Statistical Physics Approach to Understanding the Multiscale Dynamics of Earthquake Fault Systems Theory

Statistical Physics Approach to Understanding the Multiscale Dynamics of Earthquake Fault Systems Huge range of scale Phenomenology of dynamics Systems composed of large number of simple, interacting elements Uninterested in small-scale (random) behaviour Use methods of statistics (averages!)

Overview Motivation Scaling laws Fractals Correlation length Phase transitions –boiling & bubbles –fractures & microcracks Metastability, spinodal line

Limitations to Observational Approach Lack of data (shear stress, normal stress, fault geometry) Range of scales: Fault length: ~300kmFault slip ~ mFault width ~ cm

Scaling Laws log(y) = log(c) - b log(x) b>0 y = c x -b Why are scaling laws interesting? Consider interval (x0, x1) minimum of y is y1 = c (x1) -b maximum of y is y2 = c (x0) -b ratio y2/y1 = (x0/x1) -b now consider interval ( x0, x1) minimum of y is y1 = c ( x1) -b maximum of y is y2 = c ( x0) -b ratio y2/y1 = (x0/x1) -b compare with: y = e x/b, b>0 on (x0, x1), y1 = e x0/b, y2 = e x1/b y2/y1 = e (x1-x0)/b on ( x0, x1), y1 = e x0/b, y2 = e x1/b y2/y1 = e (x1-x0)/b x0 x1 λ x0 λ x1 → ratio independent of scale λ! → power-law relation ≈ scale-free process

Earthquake scaling laws Gutenberg-Richter Law Log N gr (>m) = -b m + a –m = magnitude, measured on logarithmic scale –N gr (>m) = number of earthquakes of magnitude greater than m occurring in specified interval of time & area –Valid locally & globally, even over small time intervals (e.g. 1 year) Omori law: dN as /dt = 1/t0 (1+t/t1) -p N as = number of aftershocks with m>specified value t = time after main shock Benioff strain: N = number of EQs up to time t e i = energy release of ith EQ i.e. intermediate EQ activity increases before big EQ

Fractals Fractal = self-similar = scale-free e.g. Mandelbrot set Fractals are ubiquitous in nature (topography, clouds, plants, …) Why? c.f. self-organized criticality, multifractals, etc.

Correlation Length Correlations measure structure On average, how different is f(x) for two points a distance L apart? Correlation length ~ largest structure size Correlation length → ∞ ~ all scales present = scale-free L L L LcLc Let correlation length = scale where correlation is maximal

Phase transition model… Let’s look at earthquakes as phase transitions!

Phase diagram of a pure substance: coexistence of liquid and vapor phases! Isothermal decrease in pressure Liquid boils at constant P Reduction in P, leads to isothermal expansion Formation of metastable, superheated liquid Spinodal curve: limit of stability. No superheating beyond!!! Explosive nucleation and boiling (instability) at constant P,T Vapor equilibrium curve

s’more about stability… why a spinodal line? Ideal Gas Law Van der Waals equation (of state) (real gas) volume pressure isotherms volume pressure isotherms correction term for intermolecular force, attraction between particles correction for the real volume of the gas molecules, volume enclosed within a mole of particles Incompressible fluid (liquid): at small V and low P: isotherms show large increase in P for small decrease in V compressible fluid (gas): at large V and low P: isotherms show small decrease in P for large decrease in V Metastable region: 2-phase coexistence at intermediate V and low P with horizontal isotherms Consequence of…

The spinodal line is interesting! It acts like a line of critical points for nucleating bubbles Limit of stability!

Now let’s look at brittle fracture of a solid as a phase change… Let’s look at a plot of Stress vs. Strain… Deforms elastically until failure at B Undergoes phase change at B Elastic solid rapidly loaded with constant stress ( < yield stress) Damage occurs at constant stress or pressure Elastic solid strained rapidly with a constant strain ( < yield strain) Damage occurs along constant strain path until stress is reduced to yield stress (IH)… similar to constant volume boiling (DH)

When damage occurs along a constant strain path… We call it stress relaxation! Applicable to understanding the aftershock sequence that follows an earthquake earthquake Rapid stress! If rapid stress is greater than yield stress: microcracks form, relaxing stress to yield stress

Time delay of aftershock relative to main shock = time delay of damage Why? Because it takes time to nucleate microcracks when damage occurs in form of microcracks. Damage is accelerated strain, leading to a deviation from linear elasticity.

How do we quantify derivation from linear elasticity? a damage variable!! as failure occurs as increases : brittle solid weakens due to nucleation and coalescence of Microcracks.

nucleation coalescence phase change Metastable region Increasing correlation length Spinodal Line

Metastability – an analogy Consider a ball rolling around a ‘potential well’ Gravity forces the ball to move downhill If there is friction, the ball will eventually stop in one of the depressions (A, B, C) What happens if we now perturb the balls? (~ thermal fluctuations) B is globally stable, but A & C are only metastable If we now gradually make A & C shallower, the chance of a ball staying there becomes smaller Eventually, the stable points A & C disappear – this is the limit of stability, the spinodal C B A Tomorrow, we will consider a potential that changes in time

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