Transcript Statistical Physics Approach to Understanding the Multiscale

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
Systems composed of large
number of simple, interacting
elements
Uninterested in small-scale
(random) behaviour
Use methods of statistics (averages!)
Huge range of scale
Phenomenology of dynamics
Overview
•
•
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•
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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)
y = c x-b
b>0
x0
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
→ ratio independent of scale λ!
x1
λ x0
λ x1
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
→ power-law relation ≈ scale-free process
Earthquake scaling laws
Gutenberg-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
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?
Let correlation length =
scale where correlation
is maximal
L
L Lc
L
Correlation length ~ largest structure size
Correlation length → ∞ ~ all scales present = scale-free
Phase transition model…
Let’s look at earthquakes as phase transitions!
1st order phase transition
2nd order phase transition
involves latent heat
NO latent heat involved
solid/liquid/gas
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
supercritical fluid
QuickTime™ and a
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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
Vapor equilibrium curve
Formation of metastable,
superheated liquid
Spinodal curve:
limit of stability.
No superheating
beyond!!!
Reduction in
P, leads to
isothermal
expansion
Explosive nucleation and
boiling (instability) at
constant P,T
s’more about stability… why a spinodal line?
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Consequence of…
Van der Waals equation (of state)
(real gas)
Ideal Gas Law
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isotherms
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TIFF (Uncompressed) decompressor
are needed to see this picture.
pressure
pressure
correction term for
intermolecular force,
attraction between
particles
correction for the real
volume of the gas
molecules, volume enclosed
within a mole of particles
isotherms
volume
volume
Incompressible
Metastable
region:
fluid2-phase
(liquid):
at
small
V and
at
intermediate
low
V
compressible
fluid
(gas):
atcoexistence
large
V and
low
P: P:
isotherms
isotherms
and
low
Pshow
with
horizontal
large increase
in P for
small decrease
in V
show
small
decrease
in Pisotherms
for large
decrease
in V
The spinodal line is interesting!
Limit of stability!
It acts like a line of critical points for nucleating bubbles
Now let’s look at brittle fracture of a solid as a phase change…
DamageDamage
occurs along
constant
strain
path or
until stress is reduced to yield
occurs
at constant
stress
stress (IH)…
similar to constant volume boiling (DH)
pressure
Undergoes phase change at B
Let’s look at a plot of Stress vs. Strain…
Elasticsolid
solidstrained
rapidly rapidly
loaded with
with a
Elastic
Deforms
elastically
until
failure
at B
constant
stress
(
<
yield
stress)
constant strain ( < yield strain)
When damage occurs along a constant strain path…
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We call it stress relaxation!
Applicable to understanding the aftershock sequence that follows an earthquake
Rapid stress!
Rapid stress!
earthquake
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.
Spinodal Line
Increasing correlation length
Metastable region
nucleation
coalescence
phase change
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
A
C
B
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
Tomorrow, we will consider a potential that
changes in time
the end
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