## Modelling Critical and Catastrophic Phenomena in Geoscience: A Statistical Physics Approach

Statistical Features of Earthquake Temporal Occurrence. Spatiotemporal Correlations of Earthquakes. De Rubeis, V. Loreto, L. Pietronero, P. Tabar, M. Sahimi, F. Ghasemi, K. Kaviani, M. Allamehzadeh, J. Peinke et al. Aeolian Transport and Dune Formation. Avalanches and Ripples in Sandpiles. Das, H. Chauduri, R. Bhandari, D. Ghose, P. Sen, B. A Thermomechanical Model of Earthquakes. Nurujjaman, R. The first mean field slider block model was discussed by Rundle and Klein []. The driver plate is attached to the slider block by a loader spring with spring constant k L.

The interaction of the block with the surface is controlled by friction in some form. The static frictional threshold, associated with the static coefficient of friction, determines the force required to initiate slip of the block. A variety of laws have been proposed for the dynamic friction operative during slip. The dynamic friction can be taken to be constant, a function of the slip velocity or, more generally, a function of both slip velocity and slip history rate and state friction.

This behavior is analogous to the classical hypothesis of periodic characteristic earthquakes on major faults [ Schwartz and Coppersmith , ]. Both blocks are attached to a constant velocity driver plate by loader springs, each having spring constant k L. The blocks are connected to each other with a connecting spring, where the connecting spring constant k C is not necessarily equal to the loader spring constant k L.

In between these two limits a wide range of interesting behaviors can occur. If the system is symmetric equal masses, friction, and connecting spring constants , periodic behavior is also observed. However, if the symmetry is broken i. The behavior is quasiperiodic and cannot be predicted with certainty. Similarly, the chaotic behavior of a pair of slider blocks is evidence that earthquakes might exhibit chaotic behavior.

Chaotic behavior has led to the concept of ensemble weather forecasting. The sensitivity of a forecast to initial conditions is taken to be a measure of the reliability of a forecast. Similarly for earthquakes, absolute prediction of the time and place of an event is not possible for a chaotic system. However, probabilistic forecasts are certainly possible, and it may also be possible to utilize ensemble techniques to establish the reliability of forecasts. The standard multiple slider block model consists of a square array of slider blocks as illustrated in Figure 8.

In its simplest form the blocks have equal masses, are connected to the constant velocity driver plate with loader springs spring constant k L , and are connected to each other with connecting or coupling springs spring constant k C. It is further assumed that motion of the driver plate is so slow that it is appropriate to neglect its motion during a slip event. Variations include the following:.

The required numerical codes are very similar to the MD codes used to study the behavior of solids, which simply solve Newton's second law equations for particles moving in the combined potential of all the other particles. This approach greatly simplifies computations since only one equation must be solved at a time. In the CA case, time is defined in units of Monte Carlo sweeps that can be related to actual time via the various dimensional constants in the problem [ Binder and Heermann , ].

CA simulations often have considerable physical meaning since most interesting nonlinear processes have a strong stochastic component, and the resulting probability distributions are the relationships to be compared to data. CA simulations also have the distinct advantage of being far faster to run on modern computers, so very large systems sizes can be used, and finite size effects are consequently minimized.

By contrast, MD simulations are extremely computing intensive, so simulations are frequently limited to small system sizes a few hundred particles and very short time intervals typically microseconds or less for the entire run. Morein et al. No driver plate was used, and energy was conserved.

The final state of a block after slip is then specified by a jump or transition rule. Otherwise, if masses are not neglected, the differential equations of motion Newton's second law must be solved. Stress is redistributed to q other blocks as a function of the block's distance from the slipping block. In the extreme case of redistribution from one block to all other blocks, fluctuations in stress force are averaged out, and one approaches the mean field regime, where each block is considered to interact with the mean field produced by all other blocks.

After each change in sliding velocity it is observed that stress relaxes to a new value over some characteristic sliding distance. Conditions under which these friction laws are observed to have unstable sliding include an extremely stiff testing machine and clean sliding surfaces. An example is given in Figure 9. However, recent results [ Rundle and Klein , , b ; Rundle et al. Because the elastic interactions are long range [ Klein and Unger , ; Ray and Klein , ], near mean field conditions prevail, and the fault can approach a near spinodal line [ Rundle and Klein , ].

In fact, in the case of repeated earthquakes on a fault through time the system is regarded as residing permanently in the neighborhood of the spinodal line, executing a variety of fluctuations near the spinodal line through time [ Klein et al. In this picture the order parameter for the fault can be regarded as either the stress, the slip, or the slip deficit, all of which change dramatically at the time of an earthquake on the fault.

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Equations of the form of equation 19 have been used to describe systems with multiple possible states, in which sudden transitions between the states are possible. The control parameter f determines whether the function U is symmetric under a space reflection or not.

Similarly, equilibrium is described by the following equation: which is obtained by taking the derivative of equation 19 and which is a representation of the principle of minimum free energy. These equations describe a simple model of stable sliding. As f is increased, the left well becomes shallower, and the right well becomes deeper.

When f is increased further, an inflection point will occur, and the left minimum will disappear; this is the limit of metastability. For the slider block model this limiting value of f corresponds to the static coefficient of friction, and the block will suddenly slip, thus reducing f. The local maximum is correspondingly a state of locally unstable equilibrium. This model is therefore a candidate for a process of unstable stick slip: If the system configuration initially corresponds to the metastable state, as f increases, a sudden transition will occur at some point as the system configuration moves from the higher free energy, metastable well to the lower free energy, stable well.

The spinodal line represents the classical limit of stability of the system, since the disappearance of the energy barrier means that the system now has only one local minimum, the globally stable minimum, instead of two local minima. The basic idea is to scale equation 21 so that it is uniformly valid near possible phase transitions.

The sine term is new and represents a periodic friction force. The system is driven at the plate velocity V by the sine term. In applications to real materials the sine term would presumably be replaced by a Fourier sum of terms. There might also be noise in the form of an additive random noise term, the two effects leading to a random, disordered pinning force representing friction between sliding surfaces [ Rundle et al.

The top right plot in Figure 11 illustrates how the plate motion V in equation 29 prepares the system for the next event by 1 taking the globally stable potential well just after the last event and moving the system configuration point into a state of metastable equilibrium and then 2 gradually reducing the energy barrier until the spinodal line is reached, eventually allowing the system to decay spontaneously during the unstable slip event. It can be seen from this discussion that between events, there is a period of time just after the last earthquake when there is only one globally stable system state.

At a later time the globally stable state gradually transforms to a metastable state, at which three equilibrium points exist: one metastable state, one unstable state, and one globally stable state. The latter acts as a critical point near which scaling occurs.

Examples of our simulation results, together with real data from cycles of activity in the Mammoth Lakes, California, region, are shown in Figure Similar cycles can be seen in other areas of the world [ Scholz , ]. We will thus be motivated to explore these similarities in detail to find out what observable properties should be reflected in seismicity distributions and clustering. Depending on the physical details of the configuration of the metastable well, this model may be associated either with quiescence depressed foreshock activity , or with elevated precursory activity enhanced foreshock activity.

Thus, as in the case of weather forecasting, earthquake forecasting must be considered on a statistical basis. However, there are no widely accepted earthquake forecasting algorithms currently available. A fundamental question is whether patterns of seismicity can be used to forecast future earthquakes. Promising results suggesting that this may be possible have recently been obtained.

The data on accelerated moment release given in section 2. When a threshold of the anomalous behavior was reached, a warning of the time of increased probability TIP of an earthquake was issued. Successful TIPs were issued prior to 42 of 47 events. This approach is certainly not without its critics. Independent studies have established the validity of the TIP for the Loma Prieta earthquake; however, the occurrence of recognizable precursory patterns prior to the Landers earthquake is questionable.

Also, the statistical significance of the size and time intervals of warnings in active seismic areas has been questioned. Nevertheless, seismic activation prior to a major earthquake certainly appears to be one of the most promising approaches to earthquake forecasting.

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

The correlations arise both from the threshold dynamics, as well as from the mean field long range nature of the interactions. Driven threshold systems can be considered to be examples of phase dynamical systems [ Mori and Kuramoto , ] when the rate of driving is constant, so that the integrated stress dissipation or firing rate over all sites is nearly constant, with the exception of small fluctuations.

Tiampo et al. For each box an activity rate function S x i , t b , t is defined to be the average rate of occurrence of earthquakes in box i during the period t b to t. The distribution of relative seismic intensities for southern California for the period — are given in Figure 13a. This was meant to be a retrospective forecast for the period — These random catalogs were used to construct a set of null hypotheses, since any forecast method using such a catalog cannot, by definition, produce useful information.

The likelihood is a probability measure that can be used to assess the utility of one forecast measure over another. Typically, one computes log 10 for the proposed forecast measure and compares that to the likelihood measure N for a representative null hypothesis. The ratio of these two values then yields information about which measure is more accurate in forecasting future events. Since larger values of log 10 indicate a more successful hypothesis, the logical conclusion is that the method has forecasting skill.

This forecast is reproduced in Figure Of these events the first Big Bear I occurred after the research [ Rundle et al. Earlier approaches emphasized either geologic investigations and mapping or the application of continuum methods developed in engineering, including fields such as elasticity, viscoelasticity, plasticity, fluid mechanics, and friction. Many of these models have been used to construct numerical simulations of earthquake physics, a number of which have used a cellular automaton approach.

Both approaches lead to scaling laws or power law distributions for the dynamical variables. The nucleation and coalescence of microcracks in brittle failure is analogous to the homogeneous nucleation of bubbles in a superheated liquid. A solid stressed above the yield stress is then considered to be in a metastable regime similar to that of the superheated liquid. The explosive homogeneous nucleation of bubbles in a superheated liquid occurs adjacent to the spinodal line, which is the limit of allowed superheating.

Thermal fluctuations are directly associated with the approach to a critical point or a phase change. These fluctuations are responsible for the homogeneous nucleation of a superheated liquid near a spinodal line. However, what about temperature? Experimental results are ambiguous. However, there is considerable evidence that fluctuations in stress elastic vibrations associated with microcracking play a role in brittle fracture that is analogous to the fluctuations in pressure caused by thermal fluctuations in a classical phase change.

Observations of seismic activation suggests it occurs for both large and small earthquakes. Increased numbers of earthquakes of magnitude 4 precede a 5. On the basis of results given in section 2. The model most closely associated with fault systems is the slider block model. The movement of the blocks over a surface takes place in slip events. These events scale in a manner similar to actual fault ruptures. These methods can be used to determine whether there are systematic patterns of seismicity that might be precursors to future large earthquakes.

Seismic activation, as proposed by Bufe and Varnes [] , has now been recognized to have occurred for a significant number of earthquakes with a wide range of magnitudes see Figure 2. Precursory quiescence, while more difficult to define and observe, may also occur: The power law increase in Benioff strain does not occur or cannot be recognized for all earthquakes. Studies of seismic activation to date have been largely retrospective, and forecasting and prediction of future large earthquakes based on pattern analysis methods is only just beginning.

There have been two claims of success with this methodology, the Armenian earthquake and the Loma Prieta, California, earthquake. However, a number of relatively large earthquakes have not been predicted or forecast, and the approach is mired in controversy. A key element in these algorithms is a precursory increase in seismic activity over areas similar in size to the recognized patterns of seismic activation. This approach, which does not rely on fitting any model parameters to training data, is based instead on ideas about phase dynamical systems. Here a probability for future large events can be computed from the data itself, without reference to any model.

In this method, the migration of the smallest earthquakes from one spatial region to the next through time is used as an indicator for future activity of the largest events. Maps can be computed that forecast the locations and maximum magnitudes of the largest future events. A map of this type was published by Rundle et al. In most cases, it is difficult or impossible to directly observe the current state of the system, or to have detailed knowledge of the dynamics by which the system is evolving.

Observations of natural systems may have only limited utility in forecasting or extrapolating the future evolution of the system, since such observations are taken over a very limited range of scales, and are necessarily incomplete. A comprehensive understanding of the nonlinear dynamics of earthquake fault systems can only be developed by supplementing observations with a sophisticated program of numerical simulation, leading to the development of theoretical insights and results that can then be applied to the natural system.

Even at this early stage, the models are predicting results that are being confirmed by new observations. We would like to thank Kristy Tiampo for her many contributions to this work. Volume 41 , Issue 4. If you do not receive an email within 10 minutes, your email address may not be registered, and you may need to create a new Wiley Online Library account. If the address matches an existing account you will receive an email with instructions to retrieve your username.

Open access. Reviews of Geophysics Volume 41, Issue 4. Free Access. John B. Rundle E-mail address: rundle physics. Donald L. Tools Request permission Export citation Add to favorites Track citation. Share Give access Share full text access. Share full text access. Please review our Terms and Conditions of Use and check box below to share full-text version of article. Abstract  Earthquakes and the faults upon which they occur interact over a wide range of spatial and temporal scales. In fault systems these unobservable dynamics are usually encoded [ Stein , ] in the time evolution of the Coulomb failure function, CFF x , t :.

Figure 1 Open in figure viewer PowerPoint. Cumulative number of earthquakes per year, N GR , occurring in southern California with magnitudes greater that m as a function of m. Twenty individual years are considered SCSN catalog, : a —, b —, c —, and d — If aftershocks are excluded, the background seismicity in southern California is nearly uniform in time.

Temporal Decay of Aftershocks  A universal scaling law describes the temporal decay of aftershock activity following an earthquake. This is known as the modified Omori's law and as most widely used has the form [ Scholz , ]. Figure 2 Open in figure viewer PowerPoint. In each of the four examples the data have been correlated solid lines with the power law relation given in equation 7. The values of the power law exponent s used in equation 7 are given in Table 2. Dashed straight lines represent a best fit constant rate of seismicity.

Figure 3 Open in figure viewer PowerPoint. The circles are the values given by Bowman et al. The open squares are values obtained for earthquakes in the New Madrid seismic zone by Brehm and Braile []. The error bars are limits obtained for earthquakes in the western United States by Brehm and Braile [b]. The solid squares are values obtained for earthquakes in New Zealand by Robinson []. Figure 4 Open in figure viewer PowerPoint. Figure 5 Open in figure viewer PowerPoint.

The shaded region is metastable. Shcherbakov and Turcotte [a] correlated this dependence with the relation. Figure 6 Open in figure viewer PowerPoint. The applied pressure difference across the panel was increased linearly with time. Figure 7 Open in figure viewer PowerPoint. Failure at an intermediate constant rate of stress increase takes place along path ABE. Fiber Bundle Model  Another approach to the brittle failure is applicable to composite materials. Forest Fire Model  The forest fire model [ Bak et al. Figure 8 Open in figure viewer PowerPoint. An array of blocks, each with mass m , is pulled across a surface by a driver plate at a constant velocity V.

Each block is coupled to the adjacent blocks with either leaf or coil springs spring constant k C and to the driver plate with leaf springs spring constant k L.

## Bikas K. Chakrabarti

Inertia  If the CA approach is used, the inertia mass of a block is neglected. Range of Interaction  The range of interaction specifies how many adjacent blocks to which each block is coupled via a spring or alternative mechanism for stress transmission, such as bulk elasticity. Friction Law  There are a variety of friction laws that are used; these include [ Rabinowicz et al.

Figure 9 Open in figure viewer PowerPoint. The ratio of the number of slip events N e , with area A e , to the total number of slip events N 0 is plotted against A e , the number of blocks involved in an event [ Huang et al. From Huang et al. Application to Earthquakes  In the emerging picture of earthquake physics it has been shown that earthquakes can be regarded as generalized phase transitions [ Smalley et al. Figure 10 Open in figure viewer PowerPoint. The left well is a metastable state since it is not the state of minimum potential energy.

As f is increased further, an inflection point occurs, and the left minimum disappears; this is the spinodal point. In order to do this we introduce the scaled nondimensional variables. Figure 11 Open in figure viewer PowerPoint. Traveling density wave model. Figure 12 Open in figure viewer PowerPoint.

Cycles of earthquake activity foreshock—main shock—aftershock events from a Mammoth Lakes, California, and b traveling density wave simulations [after Rundle et al. That is,. Figure 13 Open in figure viewer PowerPoint. Earthquakes that occurred between and are shown as inverted triangles smallest triangles 5. Earthquakes that have occurred since are shown as circles smallest circles 5. Larger relative likelihood values represent more probable forecast models.

Figure 14 Open in figure viewer PowerPoint. Pattern informatics method forecast for southern California for the period — [ Tiampo et al. Statistical Physics Applied to Fracture [] The statistical physics approach considers an earthquake to be a type of generalized phase transition. Thermodynamic Variables [] Temperature plays an essential role in almost all applications of statistical physics. Earthquake Nesting [] A fundamental question in earthquake physics is whether earthquakes are scale invariant.

Statistical Physics Models [] A number of models have been introduced in the past 15 years that exhibit the repetitive avalanche behavior and the power law scaling associated with fault systems. A rupture area. A e area of slip events. A F area of a burning cluster. C constant in equation 3. D fractal dimension. E 0 Young's modulus. F compressive force.

H hazard rate. L length of sample. L CG size of square box. M magnetization. N number of events. N as number of aftershocks. N e numbers of slip events. N F number of fires. N GR cumulative number of earthquakes. P pressure. P c critical pressure. P f failure pressure. P y yield pressure.

## Modelling Critical and Catastrophic Phenomena in Geoscience: a Statistical Physics Approach

R range of interaction. S activity rate function. T characteristic time. T temperature. T c temperature at the critical point. U energy potential. V driver plate velocity. Aki, K.

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