defined energy. A well defined energy helps in applying the toolbox of equilibrium physics. Previous work is suggestive of the applicability of equilibrium methods. Model does not exhibit Omori s law behavior in the aftershock rate.

19 MODEL DYNAMICS Olami-Feder-Christensen Rundle - Jackson - Brown Lattice of sites with stress σ System is loaded until a single site fails (zero velocity limit). Stress in failing site returns to residual level. Excess stress is dissipated to sites using long range stress transfer. σ diss = (1 α)[σ i σ f ] System of blocks with displacements φ (slip deficit) Stress is computed using : σ i = K L φ i + K c (φj φ i ) Loader plate in system is loaded until a single site fails (zero velocity limit). Failing block displacement is reassigned : φ f = φ i + σ i σ R K L +qk c

20 MODEL DYNAMICS Dynamics evolve identically for properly chosen set of parameters K L α = K L + qk c α - OFC stress dissipation K L - Loader plate - block spring constant K c - Block - block spring constant Total energy of RJB model is given by the following : E RJB = 1 2 [K lφ 2 i + j Range(R) K c (φ i φ j ) 2 ]

21 TESTING PROPOSED ENERGIES FOR A MAXWELL BOLTZMANN DISTRIBUTION A method is needed to test proposed energies. (6),(7) Given a system in thermal equilibrium the probability of the energy will follow a Maxwell Boltzmann distribution. P(E, β) = Ce βe Given 2 in thermal equilibrium following Maxwell-Boltzmann distribution (β1 < β2) P(E, β 1 ) P(E, β 2 ) e (β1 β2)e This can then be simplified log 10 ( P(E, β 1) P(E, β 2 ) ) = β E + b Necessary criteria for a proper energy

22 ENERGY IN RUNDLE-JACKSON-BROWN SYSTEMS Recall E RJB = 1 2 [K lφ 2 i + j Range(R) K c (φ i φ j ) 2 ]

23 AN ENERGY FOR OLAMI-FEDER-CHRISTENSEN SYSTEMS Recall : σ i = K L φ i + K c (φj φ i ) We propose the following energy function for the OFC model. E OFC N i σ 2 i

24 AN ENERGY FOR OLAMI-FEDER-CHRISTENSEN SYSTEMS Maxwell Boltzmann assumption does not hold for energy function given short range stress transfer as expected.

25 AN ENERGY FOR OLAMI-FEDER-CHRISTENSEN SYSTEMS Maxwell Boltzmann assumption holds for energy function for long range stress transfer.

26 Choose parameters for OFC and RJB model to match dynamics resulting in identical statistics. Measured slopes are within 1% each of each other; 658 for the RJB model, and 651 for the OFC model. AN ENERGY FOR OLAMI-FEDER-CHRISTENSEN SYSTEMS

27 CONCLUSIONS One can observe the basic properties of earthquake fault systems in block-spring models. It is possible to fully connect the block spring models including the OFC model. Energy like variable defined by E OFC σ 2 Step towards building equilibrium theory for OFC model No need to run computationally expensive models.

28 FURTHER WORK Why does this OFC variable behave like an energy analog. Under what conditions does this Maxwell-Boltzmann like behavior hold. Do not expect this to hold for small dissipation where system is not ergodic. Dynamics reminiscent of neural integrate and fire models Are there connections between earthquake fault models and integrate and fire models of neurons. More realistic model for an earthquake fault system Incorporate asperities into OFC model / energy.

29 SPECIAL THANKS TO COLLABORATORS Bill Klein Harvey Gould Rashi Verma Kang Liu Nick Lubbers Tyler Xuan Gu

30 ERGODICITY The ergodic hypothesis is a large part of classical statistical mechanics. A system is ergodic if it samples the state space such that a time average is equivalent to ensemble average.

31 ERGODICITY Following Thirumalai-Mountain the Thirumalai-Mountain metric can be used to determine effective ergodicity Ω = 1 N N i (< σ > t < σ > N ) 2 For an effectively ergodic system the TM metric follows : t 1 Ω

32 OFC ERGODICITY OFC system appears to be effectively ergodic given a large enough interaction in the stress transfer

33 OFC METRIC NEAREST NEIGHBOR At large residual stress noise expect ergodicity

34 OFC METRIC NEAREST NEIGHBOR At low enough residual stress noise ergodicity breaks again

35 REFERENCES I B. Gutenberg, C. F. Richter, Annali di Geofisica 9 (1956). F. Omori, Journal of the College of Science, Imperial University of Tokyo 7 (1894). R. Burridge, L. Knopoff, Bull. Seismol. Soc. Am. 57 (1967). J. B. Rundle, D. Jackson, Bull. Seismol. Soc. Am. 67(5) (1977). Z. Olami, K. Christensen, Phys. Rev. Lett. 62 (1992). R. J. B., K. W., G. S., T. D. L., Physical Review Letters 78 (1997). S. D., X. H. J., Physical Review Letters 78 (1997).