scientia et mente fractional diffusion – theory and...

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MANU E T MENTE

Fractional Diffusion – Theoryand Applications – Part III

22nd Canberra International Physics Summer School 2008

Bruce Henry(Trevor Langlands, Peter Straka, Susan Wearne, Claire Delides)

School of Mathematics and StatisticsThe University of New South Wales

Sydney NSW 2052 Australia

Fractional Diffusion – Theory and Applications – Part III – p . 1/38

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MANU E T MENTE

Continuous Time Random Walks

It was the man from Ironbark who struck the Sydney town,

He wandered over street and park, he wandered up and down.

He loitered here, he loitered there, till he was like to drop,

Until at last in sheer despair he sought a barber’s shop.

"’Ere! shave my beard and whiskers off, I’ll be a man of mark,

I’ll go and do the Sydney toff up home in Ironbark."

A.B. "Banjo" Paterson

The Bulletin, 17 December 1892


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MANU E T MENTE

CTRW Master Equations

Standard Random WalkThe step length is a fixed distance ∆x

Steps occur at discrete times separated by a fixed time

interval ∆t.


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MANU E T MENTE




interval ∆t.

Continuous Time Random Walk Montroll and Weiss (1965)

The step length is selected at random according to a step

length probability density λ(x).

Steps occur after a waiting time selected at random according

to a waiting time probability density ψ(t)


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MANU E T MENTE




interval ∆t.

Continuous Time Random Walk Montroll and Weiss (1965)

The step length is selected at random according to a step

length probability density λ(x).

Steps occur after a waiting time selected at random according

to a waiting time probability density ψ(t)

ProblemFind p(x, t|x0, t0) the conditional probability density that a RW

(Random Walker) starting from x0 at t = 0, is at x at time t.


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MANU E T MENTEArrival density

Define qn(x, t|x0, t0) conditional probability density that after n

steps a RW starting at x0 at t = 0 arrives at x at time t

qn+1(x, t|x0, 0) =

∫ +∞

−∞

(∫ t

0Ψ(x− x′, t− t′)qn(x

′, t′|x0, 0) dt′)

dx′

q0(x, t|x0, 0) = δx,x0δ(t) IC

Ψ(x− x′, t− t′) probability density that in a single step a RW

steps a distance x− x′ after waiting a time t− t′.


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MANU E T MENTEArrival density

Define qn(x, t|x0, t0) conditional probability density that after n

steps a RW starting at x0 at t = 0 arrives at x at time t

qn+1(x, t|x0, 0) =

∫ +∞

−∞

(∫ t

0Ψ(x− x′, t− t′)qn(x

′, t′|x0, 0) dt′)

dx′

q0(x, t|x0, 0) = δx,x0δ(t) IC

Ψ(x− x′, t− t′) probability density that in a single step a RW

steps a distance x− x′ after waiting a time t− t′.

The conditional probability density that a RW arrives at x at t

after any number of steps q(x, t|x0, 0) =∑∞

n=0 qn(x, t|x0, 0)

q(x, t|x0, 0) =∫ +∞−∞

∫ t0 Ψ(x′, t′)q(x− x′, t− t′|x0, 0) dt

′ dx′ + δ(t)δx,x0


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MANU E T MENTEBeing Density

Assume waiting times and step lengths are independent

Ψ(x− x′, t− t′) = λ(x− x′)ψ(t− t′).

ψ(t) =

∫ +∞

−∞Ψ(x′, t) dx′ λ(x) =

∫ ∞

0Ψ(x, t′) dt′

Survival probability that the walker does not step during time t

Φ(t) = 1 −∫ t0 ψ(t′) dt′ =

∫ ∞t ψ(t′) dt′


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MANU E T MENTEBeing Density

Assume waiting times and step lengths are independent

Ψ(x− x′, t− t′) = λ(x− x′)ψ(t− t′).

ψ(t) =

∫ +∞

−∞Ψ(x′, t) dx′ λ(x) =

∫ ∞

0Ψ(x, t′) dt′

Survival probability that the walker does not step during time t

Φ(t) = 1 −∫ t0 ψ(t′) dt′ =

∫ ∞t ψ(t′) dt′

The conditional probability density that a walker starting from

the origin at time zero is at x at time t is

p(x, t|x0, 0) =

∫ t

0q(x, t− t′|x0, 0)Φ(t′) dt′ =

∫ t

0q(x, t′|x0, 0)Φ(t− t


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MANU E T MENTECTRW Master EquationLaplace transform arrival density and being density


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q(x, u|x0, 0) =

∫ +∞

−∞Ψ(x′, u)q(x− x′, u|x0, 0) dx

′ + δx,x0 .

p(x, u|x0, 0) = q(x, u|x0, 0)Φ(u)


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MANU E T MENTECTRW Balance Equation

The expected concentration of many independent CTRW

random walkers at position x and t given the initial

concentration n(x, 0) is

n(x, t) = Φ(t)n(x, 0)+

∫ +∞

−∞

∫ t

0n(x′, t′)ψ(t− t′)λ(x−x′) dt′ dx′


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MANU E T MENTECTRW Balance Equation

The expected concentration of many independent CTRW

random walkers at position x and t given the initial

concentration n(x, 0) is

n(x, t) = Φ(t)n(x, 0)+

∫ +∞

−∞

∫ t

0n(x′, t′)ψ(t− t′)λ(x−x′) dt′ dx′

Different choices for the densities ψ(t) and λ(x) result in

different (possibly fractional) diffusion equations.


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MANU E T MENTEFrom CTRW Equations To Diffusion Equations

Decouple convolution integrals in the master equation using

Fourier transform in space and Laplace transform in time


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Consider asymptotic expansions of the transformed equation

for small values of the Fourier variable (spatial continuum limit)

and small values of the Laplace variables (long time limit)


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Consider asymptotic expansions of the transformed equation

for small values of the Fourier variable (spatial continuum limit)

and small values of the Laplace variables (long time limit)

Carry out inverse Fourier-Laplace transforms using (possibly

fractional order) differential operators.


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MANU E T MENTEFourier-Laplace CTRW Balance Equation

n(x, t) = Φ(t)n(x, 0) +

∫ +∞

−∞

∫ t

0n(x′, t′)ψ(t− t′)λ(x− x′) dt′ dx′

Φ(t) = 1 −∫ t

0ψ(t′) dt′ ⇒ Φ(u) =

1

u− ψ(u)

u

uˆn(q, u) = (1 − ψ(u))n(q, 0) + uψ(u) λ(q)n(q, u)

Asymptotic expressions for λ(q), ψ(u)?


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MANU E T MENTEStandard Diffusion

Assume step length density is an even function λ(x) = λ(−x)with finite variance σ2 =

∫

r2λ(r) dr,

λ(q) ∼ 1 − q2σ2

2+O(q4),

Example λ(x) = 1√2πσ2

exp(

− x2

2σ2

)

Gaussian density


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Assume step length density is an even function λ(x) = λ(−x)with finite variance σ2 =

∫

r2λ(r) dr,

λ(q) ∼ 1 − q2σ2

2+O(q4),

Example λ(x) = 1√2πσ2

exp(

− x2

2σ2

)

Gaussian density

Assume waiting time density has a finite mean τ , then

ψ(u) = 1 − τu+O(u2).

Example ψ(t) = 1τ exp

(

− tτ

)

Exponential density


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Leading order terms Fourier-Laplace-CTRW Balance

Equation

un(q, u) − n(q, 0) = −σ2q2

2τn(q, u).


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Equation

un(q, u) − n(q, 0) = −σ2q2

2τn(q, u).

Invert Fourier and Laplace transforms

∂n

∂t= D

∂2n

∂x2

where

D =σ2

2τ


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MANU E T MENTEExponential Density Markov PropertyThe probability of waiting a time T > t+ s conditioned on having

waited a time T > s is equivalent to the probability of waiting a time

T > t at the outset. (There is no memory in the distribution of

having already waited).


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P (T > t) =

∫ ∞

t

1

τexp

(

− t′

τ

)

dt′ = e−tτ


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P (T > t) =

∫ ∞

t

1

τexp

(

− t′

τ

)

dt′ = e−tτ

⇒ P (T > t+ s|T > s) =P (T > t+ s)

P (T > s)= e−

tτ = P (T > t).


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P (T > t) =

∫ ∞

t

1

τexp

(

− t′

τ

)

dt′ = e−tτ

⇒ P (T > t+ s|T > s) =P (T > t+ s)

P (T > s)= e−

tτ = P (T > t).

What if the waiting time density has a power law tail?


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MANU E T MENTE

An average individual who seeks a friend twice his height would fail.

On the other hand, one who has an average income will have no

trouble in discovering a richer person with twice his income, and

that richer person may, with a little diligence, locate a third party

with twice his income, etc.

Elliot Montroll and Michael Shlesinger (1984)

catastrophes can occur for no reason whatsoever

Per Bak (1996)


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MANU E T MENTEPower Law Waiting Times

Assume step length density as above λ(q) ∼ 1 − q2σ2

2 +O(q4),


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2 +O(q4),

Consider a Pareto power law waiting time density

ψ(t) =ατα

t1+αt ∈ [τ,∞], 0 < α < 1


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2 +O(q4),


ψ(t) =ατα

t1+αt ∈ [τ,∞], 0 < α < 1

Note (i) the mean waiting time is infinite, (ii) the probability of waiting a time

T > t+ s, conditioned on having waited a time T > s, is greater than the probability

of waiting a time T > t at the outset (the waiting time density has a temporal

memory) (iii) the waiting time density is scale invariant, ψ(γt) = γ−(1+α)ψ(t).


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2 +O(q4),


ψ(t) =ατα

t1+αt ∈ [τ,∞], 0 < α < 1

Note (i) the mean waiting time is infinite, (ii) the probability of waiting a time

T > t+ s, conditioned on having waited a time T > s, is greater than the probability

of waiting a time T > t at the outset (the waiting time density has a temporal

memory) (iii) the waiting time density is scale invariant, ψ(γt) = γ−(1+α)ψ(t).

Tauberian Theorem

ψ(u) ∼ 1 − Γ(1 − α)ταuα


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MANU E T MENTEAnomalous subdiffusion


Equation

uˆn(q, u) − n(q, 0) = − q2σ2

2ταΓ(1 − α)u1−αˆn(q, u)


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Equation

uˆn(q, u) − n(q, 0) = − q2σ2



∂n(x, t)

∂t= D

(

0D1−αt

∂2n(x, t)

∂x2

)

D =σ2

2ταΓ(1 − α)


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Equation

uˆn(q, u) − n(q, 0) = − q2σ2



∂n(x, t)

∂t= D

(

0D1−αt

∂2n(x, t)

∂x2

)

D =σ2

2ταΓ(1 − α)

0D1−αt denotes a Riemann-Liouville fractional derivative of

order 1 − α


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MANU E T MENTEFundamental SolutionGreen’s solution G(x, t), G(x, 0) = δ(x)

Metzler and Klafter (2000)

G(x, t) =1√

4πDtαH2,0

1,2

[

x2

4Dtα

∣

∣

∣

∣

(1 − α2 , α)

(0, 1), (12 , 1)

Special case α = 1/2 the solution in terms of Meijer G-function

G(x, t) =1

√

8πDt12

M3,00,3

[

x2

16Dt12

∣

∣

∣

∣

0, 14 ,

12

The Meijer G-functions are included as special functions in MAPLE

and Mathematica.


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MANU E T MENTEMean Square Displacement

〈x2(t)〉 =

∫ ∞

−∞x2G(x, t) dx = L−1

(

limq→0

− d2

dq2ˆG(q, u)

)

Rearrange

uˆn(q, u) − n(q, 0) = − q2σ2


using G(q, 0) = 1

ˆG(q, u) =

1

u+ q2Du1−α

Then

〈x2(t)〉 = L−1(

2Dαu−1−α)

=2D

Γ(1 + α)tα


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MANU E T MENTEThe Fundamental Solution in Fractional Subdiffusion

Carry out Fourier transform in space and Laplace transform in

time (using the known results for the Laplace transform of

Riemann-Liouville fractional derivatives).


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Find the transformed solution as the solution of an algebraic

problem in Fourier-Laplace space.


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Carry out the inverse Fourier transform (easy).


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Carry out the inverse Fourier transform (easy).

Carry our inverse Laplace transform (hard). Expand the

Laplace transform as a series expansion in Fox H-functions

and then perform a term by term inversion.


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MANU E T MENTESubdiffusion is Subordinated Diffusion

If n(x, t) is the Green’s solution of the time fractional

subdiffusion equation then

n(x, t) =

∫ ∞

0n⋆(x, τ)T (τ, t) dτ ⋆

n⋆(x, τ) is the Green’s solution of the standard diffusion

equation and L (T (τ, t)) = T (τ, u) = uα−1e−τuα

τ is called the operational time t is called the physical time


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MANU E T MENTESubdiffusion is Subordinated Diffusion

If n(x, t) is the Green’s solution of the time fractional

subdiffusion equation then

n(x, t) =

∫ ∞

0n⋆(x, τ)T (τ, t) dτ ⋆

n⋆(x, τ) is the Green’s solution of the standard diffusion

equation and L (T (τ, t)) = T (τ, u) = uα−1e−τuα

τ is called the operational time t is called the physical time

The operational time scales as the number of steps in the

walk. In the standard random walk the number of steps is

proportional to the physical time but in the CTRW with infinite

mean waiting times the number of steps is a random variable.


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MANU E T MENTEWhere Does a Power Law Waiting Time Density Come From?

Random walkers in an environment with an exponential

distribution of trap binding energies ρ(E) = 1E0e− E

E0 and

thermally activated trapping times τ = eE

kBT

Scher, Montroll (1975)


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MANU E T MENTEWhere Does a Power Law Waiting Time Density Come From?

Random walkers in an environment with an exponential

distribution of trap binding energies ρ(E) = 1E0e− E

E0 and

thermally activated trapping times τ = eE

kBT

Scher, Montroll (1975)Waiting time density

ψ(τ)dτ = ρ(E)dE = ρ(E)dE

dτdτ

=1

E0e− E

E0

(

kBT

τ

)

dτ

=1

E0τ− kT

E0

(

kBT

τ

)

dτ

=

(

kBT

E0

)

τ− kT

E0−1dτ ⇒ ψ(t) = αt−1−α


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MANU E T MENTELevy Step Length Density

Assume exponential (Markovian) waiting time density

ψ(u) ∼ 1 − τu


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MANU E T MENTELevy Step Length Density

Assume exponential (Markovian) waiting time density

ψ(u) ∼ 1 − τu

Consider a Levy (power law) step length density

λ(x) ∼ Aασα

|x|−1−α, 1 < α < 2.

The Levy step length density enables walks on all spatial

scales.

λ(q) = exp(−σα|q|α) ∼ 1 − σα|q|α


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MANU E T MENTEAnomalous Superdiffusion

Leading order terms Fourier-Laplace-CTRW Master Equation

uˆn(q, u) − u n(q, 0) = −σα|q|ατ

ˆn(q, u)


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ˆn(q, u)


∂n

∂t= D∇α

|x|n 1 < α < 2 D =σα

τ


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ˆn(q, u)


∂n

∂t= D∇α

|x|n 1 < α < 2 D =σα

τ

∇α|x| is the Riesz fractional derivative


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MANU E T MENTEFundamental SolutionGreen’s solution G(x, t), G(x, 0) = δ(x)

Metzler and Klafter (2000)

1

α |x|H1,12,2

|x|(Dt)1/α

∣

∣

∣

∣

∣

∣

(1, 1/α) (1, 1/2)

(1, 1) (1, 1/2)

Levy Stable Law


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MANU E T MENTEPseudo-Mean Square Displacement

The solution has the asymptotic behaviour

n(x, t) ∼ σαt

τ |x|1+α , 1 < α < 2.


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n(x, t) ∼ σαt

τ |x|1+α , 1 < α < 2.

The mean square displacement diverges 〈x2(t)〉 → ∞ but

〈|x|δ〉 ∼ tδα , 0 < δ < α < 2.


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n(x, t) ∼ σαt

τ |x|1+α , 1 < α < 2.

The mean square displacement diverges 〈x2(t)〉 → ∞ but

〈|x|δ〉 ∼ tδα , 0 < δ < α < 2.

Define a non-divergent pseudo mean square displacement

〈[x2(t)]〉 ∼ t2α


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MANU E T MENTEGeneralized Central Limit Theorem

A symmetric Levy stable law with a power law tail ∼ C|x|−1−α

is the limiting stable law for the distribution of the ‘normalized’

sum of random variables

X1 +X2 + . . . XN

Nα0 < α < 2

with infinite variance. The mean is infinite if 0 < α < 1 so

restrict to 1 < α < 2.


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X1 +X2 + . . . XN

Nα0 < α < 2



If the random variables are RW steps the sum is the rescaled

position.


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X1 +X2 + . . . XN

Nα0 < α < 2



If the random variables are RW steps the sum is the rescaled

position.

The solution of the space fractional diffusion equation is the

Levy stable distribution.


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MANU E T MENTESimulating Random Walks for Fractional DiffusionSet the starting position x and jump-time t to zero.

Generate a random waiting-time δt from ψ(t) and a random

jump-length δx from λ(x).


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Update the position x(t+ δt) = x(t) + δx.


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Update the jump-time t = t+ δt. For non-constant waiting

times both the position of the particle and the jump-time need

to be stored.


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Update the jump-time t = t+ δt. For non-constant waiting

times both the position of the particle and the jump-time need

to be stored.

Repeat above bullet steps


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MANU E T MENTEGeneration of Waiting-TimesTo generate a random waiting-time t with a probability density

function ψ(t)

Select a random r ∈ [0, 1] with a uniform density ρ(r) = 1


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function ψ(t)


ρ(r) dr = ρ(r(t)dr

dtdt = ψ(t) dt


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function ψ(t)



dtdt = ψ(t) dt

ρ(r(t)) = 1 ⇒ dr

dt= ψ(t)


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function ψ(t)



dtdt = ψ(t) dt

ρ(r(t)) = 1 ⇒ dr

dt= ψ(t)

∫ r

0dr′ =

∫ t

0ψ(t′) dt′


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function ψ(t)



dtdt = ψ(t) dt

ρ(r(t)) = 1 ⇒ dr

dt= ψ(t)

∫ r

0dr′ =

∫ t

0ψ(t′) dt′

r =

∫ t

0ψ(t′) dt′ invert for t.


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MANU E T MENTE

Random Walk waiting time δt step length δx constant D

Standard τ∆x 0 ≤ r < 1

2

−∆x 12≤ r < 1

∆x2

2τ

fBm τ

∆x 0 ≤ r < αnα−1

−∆x αnα−1 ≤ r < 2αnα−1

0 2αnα−1 ≤ r < 1

∆x2

τα

Subdiffusion τ“

(1 − r)−1

α − 1” ∆x 0 ≤ r < 1

2

−∆x 12≤ r < 1

∆x2

2ταΓ(1 − α)

Superdiffusion τ ∆x

„ − lnu cosφ

cos ((1 − α)φ)

«1− 1

α sin (αφ)

cosφ

∆xα

τu, v ∈ (0, 1) independent random numbers, φ = π(v − 1/2), n = t/τ .


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MANU E T MENTE

Probability Density Functions

Standard 1√4πDt

e−x2

4Dt Gaussian

Markovian

fBm 1√4πDtα e

− x2

4Dtα Gaussian

Non-Markovian

Subdiffusion 1√4πDtαH

2,01,2

2

4

x2

4Dtα

˛

˛

˛

˛

˛

˛

(1 − α/2, α)

(0, 1) (1/2, 1)

3

5 Non-Gaussian

0 < α < 1 Non-Markovian

Superdiffusion 1α|x|H

1,12,2

2

4

|x|(Dt)1/α

˛

˛

˛

˛

˛

˛

(1, 1/α) (1, 1/2)

(1, 1) (1, 1/2)

3

5 Non-Gaussian

1 < α < 2 Markovian


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–4

–2

0

2

4

6

8

10

12

x(t)

20 40 60 80 100

t

0

5

10

15

20

2 4 6 8 10 12 14 16 18 20

t

0

0.02

0.04

0.06

0.08

p(x)

–20 –10 10 20

x


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MANU E T MENTEFractional Brownian Motion α = 1/2

–4

–2

0

2

4

x(t)

20 40 60 80 100t

1.

2.

4.

7.

.1e2

.2e2

5. .1e2 .5e2 .1e3

t

0.02

0.04

0.06

0.08

0.1

0.12

0.14

p(x)

–20 –10 10 20

x


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MANU E T MENTEFractional Subdiffusion α = 1/2

–8

–6

–4

–2

0

2

x(t)

20 40 60 80 100t

2.

3.

4.

6.

8.

1. 2. 5. .1e2 .2e2

t

0

0.05

0.1

0.15

0.2

p(x)

–20 –10 10 20

x


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MANU E T MENTEFractional Superdiffusion α = 3/2 (δ = 3/4)

0

50

100

150

200

x(t)

20 40 60 80 100

t

2.

3.

4.

1. 2. 5. .1e2 .2e2

t

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

p(x)

–20 –10 0 10 20

x


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MANU E T MENTEFractional Fokker-Planck EquationsBiased CTRWs lead to fractional Fokker-Planck equations.


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Space dependent force (Barkai, Metzler, Klafter, 2000)

∂n(x, t)

∂t= 0D1−α

t D∇2n(x, t) − 0D1−αt ∇

(

1

ηf(x)n(x, t)

)

A

D = kBT

mηgeneralized Einstein relation.


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∂n(x, t)

∂t= 0D1−α

t D∇2n(x, t) − 0D1−αt ∇

(

1

ηf(x)n(x, t)

)

A

D = kBT


Time dependent force (Sokolov, Klafter, 2006)

∂n(x, t)

∂t= 0D1−α

t D∇2n(x, t) −∇(

1

ηf(t) 0D1−α

t ∇n(x, t)

)

B


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∂n(x, t)

∂t= 0D1−α

t D∇2n(x, t) − 0D1−αt ∇

(

1

ηf(x)n(x, t)

)

A

D = kBT


Time dependent force (Sokolov, Klafter, 2006)

∂n(x, t)

∂t= 0D1−α

t D∇2n(x, t) −∇(

1

ηf(t) 0D1−α

t ∇n(x, t)

)

B

Open Problems: Space and time dependent force, Model A or

Model B or . . . .? Time subordination in A may suit internal

force fields but not external force fields. Nonlinear Forces?


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MANU E T MENTEFractional Reaction-Diffusion EquationsCTRW balance eqns with temporal memory and source/sink terms

yield fractional reaction diffusion equations (Henry, Wearne, 2000)

Model A: Time fractional derivative on the spatial diffusion

term but not on the reaction terms.


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Model B: Time fractional derivative equal on both reaction and

diffusion terms. Motivated by subordination – reactions and

diffusions affected by the same operational time scales (Yuste,

Acedi, Lindenberg, 2004).


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Turing pattern formation in fractional reaction diffusion

systems (Henry, Langlands, Wearne, 2005, 2008)


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Turing pattern formation in fractional reaction diffusion

systems (Henry, Langlands, Wearne, 2005, 2008)

Problem: Should reactions modify diffusions in CTRWs?


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MANU E T MENTELinear Reactions, CTRWs and SubdiffusionHenry, Langlands, Wearne, 2006 Sokolov, Schmidt, Sagues, 2006

Linear reaction dynamics to model exponential growth (+k) or

decay (−k) during the CTRW waiting time intervals


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CTRW Balance Equation

n(x, t) = Φ(t)e±ktn(x, 0) +

Z ∞

−∞

Z t

0n(x′, t′)e±k(t−t′)ψ(t− t′)λ(x− x′) dt′ dx′


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n(x, t) = Φ(t)e±ktn(x, 0) +

Z ∞

−∞

Z t


Fractional Reaction-Subdiffusion Equation (HLW2006)

∂n

∂t= D e±kt 0D1−α

t

(

e∓kt∂2n

∂x2

)

± kn.


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n(x, t) = Φ(t)e±ktn(x, 0) +

Z ∞

−∞

Z t


Fractional Reaction-Subdiffusion Equation (HLW2006)

∂n

∂t= D e±kt 0D1−α

t

(

e∓kt∂2n

∂x2

)

± kn.

Extended to multispecies (LHW 2008) but nonlinear fractional

reaction-diffusion is an open problem.


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MANU E T MENTE

Fractional Diffusion

Space-time fractional Fokker-Planck equation (Metzler, Klafter, 2000)

∂w

∂t= D1−α

t

„

∂

∂x

V ′(x)

η+K∇µ

|x|

«

w.

Space Fractional Equation for Maximum Tsallis Entropy (Bologna, Tsallis, Grigolini, 2000)

∂p

∂t= D

∂γp

∂xγ0 < γ ≤ 2

Space-time fractional diffusion for plasmas (del-Castillo-Negrette, Carreras, Lynch, 2004)

Dβt P = χ∇α

|x|P.

Fractional Black-Scholes model for option prices (Cartea, 2006)

rV (x, t) =∂V (x, t)

∂t+

“

r + σα sec(απ

2)” ∂V

∂x− σα sec(

απ

2)Dα

xV

Fractional cable equation for nerve cells (Henry, Langlands, Wearne, 2008)

rmcm∂V

∂t=

drm

4rL(γ)D1−γ

t

„

∂2V

∂x2

«

−D1−κt (V − rmie)


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MANU E T MENTE

Thank You


scientia et mente fractional diffusion – theory and...

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