William Klein

Statistical physics approach to understanding the multiscale dynamics of earthquake fault systems

[1] Earthquakes and the faults upon which they occur interact over a wide range of spatial and temporal scales. In addition, many aspects of regional seismicity appear to be stochastic both in space and time. However, within this complexity, there is considerable self-organization. We argue that the occurrence of earthquakes is a problem that can be attacked using the fundamentals of statistical physics. Concepts of statistical physics associated with phase changes and critical points have been successfully applied to a variety of cellular automata models. Examples include sandpile models, forest fire models, and, particularly, slider block models. These models exhibit avalanche behavior very similar to observed seismicity. A fundamental question is whether variations in seismicity can be used to successfully forecast the occurrence of earthquakes. Several attempts have been made to utilize precursory seismic activation and quiescence to make earthquake forecasts, some of which show promise.

Space-time clustering and correlations of major earthquakes.

Earthquake occurrence in nature is thought to result from correlated elastic stresses, leading to clustering in space and time. We show that the occurrence of major earthquakes in California correlates with time intervals when fluctuations in small earthquakes are suppressed relative to the long term average. We estimate a probability of less than 1% that this coincidence is due to random clustering.

Repulsive potentials, clumps and the metastable glass phase

We present the results of simulations and theoretical investigations of a system of particles with a weak, long-range repulsive interaction. At fixed density the fluid phase, which is stable at high temperatures, becomes metastable and then unstable as the temperature is lowered. For low temperatures the particles from approximately equal size clumps that interact weakly with neighboring clumps. The global free energy minimum at low temperatures corresponds to a crystalline lattice clump structure. However the free energy surface has many minima associated with metastable amorphous phases. Quenches from the fluid phase to low temperatures almost always will result in an amorphous clump structure with the number of clumps dependent on quench history. The association of the amorphous phase with a large number of metastable minima is similar to a mean-field spin glass.

GEM Plate Boundary Simulations for the Plate Boundary Observatory: A Program for Understanding the Physics of Earthquakes on Complex Fault Networks via Observations, Theory and Numerical Simulation

The last five years have seen unprecedented growth in the amount and quality of geodetic data collected to characterize crustal deformation in earthquake-prone areas such as California and Japan. The installation of the Southern California Integrated Geodetic Network (SCIGN) and the Bay Area Regional Deformation (BARD) network are two examples. As part of the recently proposed Earthscope NSF/GEO/EAR/MRE initiative, the Plate Boundary Observatory (PBO) plans to place more than a thousand GPS, strainmeters, and deformation sensors along the active plate boundary of the western coast of the United States, Mexico and Canada (http://www.earthscope.org/pbo.com.html). The scientific goals of PBO include understanding how tectonic plates interact, together with an emphasis on understanding the physics of earthquakes. However, the problem of understanding the physics of earthquakes on complex fault networks through observations alone is complicated by our inability to study the problem in a manner familiar to laboratory scientists, by means of controlled, fully reproducible experiments. We have therefore been motivated to construct a numerical simulation technology that will allow us to study earthquake physics via numerical experiments. To be considered successful, the simulations must not only produce observables that are maximally similar to those seen by the PBO and other observing programs, but in addition the simulations must provide dynamical predictions that can be falsified by means of observations on the real fault networks. In general, the dynamical behavior of earthquakes on complex fault networks is a result of the interplay between the geometric structure of the fault network and the physics of the frictional sliding process. In constructing numerical simulations of a complex fault network, we will need to solve a variety of problems, including the development of analysis techniques (also called data mining), data assimilation, space-time pattern definition and analysis, and visualization needs. Using 1 Colorado Center for Chaos and Complexity, CIRES, and Department of Physics, CB 216, U.S.A. E-mail: rundle@cires.colorado.edu 2 Department of Physics, 301 E 12th St., Harvey Mudd College, Claremont, CA 91711, U.S.A. E-mail: paulrun@cires.colorado.edu 3 Department of Physics, Boston University, Boston, MA 02215, and Center for Nonlinear Science, Los Alamos National Laboratory, Los Alamos, NM 87545, U.S.A. E-mail: klein@cnls.lanl.gov 4 Colorado Center for Chaos and Complexity, and CIRES, CB 216, University of Colorado, Boulder, CO 80309, U.S.A. E-mails: jorge@cires.colorado.edu; kristy@cires.colorado.edu 5 Exploration Systems Autonomy Division, Jet Propulsion Laboratory, Pasadena, CA 91109-8099, U.S.A. E-mail: donnellan@jpl.nasa.gov 6 Department of Geology, University of California, Davis, CA 95616, U.S.A. E-mail: kellogg@geology.ucdavis.edu Pure appl. geophys. 159 (2002) 2357–2381 0033 – 4553/02/102357 – 25

New ideas about the physics of earthquakes

1.50+0.20/0 Birkhäuser Verlag, Basel, 2002 Pure and Applied Geophysics

Molecular dynamics investigation of deeply quenched liquids

It may be no exaggeration to claim that this most recent quaddrenium has seen more controversy and thus more progress in understanding the physics of earthquakes than any in recent memory. The most interesting development has clearly been the emergence of a large community of condensed matter physicists around the world who have begun working on the problem of earthquake physics. These scientists bring to the study of earthquakes an entirely new viewpoint, grounded in the physics of nucleation and critical phenomena in thermal, magnetic, and other systems. Moreover, a surprising technology transfer from geophysics to other fields has been made possible by the realization that models originally proposed to explain self-organization in earthquakes can also be used to explain similar processes in problems as disparate as brain dynamics in neurobiology (Hopfield, 1994), and charge density waves in solids (Brown and Gruner, 1994). An entirely new sub-discipline is emerging that is focused around the development and analysis of large scale numerical simulations of the dynamics of faults. At the same time, intriguing new laboratory and field data, together with insightful physical reasoning, has led to significant advances in our understanding of earthquake source physics. As a consequence, we can anticipate substantial improvement in our ability to understand the nature of earthquake occurrence. Moreover, while much research in the area of earthquake physics is fundamental in character, the results have many potential applications (Cornell et al., 1993) in the areas of earthquake risk and hazard analysis, and seismic zonation.

Near-mean-field behavior in the generalized Burridge-Knopoff earthquake model with variable-range stress transfer.

Molecular dynamics simulations of homogeneous crystalline nucleation in systems of 1300 particles have been performed as a function of quench depth for the Lennard‐Jones, r−12, and r−6 potentials. We observe that the nucleating droplet is spatially asymmetric, has a layered structure, and is ramified for deep quenches. The initial growth of the droplet occurs by the addition of layers until the droplet becomes sufficiently large to be characterized by a crystalline close‐packed structure. We also observe that the time lag between the time of formation of the nucleating droplet and the time of release of latent heat is a nonmonotonic function of quench depth. The results for deep quenches are interpreted as evidence for the influence of a pseudospinodal.

Ergodicity Breaking in Geometric Brownian Motion

Simple models of earthquake faults are important for understanding the mechanisms for their observed behavior in nature, such as Gutenberg-Richter scaling. Because of the importance of long-range interactions in an elastic medium, we generalize the Burridge-Knopoff slider-block model to include variable range stress transfer. We find that the Burridge-Knopoff model with long-range stress transfer exhibits qualitatively different behavior than the corresponding long-range cellular automata models and the usual Burridge-Knopoff model with nearest-neighbor stress transfer, depending on how quickly the friction force weakens with increasing velocity. Extensive simulations of quasiperiodic characteristic events, mode-switching phenomena, ergodicity, and waiting-time distributions are also discussed. Our results are consistent with the existence of a mean-field critical point and have important implications for our understanding of earthquakes and other driven dissipative systems.

Simulation of the Burridge-Knopoff Model of Earthquakes with Variable Range Stress Transfer

Geometric Brownian motion (GBM) is a model for systems as varied as financial instruments and populations. The statistical properties of GBM are complicated by nonergodicity, which can lead to ensemble averages exhibiting exponential growth while any individual trajectory collapses according to its time average. A common tactic for bringing time averages closer to ensemble averages is diversification. In this Letter, we study the effects of diversification using the concept of ergodicity breaking.

Born–Green hierarchy for continuum percolation

Simple models of earthquake faults are important for understanding the mechanisms for their observed behavior, such as Gutenberg-Richter scaling and the relation between large and small events, which is the basis for various forecasting methods. Although cellular automaton models have been studied extensively in the long-range stress transfer limit, this limit has not been studied for the Burridge-Knopoff model, which includes more realistic friction forces and inertia. We find that the latter model with long-range stress transfer exhibits qualitatively different behavior than both the long-range cellular automaton models and the usual Burridge-Knopoff model with nearest-neighbor springs, depending on the nature of the velocity-weakening friction force. These results have important implications for our understanding of earthquakes and other driven dissipative systems.