Heavy-tailed stochastic processes and fractional ADEs for ... · Stochastic Model • Physical (or...

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

( ) ( )t t

X t S n YΔ

= =∑

t=time

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/

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

+ +…+ −→

finite variance:iY ∼

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

+ +…+ −→

finite variance:iY ∼

as n →∞

1. center2. expand time scale

3. contract spatial scale

Classical Central Limit Theorem

Let and 2

v Dt tμ σ

= =Δ Δ

1 2 0,1nY Y Y n n Nμ σ+ +…+ ≈ +

1 2 2 0,1nY Y Y tv Dt N+ +…+ ≈ +

Let n tt

1 2 0,1nt tY Y Y Nt tμ σ

⎛ ⎞⎟⎜+ +…+ ≈ + ⎟⎜ ⎟⎜⎝ ⎠Δ Δ

1 2 0,1nY Y Y t t Nt t

μ σ+ +…+ ≈ +

location of a particle at time t

( )t t

X t YΔ

Converges in distribution to a Gaussian density

Let and 2

v Dt tμ σ

= =Δ Δ

1 2 2 0,1nY Y Y tv Dt N+ +…+ ≈ +

2( )2*21( ) ( , )

x vtDtX t C x t e

−−

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

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

∂= −

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∂

= −∂

FCnt x

∂∂− =

∂ ∂

x xC Cv C Dt x x

∂ ⎡ ∂ ∂ ⎤⎛ ⎞= − +⎜ ⎟⎢ ⎥∂ ∂ ∂⎝ ⎠⎣ ⎦

C C Cv Dt x x

∂ ∂ ∂= − +

∂ ∂ ∂

Green’s function solution to ADE2

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

= − +∂

solve for C ( )2( ) ( )ˆ ( , ) vt ik Dt ikC k t e − +=

2( )2*21( , )

x vtDtC x t e

−−

=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= ∫

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

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

∂− +

limiting stochastic process

governing equation

solution

Brownian motion

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 100110-1

2.0 (Gaussian)

x - μ

Probability distributions are described by their PDF and CDF

Area=1

Probability Density Function (PDF)

Cumulative Distribution Function (CDF)

Exponential

CDF: F(x)=1PDF: f(x)=F'(x)= e

Pareto

CDF: F(x)=1PDF: f(x)=F'(x)=

− −

( )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 1000 2000 3000 4000 5000

# samples

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 1000 2000 3000 4000 5000

# samples

Sample variance never converges

1.9 stable(0,1,0)

( ) ( )t t

X t S n YΔ

= = ∑

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/

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

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

0.41.5-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

a=.5a=.7a=.9

skewed stables

skewedstable x .5,( )

• 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= =

1 2 1, , 0nY Y Y n n Sααμ σ σ β μ+ +…+ ≈ + = =

1 2 2 1, , 0nY Y Y tv Dt Sαα σ β μ+ +…+ ≈ + = =

Let n tt

1 2 1, , 0nt tY Y Y St t

αμ σ σ β μ⎛ ⎞⎟⎜+ +…+ ≈ + = =⎟⎜ ⎟⎜⎝ ⎠Δ Δ

location of a particle at time t

( )t t

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.

1 2 0,1nt tY Y Y Nt tμ σ

⎛ ⎞⎟⎜+ +…+ ≈ + ⎟⎜ ⎟⎜⎝ ⎠Δ Δ2

Let and 2

v Dt t

μ σ= =

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

∂= −

F = fluxDd = diffusion coeff.C = concentration

Integer-Order Derivativelocal function

slope at an infinitesimally small

∂≈

Fractional-Order Derivativenon-local function

the fractional derivative at a point

depends on the values over the entire function

∂∂

• 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

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

∂∂

α = 0.9 (low heterogeneity)

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

∂∂

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

∂= −

∂ eFCn

t x∂∂

− =∂ ∂

CC v C Dt x x

⎡ ⎤⎛ ⎞∂∂ ∂= − +⎢ ⎥⎜ ⎟∂ ∂ ∂⎝ ⎠⎣ ⎦

CC Cv Dt x x

∂∂ ∂= − +

∂ ∂ ∂

Fractional ADE for SUPER-diffusive processes

CC Cv Dt x x

∂∂ ∂=− +

∂ ∂ ∂

where Cv

concentration

constant dispersion coefficient

fractional-in-spacederivative

αth derivative of concentration(where 1<α 2)≤

Space-fractional advection-dispersion equation (fADE)

∂∂

+∂∂

−=∂∂

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)C x t t C t xα α− −

CC Cv Dt x x

∂∂ ∂= − +

∂ ∂ ∂

CC Cv Dt x x

∂∂ ∂= − +

∂ ∂ ∂

governing equation

solution

Brownian motion

Gaussian density

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)

governing equation

solution

Brownian motion

Gaussian density

length distribution

?Lévy motion

space fADE

α-stable density

randomwalks

heavy tails in time

running average

numb er o f samp les

running variance

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

Continuous time random walk (CTRW)

( ) ( ( ))N t

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

{ }( )

......

( ) ( )

YJS Y Y YT J J JN n T t

X t S N t

= + + += + + +

( ) ( ( ))N t

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

( )( )( ) ( ) ... (

Random variable Density

)( ) ( ) ... ( )

f x f x f xt t t

ψ ψ ψ∗ ∗ ∗∗ ∗ ∗

1 ( ) ( , 0)( , )1 ( ) ( )

s P k tP k ss s f k

− ==

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

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

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

νσ β μ

→∞

+ +…+ −= = =∼

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 2 1, , 0nY Y Y n Sαα σ β μ+ +…+ ≈ = =

( ) 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

( )N t

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

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

ival a

t cell

cell i

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

∂∂− =

∂ ∂

C Cv C Dt x x

∂ ⎡ ∂ ∂ ⎤⎛ ⎞= − +⎜ ⎟⎢ ⎥∂ ∂ ∂⎝ ⎠⎣ ⎦

C C Cv Dt x x

∂ ∂ ∂= − +

∂ ∂ ∂

Fractional ADE for sub-diffusive processes when γ<1/2

sub- or super-diffusive processes when γ>1/2

CC Cv Dt x x

∂∂ ∂=− +

∂ ∂ ∂where C

concentration

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…

C C Cv Dt x x

∂ ∂ ∂= − +

∂ ∂ ∂

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,

solve for C….

C C Cv Dt x x

∂ ∂ ∂= − +

∂ ∂ ∂

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

( ,( , ) , , ) .t

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

∂∂ ∂= − +

∂ ∂ ∂

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

0 100 200 300 400 500 600

bromide

power law fit

exponential fit

Time (days)

action

bromide

10 100 1000Time (days)

bromide

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 )

C tL x C C xt

γβ βγ

−∂= −

∂ Γ −

,0( ) ( )(1 )

imim m

C tL x C C xt

γβγ

−∂= +

∂ Γ −

tot m m im imC C Cθ θ= +

,0( ) , ( ,0) ( )tottot tot m m

C L x C C x C xt

γβ θ∂= =

mobile

immobile

θβθ

ADE (a(x))

Immobile (βCim)Total (Ctot)

Mobile (Cm)

0 5 10 15 20 25 30 35 400

Distance from source (x)

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)

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)

100150200250300

-25 25 75 125Longitudinal Distance (meters)

g/L) model

100150200250300

g/L) model

100150200250300

g/L) model

100150200250300

g/L) model

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

governing equation

solution

Brownian motion

Gaussian density

length distribution

Lévy motion

space fADE

α-stable density

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.

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( ) ( )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.

C C Cv Dt x x

∂ ∂ ∂= − +

∂ ∂ ∂

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?

Heavy-tailed stochastic processes and fractional ADEs for ... · Stochastic Model • Physical (or...

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