Presented by Chris Rollins and Voonhui Lai Ge 277 Winter 2016 Statistical physics of fracture, friction, and earthquakes Hikaru Kawamura et al. Reviews of Modern Physics, 2012 Presented by Chris Rollins and Voonhui Lai Ge 277 Winter 2016

Motivation Are the power law behaviors observed in earthquakes scale-invariant ? How to reliably forecast large events from smaller ones based on statistical properties such as: (i) magnitude-frequency distribution and (ii) time evolution of earthquakes ? What is the appropriate model to simulate earthquake faulting ?

Burridge-Knopoff (BK) model #1 A spring-block model used to simulate stick-slip motions along fault Equation of motion of 2D BK model (dimensionless): v = Loading rate; l = Stiffness parameter; = Friction force Magnitude of event give by:

Burridge-Knopoff (BK) model #2 Only source of nonlinearity: Friction force Velocity weakening friction force decreasing function of common form: [Carlson and Langer] - rate of weakening of friction force Intrinsically discrete Other type of friction forces: rate and state friction law (RSF)

Magnitude-Frequency Relation #1 2D BK model with short range interaction First-order transition from subcritical to supercritical Smooth transition from supercritical to near-critical to subcritical Fig 12 Fig 13

Magnitude-Frequency Relation #2 2D BK model with long range interaction Observe similar transition from subcritical system to characteristics behavior Fig 15 Fig 16

Continuum Limit of BK model In previous model, dynamics becomes increasingly sensitive with decreasing grid spacing To counter, introduce viscosity term in equation of motion so that magnitude distribution becomes independent of grid spacing Subcritical Supercritical Fig 18

BK model with RSF Law #1 Friction force Condition on frictional instability depends on: b (healing term) and l (stiffness parameter) Strong Weak Aging law:

BK model with RSF Law #2 Characteristic behavior in weak frictional instability regime; Flat spectrum (or critical) in strong regime. 1D 2D Strong Strong Weak Weak Fig 20 Fig 21

Comparison between two models Observed similar transition of behavior (e.g. appearance of large events) which depends on friction parameter Heterogeneity in medium essential for criticality. 2D BK Model Amitrano, EPJ 2012

Take-aways BK model is comparable to other discrete and continuous models. There exists a transition from self-criticality to characteristic behavior, primarily dictated by friction. “Considering fault obeying RSF, the continuum limit of system lies in limit of weak frictional instability... earthquake should exhibit characteristic properties rather than critical properties.”

Supplementary Figure 1 Characteristic behavior not due to finite-size effect

Supplementary Figure 2 Transition in view of mean displacement