Statistical Physics Statistical Physics Approach to Approach to

Understanding the Understanding the Multiscale Dynamics of Multiscale Dynamics of

Earthquake Fault SystemsEarthquake Fault SystemsTheory

Statistical Statistical Physics Physics

Approach to Approach to Understanding Understanding the Multiscale the Multiscale Dynamics of Dynamics of Earthquake Earthquake

Fault SystemsFault 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: ~300km Fault slip ~ m Fault 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 = ex/b, b>0

on (x0, x1), y1 = ex0/b, y2 = ex1/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 lawsGutenberg-Richter Law• Log Ngr(>m) = -b m + a

– m = magnitude, measured on logarithmic scale

– Ngr(>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:dNas/dt = 1/t0 (1+t/t1)-p

Nas = number of aftershocks with m>specified value

t = time after main shock

Benioff strain:

N = number of EQs up to time t

ei = energy release of ith EQ

i.e. intermediate EQ activity increases before big EQ

FractalsFractal = 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

Lc

Let correlation length = scale where correlation is maximal

Phase transition model…

Let’s look at earthquakes as phase transitions!

solid/liquid/gas

involves latent heat

1st order phase transition

supercritical fluid

NO latent heat involved

2nd order phase transition

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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?

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Ideal Gas Law Van der Waals equation (of state)

(real gas)

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volume

pre

ss

ure

isotherms

volume

pre

ss

ure 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!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

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earthquake

Rapid stress!

Rapid stress!

Rapid stress!

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

Meta

stab

le r

eg

ion

Incr

ea

sin

g c

orr

ela

tion

len

gth

Spinodal LineSpinodal 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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