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STRESS2 Short course
Heavy-tailed stochastic processes and fractional ADEsfor modeling particle transport
Rina SchumerAssistant Research Professor
Division of Hydrologic SciencesDesert Research Institute
Reno, NV
Purpose
• introduce concepts STRESS working group is using to model transport processes at the Earth surface
• focus on concepts, “basic”mathematical principals
Stochastic Model
• Physical (or non-physical) transport processes are deterministic, but we conceptualize them as random.
• Use probability theory to predict the outcome of random processes.
• Emergent properties at long time
What is a particle?
• molecule • sediment grain• solute• parcel of water• price of a stock• fish• drunken sailor
Stochastic Process• A stochastic process { X(t)} is a
collection of random variables.
• X(t) tells the state of the process at time t. – Could be # of people on a bus, could be
the location of a particle
• describes the evolution of some physical process through time
Stochastic Processes
• Discrete-time process– T, the set of all t, is countable
we want to design a discrete-time process that fits our conceptual understanding of a transport process
{ , 0,1,...}nX n =
Stochastic Processes
• Continuous-time process– T is an interval on the real line
– describe long-time behavior of discrete stochastic processes
– some have governing PDEs with solutions, allowing us to model transport processes
– some are scale-invariant…
{ ( ), 0}X t t ≥
Classical Models
• Random walk, Brownian motion, diffusion, advection-dispersion equation, “normal” transport, Gaussian transport, Fickian or Boltzmannscaling
/
1
( ) ( )t t
nn
X t S n YΔ
=
= =∑
t=time
nY
1Y2Y
3Y
particle location at time t, X(t), is the sum of the lengths, Yn, of a bunch of jumpsDISCRETE PROCESS
Random Walk (drunken sailor)
Probability density of the location of a single particle at time t
location of cloud of particles at time t=
Random Walk/
1( )
t t
nn
X t YΔ
=
= ∑
To get the long term behavior of this process, we will let Δx and Δt go to 0.
This must be done in a non-trivial way (what if Δx= Δt and Δt 0?). i.e. we have to scale things properly
Law of large numbersfor sums of iid random variables
1 2 0nY Y Yn
μ+ +…+
− →
sample mean converges to the theoretical average
What’s the deviation between these two terms?
Classical Central Limit Theoremfor sums of iid random variables
( )1 212
0,1( )
nY Y Y n Nn
μ
σ
+ +…+ −→
finite variance:iY ∼
( )12
1 2 ( ) 0,1nY Y Y n n Nμ σ+ +…+ ≈ +or… rearrange to find
Add up a bunch of jumps lengths (subject to a few conditions) and we know about the likelihood of particle location– use this to take scaling limit of random walk
as n →∞
A closer look at the CLT:
( )1 212
0,1( )
nY Y Y n Nn
μ
σ
+ +…+ −→
finite variance:iY ∼
as n →∞
1. center2. expand time scale
3. contract spatial scale
Classical Central Limit Theorem
2
Let and 2
v Dt tμ σ
= =Δ Δ
( )12
1 2 0,1nY Y Y n n Nμ σ+ +…+ ≈ +
( )12
1 2 2 0,1nY Y Y tv Dt N+ +…+ ≈ +
Let n tt
=Δ
( )12
1 2 0,1nt tY Y Y Nt tμ σ
⎛ ⎞⎟⎜+ +…+ ≈ + ⎟⎜ ⎟⎜⎝ ⎠Δ Δ
( )12
1 2 0,1nY Y Y t t Nt t
μ σ+ +…+ ≈ +
Δ Δ
location of a particle at time t
/
1
( )t t
nn
X t YΔ
=
= ∑
Converges in distribution to a Gaussian density
2
Let and 2
v Dt tμ σ
= =Δ Δ
( )12
1 2 2 0,1nY Y Y tv Dt N+ +…+ ≈ +
2( )2*21( ) ( , )
2 *2
x vtDtX t C x t e
Dtπ
−−
→ =
a Brownian motion
To get the long term behavior of this process, we will rescale and let Δx and Δt go to 0.
Random walk simulation
courtesy of M.M. Meerschaert
Longer time scale
courtesy of M.M. Meerschaert
Scaling limit: Brownian motion
no jumps.courtesy of M.M. Meerschaert
Defn: Brownian MotionA stochastic processis a Brownian motion if1) X(0)=02) has stationary and
independent, finite varianceincrements
3) for every t>0, X(t) is normally distributed with mean 0 and variance
{ ( ), 0}X t t ≥
2 .tσ
{ ( ), 0}X t t ≥
Important points
1. The discrete stochastic process known as a random walk converges in the scaling limit to a
continuous time stochastic process called a Brownian motion
2. If a particle is making stationary, independent, finite variance jumps (increments), then the
random location of a particle at time t is governed by a Gaussian density.
Fick’s Law Fourier’s Law d
CF Dx
∂= −
∂
xΔ
F = fluxDd = diffusion coeff.C = concentration
Another perspective on the classical model
flow per unit area per unit time
1-D ADE Derivationmass flux conservation of mass
x x e eCF v n C n Dx∂
= −∂
xe
FCnt x
∂∂− =
∂ ∂
x xC Cv C Dt x x
∂ ⎡ ∂ ∂ ⎤⎛ ⎞= − +⎜ ⎟⎢ ⎥∂ ∂ ∂⎝ ⎠⎣ ⎦
2
2
C C Cv Dt x x
∂ ∂ ∂= − +
∂ ∂ ∂
Green’s function solution to ADE2
2
C C Cv Dt x x
∂ ∂ ∂= − +
∂ ∂ ∂I.C.: C(x,0)=δ(0)pulse
2ˆ ( , ) ˆ ˆ( ) ( , ) ( ) ( , )C k t v ik C k t D ik C k t
t∂
= − +∂
solve for C ( )2( ) ( )ˆ ( , ) vt ik Dt ikC k t e − +=
2( )2*21( , )
2 *2
x vtDtC x t e
Dtπ
−−
=deterministic solution
to a PDE is a probability density!
Fourier transform
invert
Green’s functionSolutions to non-homogeneous, linear equations of the form
( ) ( )LC x F x= −linear operator e.g. source
function2
2v Dt x x∂ ∂ ∂+ −
∂ ∂ ∂
The Green’s function G(x) satisfies
( ; ') ( ')LG x x x xδ= − − source function is a pulseGreen’s function solution
The solution to (1) is given by
( ) ( ; ') ( ') 'C x G x x F x dx= ∫
(1)
For any source function F(x), solve (1) by convolving with the Green’s function soln.
plotting C(x,t)
snapshot in time breakthrough curve
x
C(x,5)
let t=5
C(20,t)
let x=20
tshows what plume looks like in space
Time evolution of concentration at a point
Classical ADE2
2 C CDt x
∂ ∂=
∂ ∂
where CvD
===
concentration
average linear velocity
dispersion coefficient
Gaussian solutions
Cvx
∂− +
∂
limiting stochastic process
governing equation
solution
Brownian motion
ADE
Gaussian density
finite-mean waiting time distribution
infinite-mean waiting time distribution
infinite-variance jump
length distribution
finite-variance jump
length distribution
?
?
?infinite variance
=heavy tails
randomwalks
What are heavy tails?Heavy tails refer to the rate of decay of the upper (lower)
end of a probability density/distribution function
)20()|(| <<≈> − ααCrrYP n
For a random walk with heavy tailed particle jumps
10-1
10-2
10-3
10-4
10-5
10-6
10-7
1
10 100110-1
1.8
1.2
2.0 (Gaussian)
α =
1+α
1
RE
LA
TIV
E C
ON
CE
NT
RA
TIO
N
x - μ
Probability distributions are described by their PDF and CDF
x
p(x)
xp(
X<x
)0 0
1
1
Area=1
Probability Density Function (PDF)
Cumulative Distribution Function (CDF)
Exponential
CDF: F(x)=1PDF: f(x)=F'(x)= e
x
x
e λ
λλ
−
−
−
Pareto
1
CDF: F(x)=1PDF: f(x)=F'(x)=
xx
α
α
α
α
−
− −
−
( )0, 0x λ≥ > ( )0, 0x α≥ >
Important distributions for today….
Classic example of a distribution with exponential tails. In the scaling limit, the sum of any iid finite variance
jumps will get you to the same place
Classic example of a distribution with power-law tails. In the scaling limit, the
sum of any iid infinite variance jumps will get you to the same
place as the sum of paretoRVs with α<2.
Moments of power-law distributions
)20()|(| <<≈> − ααCrrYP n
if the tails of a CDF
decay as a power law…
…and α<2, then the distribution has infinite variance:
…and α<1, then the distribution has infinite mean:
2 ( )x f x dx = ∞∫( )xf x dx = ∞∫
#sample
variance Finite variance distribution
Sample variance converges to distribution variance
N(0,1)
0
0.5
1
1.5
2
2.5
3
3.5
4
0 1000 2000 3000 4000 5000
# samples
sam
ple
varia
nce
1 -1.5930532 -0.434536 0.671083 -0.491251 0.4265574 -1.856679 0.5429775 -1.090159 0.4072366 0.8048414 0.9261757 -1.191165 0.796348 -0.808973 0.6826689 -0.13856 0.650859
10 -2.15E-02 0.63242411 0.5654999 0.71068212 0.23143 0.69942313 1.4470152 0.93334414 5.63E-01 0.92139415 -0.238272 0.85573616 -0.296313 0.79869817 -1.124477 0.79030518 0.8937207 0.82751419 0.2429965 0.7951520 -0.176049 0.75349721 1.0456461 0.79399422 1.4459283 0.87555323 1.0233089 0.89070924 0.445491 0.86229725 1.209562 0.8879626 -0.50042 0.86279127 1.5313799 0.91664928 2.43E-01 0.88396329 0.8615621 0.87444630 0.3940238 0.84738531 -1.417363 0.89336932 -0.802727 0.887308
#sample
variance1 2.0173732 -1.08446 4.810673 1.98119 3.1701394 1.793982 2.2825995 -0.81369 2.5045336 0.414722 2.0257287 -0.70156 1.9760618 -1.11996 2.0280549 1.244561 1.871395
10 -0.00787 1.68131811 0.525315 1.51531112 1.585412 1.49737213 -1.16867 1.58326514 -0.07372 1.47484715 -0.18833 1.38727716 -0.31533 1.31796917 -2.65083 1.73248518 0.930852 1.67036119 0.134249 1.57756320 -0.69535 1.52873821 -0.83792 1.49333722 -1.63973 1.55142223 1.545988 1.58895224 -0.18058 1.5218625 1.066792 1.50153126 -0.53109 1.45538927 1.418536 1.46904728 -0.04284 1.41534829 -1.64665 1.4691730 -0.21179 1.42050931 1.548407 1.44802632 -1.72853 1.502862
Infinite variance distribution
0
0.5
1
1.52
2.5
3
3.5
4
0 1000 2000 3000 4000 5000
# samples
sam
ple
varia
nce
Sample variance never converges
1.9 stable(0,1,0)
/
1
( ) ( )t t
nn
X t S n YΔ
=
= = ∑
nY
1Y2Y
3Y
particle location at time t, X(t), is the sum of the lengths, Yn, of a bunch of jumps
Lagrangian modelHeavy tailed Random Walk
t=time
this time, jump lengths will be iid random variables with
infinite variance
Random Walk/
1( )
t t
nn
X t YΔ
=
= ∑
To get the long term behavior of this process, we will let Δx and Δt go to 0.
This must be done in a non-trivial way (what if Δx= Δt and Δt 0?). i.e. we have to scale things properly
this time, jump lengths will be iid random variables with
infinite variance
Limit Theoremsfor sums of iid random variables
( )1 21n2
lim 0,1( )
nY Y Y n X Nsdev n
μ→∞
+ +…+ −= ∼
( )1 21n
lim 1, , 0nY Y Y n X Sn
αα
μσ β μ
σ→∞
+ +…+ −= = =∼
finite variance:Y ∼
more general
α-stable density3. contract spatial scale
2. expand time scale
α-stable densities(lots of info in the classic reference by Samorodnitsky and Taqqu)
• in general, can not be written in closed form. written as Fourier transform- inverted numerically to view densities
• When α=2, β 0, Gaussian. When α=1, Cauchy• sums of stables are stable in the limit• spread is not standard deviation
( )1, , 0Sα σ β μ= =
spread skewness shift
ˆ( ) exp ( ( ) (1 ) ( ) )s ( 1)P k ik ik ignα α α αβσ β σ α⎡ ⎤= + − − −⎣ ⎦
,
4− 2− 0 2 40
0.1
0.2
0.3
0.41.5-stable densities
fα x 1−, ( )
fα x .5−, ( )
fα x 0, ( )
fα x .5, ( )
fα x 1, ( )
Totally skewed
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.5
1
1.5
2
a=.5a=.7a=.9
skewed stables
skewedstable x .5,( )
skewedstable x .7,( )
skewedstable x .9,( )
x
• Totally skewed densities are limits of jump densities where you have long jumps in one direction but not the other.
Generalized Central Limit Theorem
Let and 2
v Dt t
αμ σ= =
Δ Δ
( )1
1 2 1, , 0nY Y Y n n Sααμ σ σ β μ+ +…+ ≈ + = =
( )1
1 2 2 1, , 0nY Y Y tv Dt Sαα σ β μ+ +…+ ≈ + = =
Let n tt
=Δ
( )1
1 2 1, , 0nt tY Y Y St t
α
αμ σ σ β μ⎛ ⎞⎟⎜+ +…+ ≈ + = =⎟⎜ ⎟⎜⎝ ⎠Δ Δ
location of a particle at time t
/
1
( )t t
nn
X t YΔ
=
= ∑
Converges in distribution to an α-stable density
( ) ( , )X t C x t→ =
a Lévy motion
To get the long term behavior of this process, we will rescale and let Δx and Δt go to 0.
( )12
1 2 0,1nt tY Y Y Nt tμ σ
⎛ ⎞⎟⎜+ +…+ ≈ + ⎟⎜ ⎟⎜⎝ ⎠Δ Δ2
Let and 2
v Dt t
μ σ= =
Δ Δ
( )1
1 2 2 1, , 0nY Y Y tv Dt Sαα σ β μ+ +…+ ≈ + = =
no closed form; usually in Fourier space
Heavy tailed random walk simulation
Longer time scale
Scaling limit: Stable Lévy motion
includes jumps
Defn: Lévy Motion
A stochastic processis a Lévy motion if1) X(0)=02) has stationary and
independent increments3) for every t>0, X(t) is α-stable
distributed with mean 0 and spread
{ ( ), 0}X t t ≥
2 .tσ
{ ( ), 0}X t t ≥
Important points
The discrete stochastic process known as a random walk convergesin the scaling limit to a continuous time stochastic process called a Lévy Motion
Brownian motion is a subset of Lévy motion that arises when the jump length distribution has finite variance
If a particle is making stationary, independent jumps (increments), then the random location of a particle at time t is governed by an α-stable density.
The Gaussian distribution is α-stable with α=2
Fractional Fick’s Law 1
1
CF Dx
α
α
−
−
∂= −
∂
F = fluxDd = diffusion coeff.C = concentration
xΔ
Integer-Order Derivativelocal function
slope at an infinitesimally small
point
x
C(x)
Cx
∂≈
∂
Fractional-Order Derivativenon-local function
the fractional derivative at a point
depends on the values over the entire function
x
C(x)
Cx
α
α
∂∂
• the fractional derivative is a weighted average of all of the values over the function
• the order of the fractional derivative indicates how those weights decay with distance from x
i
Prob
abili
ty
0.1
1
0.01
0.9000.900
0.0450.045
0.0170.0170.0090.009
0.0050.005
i+3i+2i+1
0.0030.003
i+4 i+5 i+6
conceptual model:Probability that a particle jumps forward n boxes
x
α
α
∂∂
α = 0.9 (low heterogeneity)
Prob
abili
ty
0.01
i
0.1
1
0.1000.100
0.0450.0450.0290.029 0.0200.020 0.0160.016
0.0130.013
i+4i+3i+2 i+5i+1 i+6
x
α
α
∂∂
α = 0.1 (high heterogeneity)
conceptual model:Probability that a particle jumps forward n boxes
1-D Fractional ADE Derivation
mass flux conservation of mass 1
1e eCF v n C n D
x
α
α
−
−
∂= −
∂ eFCn
t x∂∂
− =∂ ∂
1
1
CC v C Dt x x
α
α
−
−
⎡ ⎤⎛ ⎞∂∂ ∂= − +⎢ ⎥⎜ ⎟∂ ∂ ∂⎝ ⎠⎣ ⎦
CC Cv Dt x x
α
α
∂∂ ∂= − +
∂ ∂ ∂
Fractional ADE for SUPER-diffusive processes
CC Cv Dt x x
α
α
∂∂ ∂=− +
∂ ∂ ∂
where Cv
DC
x
α
α
===
∂=
∂
concentration
average linear velocity
constant dispersion coefficient
fractional-in-spacederivative
αth derivative of concentration(where 1<α 2)≤
Space-fractional advection-dispersion equation (fADE)
α
α
xCD
xCv
tC
∂∂
+∂∂
−=∂∂
CikDCikvdtCd ˆ)(ˆ)(ˆ
α+−=
invert
( )tikDtikvC α)()(expˆ +−=
),( txC
Fourier transform
is an α-stable stable density with mean vt
solve for C
deterministic solution to a PDE is a probability density!
Space-fractional ADE Characteristics
• Spatial snapshots– α-stable concentration profiles are
skewed with long right tail– plume leading edge decays as a power
law x-α-1
– snapshot width spreads like t1/α
– total mass remains constant over time
• Flux at position x– asymmetric breakthrough curves– long leading edge – breakthrough curve width grows like x1/α
– area under breakthrough curves remains constant
Important points
• Fractional dispersion leads to skewed, heavy-tailed breakthrough curves
• Fractional ADE has α-stable solutions
• scaling
1 12 2( , ) ( ,1)C x t t C t x
− −=
1 1
( , ) ( ,1)C x t t C t xα α− −
=
CC Cv Dt x x
α
α
∂∂ ∂= − +
∂ ∂ ∂
2
2
CC Cv Dt x x
∂∂ ∂= − +
∂ ∂ ∂
limiting stochastic process
governing equation
solution
Brownian motion
ADE
Gaussian density
finite-mean waiting time distribution
infinite-mean waiting time distribution
infinite-variance jump
length distribution
finite-variance jump
length distribution
?
?Lévy motion
space fADE
α-stable density
randomwalks
Why might we model a transport process with a random walk that has heavy tailed jump lengths?
Tracer studies in sand and gravel bed streams reveal hop length distributions with heavy tails (Bradley et al., 2009)
Super-diffusive transport of solute in aquifers and streams (Benson et al, 2000a,
2000b, 2001)
Transport on hillslopes may be non-local; sediment flux must be calculated using not just the local gradient, but also upstream topography (Foufoula-Georgiou et al., 2009)
limiting stochastic process
governing equation
solution
Brownian motion
ADE
Gaussian density
finite-mean waiting time distribution
infinite-mean waiting time distribution
finite-variance jump
length distribution
?
infinite-variance jump
length distribution
?Lévy motion
space fADE
α-stable density
randomwalks
heavy tails in time
running average
0
2
4
6
8
10
12
numb er o f samp les
running variance
0
1000
2000
3000
4000
5000
6000
7000
8000
number o f samples
0.878805 0.878805 167.475719.18048 10.02964 109.83021.182213 7.080501 84.781310.280155 5.380414 68.022640.670505 4.438432 56.89110.586423 3.796431 48.2081.431849 3.458633 93.4198723.87406 6.010562 83.131782.474393 5.617654 76.379330.633261 5.119215 70.571140.632889 4.711367 69.872612.99414 5.401598 65.421681.178645 5.076756 60.57923.445996 4.960273 57.632970.409136 4.656864 54.203782.086231 4.496199 51.865620.27212 4.247724 48.92196
2.858282 4.170533 46.699251.103217 4.009095 44.24144.018116 4.009546 42.520830.796826 3.856559 41.11975
=rand()/rand()
Infinite mean distributions
Time is not explicitly considered in the random walk model….each time step, Δt, a jump occurred so that the number of jumps was
now we want a model that incorporates random time intervals between jumps representing immobile periods
n tt
=Δ
Continuous time random walk (CTRW)
( )
1
( ) ( ( ))N t
nn
X t S N t Y=
= = ∑1 2( ) ... nS n Y Y Y= + + +
1( ) ... nT n J J= + +
{ }( ) max : ( )N t n T n t= ≤
DISCRETE
CTRW
{ }( )
1 2
1 2
......
max :
( ) ( )
n
n
n n
n n
t n
YJS Y Y YT J J JN n T t
X t S N t
= + + += + + +
= ≤
=
( )
1
( ) ( ( ))N t
nn
X t S N t Y=
= = ∑
iid particle jump lengths
iid inter-jump waiting times
particle location after nth jump
time of nth jump
number of jumps by time t
location of particle at time t
1 2
1 2
( )( )( ) ( ) ... (
Random variable Density
)( ) ( ) ... ( )
( , )
n
n
f xt
f x f x f xt t t
P x t
ψ
ψ ψ ψ∗ ∗ ∗∗ ∗ ∗
1 ( ) ( , 0)( , )1 ( ) ( )
s P k tP k ss s f k
ψψ
− ==
−
waiting timedensity
Governing equation for CTRWs(discrete stochastic process given by its Fourier-Laplace transform)
initialcondition
product of waiting time densityand jump length density
probability ofparticle location
This equation can be used in discrete form…. we want to take a scaling limit
Limit Theorems
( )1 2 ( )1n
lim 1, , 0N tY Y Y nX S
nα
α
μσ β μ
σ→∞
+ +…+ −= = =∼
We still need to use a limit theorem to add up jumps…but we’ve got the added complication of number of
events by time t being random
If waiting times have finite mean v then LLN shows that the nth jump happens at time so that
1 2 /( ) ... tX t Y Y Y ν= + + +nt T nν= ≈
same result as a classical random walk
Limit Theorems
( )1 2 ( )1n
lim 1, , 0N tY Y Y nX S
nα
α
μσ β μ
σ→∞
+ +…+ −= = =∼
We still need to use a limit theorem to add up jumps…but we’ve got the added complication of number of
events by time t being random
If waiting times have infinite mean, i.e. waiting times Jnare heavy tailed with power law index 0<γ<1 then
( )1 21n
lim 1, , 0nJ J J n W Sn
γγ
νσ β μ
→∞
+ +…+ −= = =∼
( )1
1 2 1, , 0nt J J J n Sγγ σ β μ= + +…+ ≈ = =
rearrange… the time of the nth jump looks like…
and recall, the sum of jump lengths looks like
( )1
1 2 1, , 0nY Y Y n Sαα σ β μ+ +…+ ≈ = =
Z
W
( ) for big n t W nγ≈
Finally, particle location at long time looks like
( )( )X t t W Zγ α≈
γ-stabledensity α-stable
density
There’s more, but you get the idea?TWO PROCESSES AT WORK
location of a CTRW particle at time t
( )
1
( )N t
nn
X t Y=
= ∑To get the long term behavior of this process, we will rescale and let Δx and Δt go to 0.
( ) ( , )X t C k s→ =
Converges in distribution to a subordinated α-stable density
( ) 1
0
Du ik use s e duα γγ
∞− −∫
CTRW simulation with heavy tail waiting times
Longer time scale
Scaling limit: Subordinated motion
Limit retains long waiting times.
Important points
A discrete CTRW with finite mean waiting time behaves in the scaling limit as a classical random walk (converges to BM or LM)
A discrete CTRW with infinite mean waiting times and finite variance jumps converges in the scaling limit to a continuous time stochastic process called a subordinated Brownian motion
A discrete CTRW with infinite mean waiting times and infinite variance jumps converges in the scaling limit to a continuous time stochastic process called a subordinated Lévy motion
Subordination randomizes time… time does not proceed linearly, but according to an operational time
Time o
f arr
ival a
t cell
i
t-2
t-3
cell i
t-1
t
for how long have the particles that will land in cell i+1 been in cell i?
conceptual model:memory in time
1-D time-fractional ADE Derivation
mass flux conservation of mass
e eCF v n C n Dx
∂= −
∂ eFCn
t x
γ
γ
∂∂− =
∂ ∂
C Cv C Dt x x
γ
γ
∂ ⎡ ∂ ∂ ⎤⎛ ⎞= − +⎜ ⎟⎢ ⎥∂ ∂ ∂⎝ ⎠⎣ ⎦
2
2
C C Cv Dt x x
γ
γ
∂ ∂ ∂= − +
∂ ∂ ∂
Fractional ADE for sub-diffusive processes when γ<1/2
sub- or super-diffusive processes when γ>1/2
2
2
CC Cv Dt x x
γ
γ
∂∂ ∂=− +
∂ ∂ ∂where C
vDC
x
γ
γ
===
∂=
∂
concentration
average linear velocity
constant dispersion coefficient
fractional-in-timederivative
γth derivative of concentration(where 0<γ 1)≤
Time-fractional advection-dispersion equation
invert ),( txC
Fourier-Laplace transform
is a subordinated Gaussian density
solve for C…
2
2
C C Cv Dt x x
γ
γ
∂ ∂ ∂= − +
∂ ∂ ∂
1 2ˆ ˆ ˆˆ ( )os C C s vikC D ik Cγ γ −− = − +
Space/time-fractional advection-dispersion equation
invert
Fourier-Laplace transform
USUALLY CAN NOT BE DONE
instead, use method for solving Cauchy problems (Baeumer and Meerschaert,
2001)
solve for C….
C C Cv Dt x x
γ α
γ α
∂ ∂ ∂= − +
∂ ∂ ∂
1ˆ ˆ ˆ ˆ( ) ( )os C C s v ik C D ik Cγ γ α−− = − +
Solutions to fractional in time ADEs are transforms of their conservative counterparts
( )0
( ,( , ) , , ) .t
mu
C x t f t x u duγ β=
= ∫ c
If the solution to ( , ) ( ) ( , )x t L x x tt
∂=
∂c c
is the probability density ( , )x tc
then the solution to the non-conservative mobile transport equation is buried in here is a α-stable
density with scaling parameter 0<γ<1, known as a stable
subordinator
like the Gaussian or a-stable
Space/time-fractional ADE Characteristics
• Spatial snapshots– Subordinated α-stable concentration profiles are skewed
with long right tail if α<2– plume leading edge decays as a power law x-α-1 if α<2– snapshot width spreads like tγ/α
– total mass remains constant over time
• Flux at position x– breakthrough curve tail decays as t-γ-1
– breakthrough curve width grows like xγ/α
– area under breakthrough curves remains constant with time
Important points
• Concept of “operational” time
• Solutions to time fADE are subordinated densities
• scaling
( , ) ( ,1)C x t t C t xγ γα α
− −=
CC Cv Dt x x
αγ
γ α
∂∂ ∂= − +
∂ ∂ ∂
Mobile-immobile fADEs• In some applications, you can not measure immobile
particles• Mobile zone (measureable) mass decays over time—
sometimes as a power law
MADE Site mobile mass loss
power law fit0
0.5
1
1.5
2
2.5
0 100 200 300 400 500 600
bromide
power law fit
exponential fit
Time (days)
Mas
s Fr
action
bromide
bromide
1
10
10 100 1000Time (days)
bromide
0.110
Mas
s Fr
action
power law fit
exponential fit
1 ( ) ( , 0)( , )1 ( , )
s P k tP k ss p k s
ψ− ==
−
CTRW can be broken into its mobile and immobile components
( ) ( )( , 0) ( ) ( , 0)
1 ( , ) 1 ( , )P k t s P k t
s p k s s p k sψ= =
= −− −
mobile immobile
Mobile, immobile, total fractional in time ADEs
,0( ) ( )(1 )
mm m
C tL x C C xt
γ γ
γβ βγ
−∂= −
∂ Γ −
,0( ) ( )(1 )
imim m
C tL x C C xt
γ γ
γβγ
−∂= +
∂ Γ −
tot m m im imC C Cθ θ= +
,0( ) , ( ,0) ( )tottot tot m m
C L x C C x C xt
γ
γβ θ∂= =
∂
mobile
immobile
total
im
m
θβθ
=
ADE (a(x))
Immobile (βCim)Total (Ctot)
Mobile (Cm)
0 5 10 15 20 25 30 35 400
0.05
0.1
0.15
0.2
Distance from source (x)
Rela
tive
mas
s in
eac
h ph
ase
Mobile/immobile/total solute transport
ADE (a(x))
Immobile (βCim)
late time tail
tot imC C t γ−∼ ∼
Total (Ctot)
Mobile (Cm)
late time tail ( 1)
mC t γ− +∼
time since injection (t)
brea
kthr
ough
cur
ve C
(t)
10-6
10-1
10-2
10-3
10-4
10-5
100
103102101100 .
Breakthrough curves with power law tails
a) Snapshot 4 (day 202)
d) Snapshot 7 (day 503)c) Snapshot 6 (day 370)
b) Snapshot 5 (day 279)
050
100150200250300
-25 25 75 125Longitudinal Distance (meters)
Brom
ide
Conc
. (m
g/L) model
data
050
100150200250300
-25 25 75 125Longitudinal Distance (meters)
Brom
ide
Conc
. (m
g/L) model
data
050
100150200250300
-25 25 75 125Longitudinal Distance (meters)
Brom
ide
Conc
. (m
g/L) model
data
050
100150200250300
-25 25 75 125Longitudinal Distance (meters)
Brom
ide
Conc
. (m
g/L) model
data
Scaling of MADE Site plume
Important points
CTRW can be divided into its mobile and immobile components
Taking scaling limits of the mobile and immobile components leads to equations for mobile particles and immobile particles
Mobile particle equation solutions lose mass over time when particles have infinite mean waiting times
Use mobile fADE if you can not measure immobile phase
limiting stochastic process
governing equation
solution
Brownian motion
ADE
Gaussian density
finite-mean waiting time distribution
infinite-mean waiting time distribution
finite-variance jump
length distribution
infinite-variance jump
length distribution
Lévy motion
space fADE
α-stable density
CTRW
subordinatedBrownian motion
time fADE
subordinatedGaussian density
subordinatedLévy motion
space/time fADE
subordinatedα-stable density
VocabularyErgodic stochastic processdistribution of the sum of random
variables reaches a limit that does not depend on its initial conditions
Pre-ergodic stochastic processe.g. CTRW is commonly used without
taking limits….if you know the exact waiting time density, why not use it?
Vocabulary
Self-similar stochastic processhas stationary increments and is invariant
if the proper scaling index is used(e.g. 1/2 for BM and 1/α for LM)
FRACTALS and their relationship with some stochastic processes
Fractal – an object in which properly scaled portions are identical (in a deterministic or statistical sense) to the original object
Fractals can be deterministic or random
Path of a Brownian motion
random fractal with Hausdorff dimension 3/2 (Mandelbrot, 1982)courtesy of M.M. Meerschaert
Path of a Levy motion
Random graph of fractal dimension 2-1/α includes jumps.
courtesy of M.M. Meerschaert
Fractals are ubiquitous in nature.
What kind of stochastic process governs particle transport through
fractal networks or on fractal structures?
What if we relax assumptions of the classical random walk model?
stationary, independent, finite variance increments
+ hidden assumption of regular jump times (finite mean interarrivals)
non-stationaryincrements?
infinite variance ( heavy-tailed jumps)?
long-range correlation?
infinite mean interarrivals( heavy tailed wait times)?
Fractional Brownian (Levy) motion
Scaling parameter for fBM is the Hurst coefficient
Random walks with long range correlation in the increments discrete
continuous
scaling limit
if H = 1 / 2 Brownian motionif H > 1 / 2, increments of the process are positively correlated if H < 1 / 2, increments of the process are negatively correlated
Non-stationarity
2
2( ) ( )C C Cv t D tt x x
∂ ∂ ∂= − +
∂ ∂ ∂
( ) ( )C v x D xt x x
∂ ∂ ∂⎛ ⎞= − +⎜ ⎟∂ ∂ ∂⎝ ⎠
Pseudo-differential operators for non-stationarity in space…..e.g.
( )
( )
x
x
C C Cv Dt x x
α
α
∂ ∂ ∂= − +
∂ ∂ ∂
or
For non-stationarity in time…..e.g.
Lots of other non-local models
• Cushman and Ginn (1993) describe a non-local space/time convolution flux…if the kernel in the convolution term is a power-law, get fADE….
• This generalization allows “continuously evolving heterogeneity”….
Multifractals are used to describe complex geometry
• distribution describing different heterogeneity scales can follow different power laws
• Are there simple stochastic processes with well-known governing equations that produce multifractals?
