SCIENTIA
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
SCIENTIA
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
Fractional Diffusion – Theory and Applications – Part III – p . 2/38
SCIENTIA
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.
Fractional Diffusion – Theory and Applications – Part III – p . 3/38
SCIENTIA
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.
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)
Fractional Diffusion – Theory and Applications – Part III – p . 3/38
SCIENTIA
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.
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.
Fractional Diffusion – Theory and Applications – Part III – p . 3/38
SCIENTIA
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′.
Fractional Diffusion – Theory and Applications – Part III – p . 4/38
SCIENTIA
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
Fractional Diffusion – Theory and Applications – Part III – p . 4/38
SCIENTIA
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′
Fractional Diffusion – Theory and Applications – Part III – p . 5/38
SCIENTIA
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
Fractional Diffusion – Theory and Applications – Part III – p . 5/38
SCIENTIA
MANU E T MENTECTRW Master EquationLaplace transform arrival density and being density
Fractional Diffusion – Theory and Applications – Part III – p . 6/38
SCIENTIA
MANU E T MENTECTRW Master EquationLaplace transform arrival density and being density
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)
Fractional Diffusion – Theory and Applications – Part III – p . 6/38
SCIENTIA
MANU E T MENTECTRW Master EquationLaplace transform arrival density and being density
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)
=
∫ +∞
−∞Ψ(x′, u)Φ(u)q(x− x′, u|x0, 0) dx
′ + Φ(u)δx,x0
Fractional Diffusion – Theory and Applications – Part III – p . 6/38
SCIENTIA
MANU E T MENTECTRW Master EquationLaplace transform arrival density and being density
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)
=
∫ +∞
−∞Ψ(x′, u)Φ(u)q(x− x′, u|x0, 0) dx
′ + Φ(u)δx,x0
=
∫ +∞
−∞Ψ(x′, u)p(x− x′, u|x0, 0) dx
′ + Φ(u)δx,x0 .
Fractional Diffusion – Theory and Applications – Part III – p . 6/38
SCIENTIA
MANU E T MENTECTRW Master EquationLaplace transform arrival density and being density
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)
=
∫ +∞
−∞Ψ(x′, u)Φ(u)q(x− x′, u|x0, 0) dx
′ + Φ(u)δx,x0
=
∫ +∞
−∞Ψ(x′, u)p(x− x′, u|x0, 0) dx
′ + Φ(u)δx,x0 .
Inverse Laplace transform yields Master Equation
p(x, t|x0, 0) = Φ(t)δx,x0+
∫ t
0ψ(t−t′)
∫ +∞
−∞λ(x−x′)p(x′, t′|x0, 0) dx
′ dt′.
Fractional Diffusion – Theory and Applications – Part III – p . 6/38
SCIENTIA
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′
Fractional Diffusion – Theory and Applications – Part III – p . 7/38
SCIENTIA
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.
Fractional Diffusion – Theory and Applications – Part III – p . 7/38
SCIENTIA
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
Fractional Diffusion – Theory and Applications – Part III – p . 8/38
SCIENTIA
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
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)
Fractional Diffusion – Theory and Applications – Part III – p . 8/38
SCIENTIA
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
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.
Fractional Diffusion – Theory and Applications – Part III – p . 8/38
SCIENTIA
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)?
Fractional Diffusion – Theory and Applications – Part III – p . 9/38
SCIENTIA
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
Fractional Diffusion – Theory and Applications – Part III – p . 10/38
SCIENTIA
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
Assume waiting time density has a finite mean τ , then
ψ(u) = 1 − τu+O(u2).
Example ψ(t) = 1τ exp
(
− tτ
)
Exponential density
Fractional Diffusion – Theory and Applications – Part III – p . 10/38
SCIENTIA
MANU E T MENTEStandard Diffusion
Leading order terms Fourier-Laplace-CTRW Balance
Equation
un(q, u) − n(q, 0) = −σ2q2
2τn(q, u).
Fractional Diffusion – Theory and Applications – Part III – p . 11/38
SCIENTIA
MANU E T MENTEStandard Diffusion
Leading order terms Fourier-Laplace-CTRW Balance
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
Fractional Diffusion – Theory and Applications – Part III – p . 11/38
SCIENTIA
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).
Fractional Diffusion – Theory and Applications – Part III – p . 12/38
SCIENTIA
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).
P (T > t) =
∫ ∞
t
1
τexp
(
− t′
τ
)
dt′ = e−tτ
Fractional Diffusion – Theory and Applications – Part III – p . 12/38
SCIENTIA
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).
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).
Fractional Diffusion – Theory and Applications – Part III – p . 12/38
SCIENTIA
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).
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?
Fractional Diffusion – Theory and Applications – Part III – p . 12/38
SCIENTIA
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)
Fractional Diffusion – Theory and Applications – Part III – p . 13/38
SCIENTIA
MANU E T MENTEPower Law Waiting Times
Assume step length density as above λ(q) ∼ 1 − q2σ2
2 +O(q4),
Fractional Diffusion – Theory and Applications – Part III – p . 14/38
SCIENTIA
MANU E T MENTEPower Law Waiting Times
Assume step length density as above λ(q) ∼ 1 − q2σ2
2 +O(q4),
Consider a Pareto power law waiting time density
ψ(t) =ατα
t1+αt ∈ [τ,∞], 0 < α < 1
Fractional Diffusion – Theory and Applications – Part III – p . 14/38
SCIENTIA
MANU E T MENTEPower Law Waiting Times
Assume step length density as above λ(q) ∼ 1 − q2σ2
2 +O(q4),
Consider a Pareto power law waiting time density
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).
Fractional Diffusion – Theory and Applications – Part III – p . 14/38
SCIENTIA
MANU E T MENTEPower Law Waiting Times
Assume step length density as above λ(q) ∼ 1 − q2σ2
2 +O(q4),
Consider a Pareto power law waiting time density
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
Fractional Diffusion – Theory and Applications – Part III – p . 14/38
SCIENTIA
MANU E T MENTEAnomalous subdiffusion
Leading order terms Fourier-Laplace-CTRW Balance
Equation
uˆn(q, u) − n(q, 0) = − q2σ2
2ταΓ(1 − α)u1−αˆn(q, u)
Fractional Diffusion – Theory and Applications – Part III – p . 15/38
SCIENTIA
MANU E T MENTEAnomalous subdiffusion
Leading order terms Fourier-Laplace-CTRW Balance
Equation
uˆn(q, u) − n(q, 0) = − q2σ2
2ταΓ(1 − α)u1−αˆn(q, u)
Invert Fourier and Laplace transforms
∂n(x, t)
∂t= D
(
0D1−αt
∂2n(x, t)
∂x2
)
D =σ2
2ταΓ(1 − α)
Fractional Diffusion – Theory and Applications – Part III – p . 15/38
SCIENTIA
MANU E T MENTEAnomalous subdiffusion
Leading order terms Fourier-Laplace-CTRW Balance
Equation
uˆn(q, u) − n(q, 0) = − q2σ2
2ταΓ(1 − α)u1−αˆn(q, u)
Invert Fourier and Laplace transforms
∂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
Fractional Diffusion – Theory and Applications – Part III – p . 15/38
SCIENTIA
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.
Fractional Diffusion – Theory and Applications – Part III – p . 16/38
SCIENTIA
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
2ταΓ(1 − α)u1−αˆn(q, u)
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
Fractional Diffusion – Theory and Applications – Part III – p . 17/38
SCIENTIA
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).
Fractional Diffusion – Theory and Applications – Part III – p . 18/38
SCIENTIA
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).
Find the transformed solution as the solution of an algebraic
problem in Fourier-Laplace space.
Fractional Diffusion – Theory and Applications – Part III – p . 18/38
SCIENTIA
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).
Find the transformed solution as the solution of an algebraic
problem in Fourier-Laplace space.
Carry out the inverse Fourier transform (easy).
Fractional Diffusion – Theory and Applications – Part III – p . 18/38
SCIENTIA
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).
Find the transformed solution as the solution of an algebraic
problem in Fourier-Laplace space.
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.
Fractional Diffusion – Theory and Applications – Part III – p . 18/38
SCIENTIA
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
Fractional Diffusion – Theory and Applications – Part III – p . 19/38
SCIENTIA
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.
Fractional Diffusion – Theory and Applications – Part III – p . 19/38
SCIENTIA
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)
Fractional Diffusion – Theory and Applications – Part III – p . 20/38
SCIENTIA
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
Fractional Diffusion – Theory and Applications – Part III – p . 20/38
SCIENTIA
MANU E T MENTELevy Step Length Density
Assume exponential (Markovian) waiting time density
ψ(u) ∼ 1 − τu
Fractional Diffusion – Theory and Applications – Part III – p . 21/38
SCIENTIA
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
Fractional Diffusion – Theory and Applications – Part III – p . 21/38
SCIENTIA
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)
Fractional Diffusion – Theory and Applications – Part III – p . 22/38
SCIENTIA
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)
Invert Fourier and Laplace transforms
∂n
∂t= D∇α
|x|n 1 < α < 2 D =σα
τ
Fractional Diffusion – Theory and Applications – Part III – p . 22/38
SCIENTIA
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)
Invert Fourier and Laplace transforms
∂n
∂t= D∇α
|x|n 1 < α < 2 D =σα
τ
∇α|x| is the Riesz fractional derivative
Fractional Diffusion – Theory and Applications – Part III – p . 22/38
SCIENTIA
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
Fractional Diffusion – Theory and Applications – Part III – p . 23/38
SCIENTIA
MANU E T MENTEPseudo-Mean Square Displacement
The solution has the asymptotic behaviour
n(x, t) ∼ σαt
τ |x|1+α , 1 < α < 2.
Fractional Diffusion – Theory and Applications – Part III – p . 24/38
SCIENTIA
MANU E T MENTEPseudo-Mean Square Displacement
The solution has the asymptotic behaviour
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.
Fractional Diffusion – Theory and Applications – Part III – p . 24/38
SCIENTIA
MANU E T MENTEPseudo-Mean Square Displacement
The solution has the asymptotic behaviour
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
Fractional Diffusion – Theory and Applications – Part III – p . 24/38
SCIENTIA
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.
Fractional Diffusion – Theory and Applications – Part III – p . 25/38
SCIENTIA
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.
If the random variables are RW steps the sum is the rescaled
position.
Fractional Diffusion – Theory and Applications – Part III – p . 25/38
SCIENTIA
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.
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.
Fractional Diffusion – Theory and Applications – Part III – p . 25/38
SCIENTIA
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).
Fractional Diffusion – Theory and Applications – Part III – p . 26/38
SCIENTIA
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).
Update the position x(t+ δt) = x(t) + δx.
Fractional Diffusion – Theory and Applications – Part III – p . 26/38
SCIENTIA
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).
Update the position x(t+ δt) = x(t) + δx.
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.
Fractional Diffusion – Theory and Applications – Part III – p . 26/38
SCIENTIA
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).
Update the position x(t+ δt) = x(t) + δx.
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
Fractional Diffusion – Theory and Applications – Part III – p . 26/38
SCIENTIA
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
Fractional Diffusion – Theory and Applications – Part III – p . 27/38
SCIENTIA
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
ρ(r) dr = ρ(r(t)dr
dtdt = ψ(t) dt
Fractional Diffusion – Theory and Applications – Part III – p . 27/38
SCIENTIA
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
ρ(r) dr = ρ(r(t)dr
dtdt = ψ(t) dt
ρ(r(t)) = 1 ⇒ dr
dt= ψ(t)
Fractional Diffusion – Theory and Applications – Part III – p . 27/38
SCIENTIA
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
ρ(r) dr = ρ(r(t)dr
dtdt = ψ(t) dt
ρ(r(t)) = 1 ⇒ dr
dt= ψ(t)
∫ r
0dr′ =
∫ t
0ψ(t′) dt′
Fractional Diffusion – Theory and Applications – Part III – p . 27/38
SCIENTIA
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
ρ(r) dr = ρ(r(t)dr
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.
Fractional Diffusion – Theory and Applications – Part III – p . 27/38
SCIENTIA
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 .
Fractional Diffusion – Theory and Applications – Part III – p . 28/38
SCIENTIA
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
Fractional Diffusion – Theory and Applications – Part III – p . 29/38
SCIENTIA
MANU E T MENTEStandard Diffusion
–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
Fractional Diffusion – Theory and Applications – Part III – p . 30/38
SCIENTIA
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
Fractional Diffusion – Theory and Applications – Part III – p . 31/38
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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
Fractional Diffusion – Theory and Applications – Part III – p . 32/38
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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
Fractional Diffusion – Theory and Applications – Part III – p . 33/38
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MANU E T MENTEFractional Fokker-Planck EquationsBiased CTRWs lead to fractional Fokker-Planck equations.
Fractional Diffusion – Theory and Applications – Part III – p . 34/38
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MANU E T MENTEFractional Fokker-Planck EquationsBiased CTRWs lead to fractional Fokker-Planck equations.
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.
Fractional Diffusion – Theory and Applications – Part III – p . 34/38
SCIENTIA
MANU E T MENTEFractional Fokker-Planck EquationsBiased CTRWs lead to fractional Fokker-Planck equations.
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.
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
Fractional Diffusion – Theory and Applications – Part III – p . 34/38
SCIENTIA
MANU E T MENTEFractional Fokker-Planck EquationsBiased CTRWs lead to fractional Fokker-Planck equations.
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.
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?
Fractional Diffusion – Theory and Applications – Part III – p . 34/38
SCIENTIA
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.
Fractional Diffusion – Theory and Applications – Part III – p . 35/38
SCIENTIA
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.
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).
Fractional Diffusion – Theory and Applications – Part III – p . 35/38
SCIENTIA
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.
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).
Turing pattern formation in fractional reaction diffusion
systems (Henry, Langlands, Wearne, 2005, 2008)
Fractional Diffusion – Theory and Applications – Part III – p . 35/38
SCIENTIA
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.
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).
Turing pattern formation in fractional reaction diffusion
systems (Henry, Langlands, Wearne, 2005, 2008)
Problem: Should reactions modify diffusions in CTRWs?
Fractional Diffusion – Theory and Applications – Part III – p . 35/38
SCIENTIA
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
Fractional Diffusion – Theory and Applications – Part III – p . 36/38
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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
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′
Fractional Diffusion – Theory and Applications – Part III – p . 36/38
SCIENTIA
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
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′
Fractional Reaction-Subdiffusion Equation (HLW2006)
∂n
∂t= D e±kt 0D1−α
t
(
e∓kt∂2n
∂x2
)
± kn.
Fractional Diffusion – Theory and Applications – Part III – p . 36/38
SCIENTIA
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
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′
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.
Fractional Diffusion – Theory and Applications – Part III – p . 36/38
SCIENTIA
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)
Fractional Diffusion – Theory and Applications – Part III – p . 37/38
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MANU E T MENTE
Thank You
Fractional Diffusion – Theory and Applications – Part III – p . 38/38