Download Fractional Diffusion – Theory and Applications – Part I

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Fractional Diffusion – Theory
and Applications – Part I
22nd Canberra International Physics Summer School 2008
Bruce Henry
(Trevor Langlands, Peter Straka, Susan Wearne, Claire Delides)
School of Mathematics and Statistics
The University of New South Wales
Sydney NSW 2052 Australia
Fractional Diffusion – Theory and Applications – Part I – p. 1/3
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The mathematical description of diffusion has many formulations:
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conservation of mass and constitutive laws
probabilistic models based on random walks and central limit
theorems
microscopic stochastic models based on Brownian motion and
Langevin equations
mesoscopic stochastic models based on master equations
and Fokker-Planck equations
Fractional Diffusion – Theory and Applications – Part I – p. 2/3
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The mathematical description of diffusion has many formulations:
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conservation of mass and constitutive laws
probabilistic models based on random walks and central limit
theorems
microscopic stochastic models based on Brownian motion and
Langevin equations
mesoscopic stochastic models based on master equations
and Fokker-Planck equations
The mean square displacement h(X − hXi)2 i of a diffusing
particle scales linearly with time
Fractional Diffusion – Theory and Applications – Part I – p. 2/3
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Numerous experimental measurements in spatially complex
systems have revealed anomalous diffusion in which
The mean square displacement scales as a fractional order
power law in time
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Fractional Diffusion – Theory and Applications – Part I – p. 3/3
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Numerous experimental measurements in spatially complex
systems have revealed anomalous diffusion in which
The mean square displacement scales as a fractional order
power law in time
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Over the past two decades a new mathematical description has
been formulated, linked together by tools of fractional calculus:
fractional constitutive laws
probabilistic models based on continuous time random walks
and generalized central limit theorems
fractional Langevin equations, fractional Brownian motions
fractional diffusion, fractional Fokker-Planck equations
Fractional Diffusion – Theory and Applications – Part I – p. 3/3
A Triptych Overview
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Fractional Diffusion – Theory and Applications – Part I – p. 4/3
A Triptych Overview
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Brownian motion
standard random walks
standard diffusion
Fractional Diffusion – Theory and Applications – Part I – p. 4/3
A Triptych Overview
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Brownian motion
standard random walks
standard diffusion
fractional calculus
Fractional Diffusion – Theory and Applications – Part I – p. 4/3
A Triptych Overview
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Brownian motion
standard random walks
standard diffusion
fractional calculus
fractional Brownian motion
continuous time random walks
fractional diffusion
Fractional Diffusion – Theory and Applications – Part I – p. 4/3
Mathematical Models for Diffusion
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Having found motion in the particles
of the pollen of all the living plants
which I had examined, I was led
next to inquire whether this property
continued after the death of the
plant, and for what length of time it
was retained.
Robert Brown (1828)
Fractional Diffusion – Theory and Applications – Part I – p. 5/3
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Einstein’s Relations (1905)
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It is possible that the movements to be discussed here are
identical with the so-called “Brownian molecular motion” ;
however, the information available to me regarding the latter is
so lacking in precision, that I can form no judgment in the
matter
Fractional Diffusion – Theory and Applications – Part I – p. 6/3
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Einstein’s Relations (1905)
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It is possible that the movements to be discussed here are
identical with the so-called “Brownian molecular motion” ;
however, the information available to me regarding the latter is
so lacking in precision, that I can form no judgment in the
matter
The motion of the suspended particle arises as a
consequence of random buffeting from the thermal motions of
the enormous numbers of molecules that comprise the fluid
Fractional Diffusion – Theory and Applications – Part I – p. 6/3
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Einstein’s Relations (1905)
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It is possible that the movements to be discussed here are
identical with the so-called “Brownian molecular motion” ;
however, the information available to me regarding the latter is
so lacking in precision, that I can form no judgment in the
matter
The motion of the suspended particle arises as a
consequence of random buffeting from the thermal motions of
the enormous numbers of molecules that comprise the fluid
kB T
=
hr2 (t)i = 2Dt D = 6NRT
Einstein Relations
πaη
γ
Fractional Diffusion – Theory and Applications – Part I – p. 6/3
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Einstein’s Relations (1905)
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It is possible that the movements to be discussed here are
identical with the so-called “Brownian molecular motion” ;
however, the information available to me regarding the latter is
so lacking in precision, that I can form no judgment in the
matter
The motion of the suspended particle arises as a
consequence of random buffeting from the thermal motions of
the enormous numbers of molecules that comprise the fluid
kB T
=
hr2 (t)i = 2Dt D = 6NRT
Einstein Relations
πaη
γ
γ = 6πηa (Stokes), T temperature of fluid, R = N kB gas constant,
a particle radius, η fluid viscosity, N Avogadro’s number
Fractional Diffusion – Theory and Applications – Part I – p. 6/3
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Langevin’s Equation (1908)
d2 x
m 2 =
dt
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F (t)
|{z}
random force
−
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dx
γ
dt
|{z}
viscous drag
Fractional Diffusion – Theory and Applications – Part I – p. 7/3
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Langevin’s Equation (1908)
d2 x
m 2 =
dt
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F (t)
|{z}
random force
hF (t)i = 0,
hF (0)F (t)i = Dδ(t)
−
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dx
γ
dt
|{z}
viscous drag
hx(t)F (t)i = hx(t)ihF (t)i = 0
Fractional Diffusion – Theory and Applications – Part I – p. 7/3
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Langevin’s Equation (1908)
d2 x
m 2 =
dt
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F (t)
|{z}
random force
−
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dx
γ
dt
|{z}
viscous drag
hF (t)i = 0, hF (0)F (t)i = Dδ(t) hx(t)F (t)i = hx(t)ihF (t)i = 0
* +
1
dx 2
1
m
= kB T Equipartition of energy (1D) – Boltzmann (1896)
2
dt
2
Fractional Diffusion – Theory and Applications – Part I – p. 7/3
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Langevin’s Equation (1908)
d2 x
m 2 =
dt
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F (t)
|{z}
dx
γ
dt
|{z}
−
random force
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viscous drag
hF (t)i = 0, hF (0)F (t)i = Dδ(t) hx(t)F (t)i = hx(t)ihF (t)i = 0
* +
1
dx 2
1
m
= kB T Equipartition of energy (1D) – Boltzmann (1896)
2
dt
2
d 2
2kB T γ hx i =
1 − exp(− t)
dt
γ
m
⇒
|{z}
“ ”
t≫
m
γ
d 2
2kB T
hx i ≈
dt
γ
≈10−8
Fractional Diffusion – Theory and Applications – Part I – p. 7/3
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Langevin’s Equation (1908)
d2 x
m 2 =
dt
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F (t)
|{z}
dx
γ
dt
|{z}
−
random force
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viscous drag
hF (t)i = 0, hF (0)F (t)i = Dδ(t) hx(t)F (t)i = hx(t)ihF (t)i = 0
* +
1
dx 2
1
m
= kB T Equipartition of energy (1D) – Boltzmann (1896)
2
dt
2
d 2
2kB T γ hx i =
1 − exp(− t)
dt
γ
m
hx2 i ∼ 2Dt
⇒
|{z}
“ ”
t≫
m
γ
d 2
2kB T
hx i ≈
dt
γ
≈10−8
Fractional Diffusion – Theory and Applications – Part I – p. 7/3
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Random Walks
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A man starts from a point O and walks ℓ yards in a straight
line; he then turns through any angle whatever and walks
another ℓ yards in a second straight line. He repeats this
process n times. I require the probability that after these n
stretches he is at a distance between r and r + δr from his
starting point, O.
Karl Pearson (1905)
Fractional Diffusion – Theory and Applications – Part I – p. 8/3
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Random Walks
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A man starts from a point O and walks ℓ yards in a straight
line; he then turns through any angle whatever and walks
another ℓ yards in a second straight line. He repeats this
process n times. I require the probability that after these n
stretches he is at a distance between r and r + δr from his
starting point, O.
Karl Pearson (1905)
If n be very great, the probability sought is
2 −r2 /n
e
r dr
n
Lord Rayleigh (1905)
Fractional Diffusion – Theory and Applications – Part I – p. 8/3
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The lesson of Lord Rayleigh’s solution is that in open country
the most probable place to find a drunken man who is at all
capable of keeping on his feet is somewhere near his starting
point!
Karl Pearson (1905)
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Fractional Diffusion – Theory and Applications – Part I – p. 9/3
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The lesson of Lord Rayleigh’s solution is that in open country
the most probable place to find a drunken man who is at all
capable of keeping on his feet is somewhere near his starting
point!
Karl Pearson (1905)
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... the consideration of true prices permits the statement of the
fundamental principle – The mathematical expectation of the
speculator is zero.
Louis Bachelier (1900)
Fractional Diffusion – Theory and Applications – Part I – p. 9/3
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Random Walks and the Binomial Distribution
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A particle starts from an origin and at each time step ∆t the
particle has an equal probability of jumping an equal distance
∆x to the left or the right. What is the probability Pm,n that the
particle will be at position x = m∆x at time t = n∆t?
Fractional Diffusion – Theory and Applications – Part I – p. 10/3
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Random Walks and the Binomial Distribution
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A particle starts from an origin and at each time step ∆t the
particle has an equal probability of jumping an equal distance
∆x to the left or the right. What is the probability Pm,n that the
particle will be at position x = m∆x at time t = n∆t?
1/2
m−1
m
m+1
1/2
Fractional Diffusion – Theory and Applications – Part I – p. 10/3
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Random Walks and the Binomial Distribution
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A particle starts from an origin and at each time step ∆t the
particle has an equal probability of jumping an equal distance
∆x to the left or the right. What is the probability Pm,n that the
particle will be at position x = m∆x at time t = n∆t?
1/2
m−1
m
m+1
1/2
1
1
Pm,n = Pm−1,n−1 + Pm+1,n−1 ,
2
2
k
X
P0,0 = 1,
Pj,k = 1
k = 0, 1, 2, . . . n
j=−k
Fractional Diffusion – Theory and Applications – Part I – p. 10/3
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Enumeration
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Suppose an n step walk from 0 to m has k steps to the right
and n − k = k − m steps to the left.
Fractional Diffusion – Theory and Applications – Part I – p. 11/3
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Enumeration
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Suppose an n step walk from 0 to m has k steps to the right
and n − k = k − m steps to the left.


n
n!


There are C(n, k) =
=
ways of distributing
k!(n − k)!
k
these k steps among the n steps.
Fractional Diffusion – Theory and Applications – Part I – p. 11/3
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Enumeration
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Suppose an n step walk from 0 to m has k steps to the right
and n − k = k − m steps to the left.


n
n!


There are C(n, k) =
=
ways of distributing
k!(n − k)!
k
these k steps among the n steps.
There are 2n possible paths in an n step walk.
Fractional Diffusion – Theory and Applications – Part I – p. 11/3
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Enumeration
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Suppose an n step walk from 0 to m has k steps to the right
and n − k = k − m steps to the left.


n
n!


There are C(n, k) =
=
ways of distributing
k!(n − k)!
k
these k steps among the n steps.
There are 2n possible paths in an n step walk.
C(n, k)
p(m(k), n) =
2n
Fractional Diffusion – Theory and Applications – Part I – p. 11/3
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Enumeration
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Suppose an n step walk from 0 to m has k steps to the right
and n − k = k − m steps to the left.


n
n!


There are C(n, k) =
=
ways of distributing
k!(n − k)!
k
these k steps among the n steps.
There are 2n possible paths in an n step walk.
C(n, k)
p(m(k), n) =
2n
But
n+m
k=
2
⇒
p(m, n) =
2n
n!
n+m
!
2
n−m
2
!
Fractional Diffusion – Theory and Applications – Part I – p. 11/3
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Random Walks and the Normal Distribution
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In n step walks from 0 to m the av. # steps to the right is
n
hki = .
2
n+m n
m
− =
How are fluctuations X = k − hki =
2
2
2
distributed?
Fractional Diffusion – Theory and Applications – Part I – p. 12/3
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Random Walks and the Normal Distribution
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In n step walks from 0 to m the av. # steps to the right is
n
hki = .
2
n+m n
m
− =
How are fluctuations X = k − hki =
2
2
2
distributed?
n!
n!
p(m(X), n) = n n+m n−m = n
n
n
(
−
X)!(
+
X)!2
2
!
!
2
2
2
2
Fractional Diffusion – Theory and Applications – Part I – p. 12/3
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Random Walks and the Normal Distribution
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In n step walks from 0 to m the av. # steps to the right is
n
hki = .
2
n+m n
m
− =
How are fluctuations X = k − hki =
2
2
2
distributed?
n!
n!
p(m(X), n) = n n+m n−m = n
n
n
(
−
X)!(
+
X)!2
2
!
!
2
2
2
2
√
n −n
n!
≈
2πnn
e }⇒
|
{z
q
Stirling’s approx.
2
nπ
P (X, n) =
1−
1
n
2X ( 2 −X+ 2 )
n
1+
1
n
2X ( 2 +X+ 2 )
n
Fractional Diffusion – Theory and Applications – Part I – p. 12/3
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Re-write
P (X, n) =
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exp
( n2
−X +
1
2 ) ln(1
−
q
2
nπ
2X
n )
+
( n2
+X +
1
2 ) ln(1
Carry out a series expansion of the log terms in powers of
P (X, n) ∼
r
2 − 2X 2
e n =
nπ
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r
+
2X
n )
2X
n
2 − m2
e 2n
nπ
Recall m is the final position of the walker.
The probability density function for unbiased standard
random walks in the long time limit is the Gaussian or
normal distribution.
Fractional Diffusion – Theory and Applications – Part I – p. 13/3
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Random Walks in the Continuum Approximation
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Write P (m, n) = P (x, t),
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x = m∆x, t = n∆t
1
1
P (x, t) = P (x − ∆x, t − ∆t) + P (x + ∆x, t − ∆t)
2
2
Fractional Diffusion – Theory and Applications – Part I – p. 14/3
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Random Walks in the Continuum Approximation
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Write P (m, n) = P (x, t),
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x = m∆x, t = n∆t
1
1
P (x, t) = P (x − ∆x, t − ∆t) + P (x + ∆x, t − ∆t)
2
2
Taylor expansions
P (x ± ∆x, t − ∆t)
≈
∂P
(∆x)2 ∂ 2 P
(∆t)2 ∂ 2 P
∂P
− ∆t
+
+
P (x, t) ± ∆x
∂x
∂t
2
∂x2
2
∂t2
∂2P
∓∆t∆x
+ O((∆t)3 ) + O((∆x)3 ),
∂x∂t
Fractional Diffusion – Theory and Applications – Part I – p. 14/3
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Random Walks in the Continuum Approximation
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Write P (m, n) = P (x, t),
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x = m∆x, t = n∆t
1
1
P (x, t) = P (x − ∆x, t − ∆t) + P (x + ∆x, t − ∆t)
2
2
Taylor expansions
P (x ± ∆x, t − ∆t)
≈
∂P
(∆x)2 ∂ 2 P
(∆t)2 ∂ 2 P
∂P
− ∆t
+
+
P (x, t) ± ∆x
∂x
∂t
2
∂x2
2
∂t2
∂2P
∓∆t∆x
+ O((∆t)3 ) + O((∆x)3 ),
∂x∂t
Retain leading order terms in ∆t and ∆x then
∂P
∂2P
=D 2,
∂t
∂x
(∆x)2
= constant
D=
lim
∆t→0,∆x→0 2∆t
Fractional Diffusion – Theory and Applications – Part I – p. 14/3
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Fundamental Solution and Mean Square Displacement
Green’s solution G(x, t), G(x, 0) = δ(x).
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Fractional Diffusion – Theory and Applications – Part I – p. 15/3
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Fundamental Solution and Mean Square Displacement
Green’s solution G(x, t), G(x, 0) = δ(x).
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dĜ(q, t)
= −Dq 2 Ĝ(q, t), Ĝ(q, 0) = δ̂(q) = 1
Fourier transform
dt
Fractional Diffusion – Theory and Applications – Part I – p. 15/3
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Fundamental Solution and Mean Square Displacement
Green’s solution G(x, t), G(x, 0) = δ(x).
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dĜ(q, t)
= −Dq 2 Ĝ(q, t), Ĝ(q, 0) = δ̂(q) = 1
Fourier transform
dt
Fourier solution
Ĝ(q, t) = e
−Dq 2 t
Fractional Diffusion – Theory and Applications – Part I – p. 15/3
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Fundamental Solution and Mean Square Displacement
Green’s solution G(x, t), G(x, 0) = δ(x).
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dĜ(q, t)
= −Dq 2 Ĝ(q, t), Ĝ(q, 0) = δ̂(q) = 1
Fourier transform
dt
Fourier solution
Ĝ(q, t) = e
−Dq 2 t
Inverse Fourier transform
1
G(x, t) =
2π
Z
+∞
−Dq 2 t + iqx
|
{z
}
e complete the square
−∞
dq
x2
1
− 4Dt
e
=√
4πDt
Fractional Diffusion – Theory and Applications – Part I – p. 15/3
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Fundamental Solution and Mean Square Displacement
Green’s solution G(x, t), G(x, 0) = δ(x).
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dĜ(q, t)
= −Dq 2 Ĝ(q, t), Ĝ(q, 0) = δ̂(q) = 1
Fourier transform
dt
Fourier solution
Ĝ(q, t) = e
−Dq 2 t
Inverse Fourier transform
1
G(x, t) =
2π
Z
+∞
−Dq 2 t + iqx
|
{z
}
e complete the square
−∞
dq
x2
1
− 4Dt
e
=√
4πDt
Mean square displacement
Z +∞
d2
2
2
2
hx i =
x G(x, t) dx or hx i = lim − 2 Ĝ(q, t) = 2Dt.
q→0 dq
−∞
Fractional Diffusion – Theory and Applications – Part I – p. 15/3
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Central Limit Theorem
Each step of a random walk is a random variable ∆xi . The
mean of ∆xi after N steps is the position x divided by N .
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Fractional Diffusion – Theory and Applications – Part I – p. 16/3
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Central Limit Theorem
Each step of a random walk is a random variable ∆xi . The
mean of ∆xi after N steps is the position x divided by N .
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If we sample (infinitely many) random walks of length N
(sufficiently large) then the sampling distribution of random
variables X = Nx should approach a normal distribution
µ)2
1
(x −
√
P (X ∈ dx) =
exp −
2
2σ 2
2πσ
with µ = hXi equal to the mean of the sampled set and
variance σ 2 = hX 2 i − hXi2 equal to the variance of the
sampled set divided by N .
Fractional Diffusion – Theory and Applications – Part I – p. 16/3
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Central Limit Theorem
Each step of a random walk is a random variable ∆xi . The
mean of ∆xi after N steps is the position x divided by N .
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If we sample (infinitely many) random walks of length N
(sufficiently large) then the sampling distribution of random
variables X = Nx should approach a normal distribution
µ)2
1
(x −
√
P (X ∈ dx) =
exp −
2
2σ 2
2πσ
with µ = hXi equal to the mean of the sampled set and
variance σ 2 = hX 2 i − hXi2 equal to the variance of the
sampled set divided by N .
Consider µ = 0 and σ 2 = 2Dt.
Fractional Diffusion – Theory and Applications – Part I – p. 16/3
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Diffusion Equation
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If there are N non-interacting walkers then they all have the
same probability P (x, t) of being at x at time t and hence the
number per unit volume (concentration) at x at time t is
c(x, t) = N P (x, t)
∂2c
∂c
=D 2
∂t
∂x
Fractional Diffusion – Theory and Applications – Part I – p. 17/3
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Diffusion Equation
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If there are N non-interacting walkers then they all have the
same probability P (x, t) of being at x at time t and hence the
number per unit volume (concentration) at x at time t is
c(x, t) = N P (x, t)
∂2c
∂c
=D 2
∂t
∂x
The diffusion equation can be derived from conservation of
mass and Fick’s Law.
∂c
∂q
=−
|∂t {z ∂x}
conservation of mass
∂c
q=−
| {z ∂x}
Fick’s law
Fractional Diffusion – Theory and Applications – Part I – p. 17/3
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Fick’s Law (1855)
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Define q(x, t) as the number of particles per unit time that
pass through a test area perpendicular to the flow in the
positive x direction.
Fractional Diffusion – Theory and Applications – Part I – p. 18/3
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Fick’s Law (1855)
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Define q(x, t) as the number of particles per unit time that
pass through a test area perpendicular to the flow in the
positive x direction.
The net flow of diffusing particles is from regions of high
concentration to regions of low concentration and the
magnitude of this flow is proportional to the concentration
gradient.
∂c
q(x, t) = −D
| {z }
∂x
flux
Fractional Diffusion – Theory and Applications – Part I – p. 18/3
SCIENTIA
Conservation of Mass
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A
x
0
1
0
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x+ δx
1 0
1
net number of particles
number of particles
number of particles
C
C B
C B
B
B in volume V at time C = B in volume V at time C+B entering volume V between C
A
A @
A @
@
times t, t + δt
t
t + δt
(c(x, t + δt)Aδx) = (c(x, t)Aδx) + (q(x, t)Aδt − q(x + δx, t)Aδt)
c(x, t + δt) − c(x, t)
q(x + δx) − q(x, t)
⇒
=−
δt
δx
∂c
∂q
⇒
=−
∂t
∂x
Fractional Diffusion – Theory and Applications – Part I – p. 19/3
SCIENTIA
Generalizations
If the media is spatially heterogeneous then replace the diffusion
constant in Fick’s law with a space dependent function.
∂
∂c
=
∂t
∂x
∂c
D(x)
∂x
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diffusion 1D
Fractional Diffusion – Theory and Applications – Part I – p. 20/3
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Generalizations
If the media is spatially heterogeneous then replace the diffusion
constant in Fick’s law with a space dependent function.
∂
∂c
=
∂t
∂x
∂
∂c
=
∂t
∂x
∂c
D(x)
∂x
∂c
D(x)
∂x
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diffusion 1D
∂
−
(v(x) c)
∂x
advection-diffusion 1 dim
Fractional Diffusion – Theory and Applications – Part I – p. 20/3
SCIENTIA
Generalizations
If the media is spatially heterogeneous then replace the diffusion
constant in Fick’s law with a space dependent function.
∂
∂c
=
∂t
∂x
∂
∂c
=
∂t
∂x
∂c
D(x)
∂x
∂c
D(x)
∂x
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diffusion 1D
∂
−
(v(x) c)
∂x
∂
∂c
∂
∂u
∂c
=
D(x)
−χ
c
∂t
∂x
∂x
∂x
∂x
advection-diffusion 1 dim
cemotactic-diffusion 1 dim
Fractional Diffusion – Theory and Applications – Part I – p. 20/3
SCIENTIA
Generalizations
If the media is spatially heterogeneous then replace the diffusion
constant in Fick’s law with a space dependent function.
∂
∂c
=
∂t
∂x
∂
∂c
=
∂t
∂x
∂c
D(x)
∂x
∂c
D(x)
∂x
∂
−
(v(x) c)
∂x
1 ∂
∂c
= d−1
∂t
r
∂r
r
d−1
∂c
D(r)
∂r
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diffusion 1D
∂
∂c
∂
∂u
∂c
=
D(x)
−χ
c
∂t
∂x
∂x
∂x
∂x
MA
advection-diffusion 1 dim
cemotactic-diffusion 1 dim
radially symmetric d dim
Fractional Diffusion – Theory and Applications – Part I – p. 20/3
SCIENTIA
Master Equations and the Fokker-Planck Equation
P (x, t) =
Z
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+∞
−∞
λ(∆x)P (x − ∆x, t − ∆t) d∆x
P (x, t) probability density for walker to be at x at time t
λ(∆x) probability density function for a jump of length ∆x
∆t discrete time between jumps
For spatially heterogeneous media generalize λ = λ(∆x, x) the
step length density varies with position
Z +∞
P (x, t) =
λ(∆x, x − ∆x)P (x − ∆x, t − ∆t) d∆x
−∞
Fractional Diffusion – Theory and Applications – Part I – p. 21/3
SCIENTIA
Continuum limit ∆t → 0 and ∆x → 0
P |(x,t−∆t)
˛
∂P ˛˛
+ ∆t
∂t ˛(x,t−∆t)
Z
Z
Z
≈
×
+∞
MA
NU
+∞
˛
∆x2
∂λ ˛˛
+
λ|(∆x,x) − ∆x
˛
∂x
2
−∞
(∆x,x)
˛
∂P ˛˛
(∆x)2
P |(x,t−∆t) − ∆x
+
∂x ˛(x,t−∆t)
2
Z
λ(∆x, x) d∆x
=
1,
∆x λ(∆x, x) d∆x
=
h∆x(x)i,
∆x2 λ(∆x, x) d∆x
=
h∆x2 (x)i.
∂2λ ˛
˛
˛
2
∂x ˛
(∆x,x)
NTE
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!
!
˛
˛
˛
2
∂x ˛(x,t−∆t)
∂2P
−∞
+∞
−∞
+∞
−∞
Retain leading order terms to derive the general Fokker-Planck
equation
Fractional Diffusion – Theory and Applications – Part I – p. 22/3
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General Fokker-Planck Equation
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∂2
∂
∂P
=
(D(x)P (x, t)) −
(v(x)P (x, t))
∂t
∂x2
∂x
h∆x(x)i
v(x) =
drift
∆t
h∆x2 (x)i
D(x) =
diffusivity
2∆t
Fractional Diffusion – Theory and Applications – Part I – p. 23/3
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General Fokker-Planck Equation
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∂2
∂
∂P
=
(D(x)P (x, t)) −
(v(x)P (x, t))
∂t
∂x2
∂x
h∆x(x)i
v(x) =
drift
∆t
h∆x2 (x)i
D(x) =
diffusivity
2∆t
Compare the generalized Fickian diffusion equation
∂
∂P (x, t)
∂
∂P
=
D(x)
−
(v(x)P (x, t))
∂t
∂x
∂x
∂x
Fractional Diffusion – Theory and Applications – Part I – p. 23/3
SCIENTIA
The Chapman-Kolmogorov Equation, Markov Processes
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In the limit where the time increment approaches zero the
sequence of jumps {Xt } in a random walk defines a
stochastic process. A realization defines a trajectory x(t).
Fractional Diffusion – Theory and Applications – Part I – p. 24/3
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The Chapman-Kolmogorov Equation, Markov Processes
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In the limit where the time increment approaches zero the
sequence of jumps {Xt } in a random walk defines a
stochastic process. A realization defines a trajectory x(t).
Markov property if at any t the distribution of all Xu , u > t only
depends on Xt and not on any Xs , s < t.
Fractional Diffusion – Theory and Applications – Part I – p. 24/3
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The Chapman-Kolmogorov Equation, Markov Processes
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In the limit where the time increment approaches zero the
sequence of jumps {Xt } in a random walk defines a
stochastic process. A realization defines a trajectory x(t).
Markov property if at any t the distribution of all Xu , u > t only
depends on Xt and not on any Xs , s < t.
If Markov then the conditional probability q(x, t|x′ , t′ ) that Xt
lies in x, x + dx given that Xt′ lies in x′ , x′ + dx′ satisfies
Z
q(x, t|x′′ , t′′ ) = q(x, t|x′ , t′ )q(x′ , t′ |x′′ , t′′ ) dx′ Bachelier (1990)
Note the times t > t′ > t′′ are discrete.
Fractional Diffusion – Theory and Applications – Part I – p. 24/3
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Wiener Process, Brownian Motion
MA
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One solution of Bachelier’s equation is
(x−x′ )2
1
−
e 2(t−t′ ) , t > t′
q(x, t|x′ , t′ ) = p
2π(t − t′ )
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Wiener process
Fractional Diffusion – Theory and Applications – Part I – p. 25/3
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Wiener Process, Brownian Motion
MA
NU
One solution of Bachelier’s equation is
(x−x′ )2
1
−
e 2(t−t′ ) , t > t′
q(x, t|x′ , t′ ) = p
2π(t − t′ )
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Wiener process
Limiting behaviour of a random walk in the limit where the time
increment approaches zero (Wiener, 1923) – also referred to
as Brownian motion Bt .
Fractional Diffusion – Theory and Applications – Part I – p. 25/3
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Wiener Process, Brownian Motion
MA
NU
One solution of Bachelier’s equation is
(x−x′ )2
1
−
e 2(t−t′ ) , t > t′
q(x, t|x′ , t′ ) = p
2π(t − t′ )
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Wiener process
Limiting behaviour of a random walk in the limit where the time
increment approaches zero (Wiener, 1923) – also referred to
as Brownian motion Bt .
(i) Realizations xB (t) of Bt are continuous but nowhere
differentiable – xB (t) versus t is a fractal graph with dimension
d = 3/2. (ii) The increments Bt − Bt′ are normally distributed
with mean 0 and variance t − t′ for t > t′ . (iii) The increments
Bt − Bt′ and Bs − Bs′ are independent for t > t′ ≥ s ≥ s′ ≥ 0.
Fractional Diffusion – Theory and Applications – Part I – p. 25/3
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Standard diffusion
MA
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The mean square displacement scales linearly with time and
the probability density function is the Gaussian normal
distribution
Fractional Diffusion – Theory and Applications – Part I – p. 26/3
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Standard diffusion
MA
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The mean square displacement scales linearly with time and
the probability density function is the Gaussian normal
distribution
Theoretical support from random walks, central limit theorem,
Langevin equation, diffusion equation, master equations,
Weiner processes.
Fractional Diffusion – Theory and Applications – Part I – p. 26/3
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Standard diffusion
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The mean square displacement scales linearly with time and
the probability density function is the Gaussian normal
distribution
Theoretical support from random walks, central limit theorem,
Langevin equation, diffusion equation, master equations,
Weiner processes.
Experimental support widespread, e.g., Perrin (1909)
measured mean square displacements and used Einstein’s
relations to determine Avogadro’s number, the constant
number of molecules in any mole of substance, thus
consolidating the atomistic description of nature.
Fractional Diffusion – Theory and Applications – Part I – p. 26/3
SCIENTIA
In the theory of Brownian motion the first concern has always
been the calculation of the mean square displacement of the
particle, because this could be immediately observed.
George Uhlenbeck and Leonard Ornstein (1930)
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Fractional Diffusion – Theory and Applications – Part I – p. 27/3
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In the theory of Brownian motion the first concern has always
been the calculation of the mean square displacement of the
particle, because this could be immediately observed.
George Uhlenbeck and Leonard Ornstein (1930)
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NU
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In the atmosphere a spreading dot will not serve as an
element from which general distributions can be built up.
Lewis Richardson (1926)
Fractional Diffusion – Theory and Applications – Part I – p. 27/3
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Anomalous Diffusion
MA
NU
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There have been numerous experimental measurements of
anomalous diffusion – the mean square displacement
does not scale linearly with time.
D
E
h∆X 2 (t)i = (X(t) − hX(t)i)2 ≁ t
Fractional Diffusion – Theory and Applications – Part I – p. 28/3
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Anomalous Diffusion
MA
NU
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There have been numerous experimental measurements of
anomalous diffusion – the mean square displacement
does not scale linearly with time.
D
E
h∆X 2 (t)i = (X(t) − hX(t)i)2 ≁ t
Anomalous diffusion is ‘normal’ in spatially disordered
systems, porous media, fractal media, turbulent fluids and
plasmas, biological media with traps, binding sites or
macro-molecular crowding, stock price movements.
Fractional Diffusion – Theory and Applications – Part I – p. 28/3
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Anomalous Scaling
h∆X 2 i ∼ t (ln t)κ
MA
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ultraslow diffusion
1<κ<4
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Sinai diffusion
deterministic diffusion
h∆X 2 i ∼ tα
subdiffusion
disordered solids
0<α<1
biological media
fractal media
porous media
8
< tα
2
h∆X i ∼
: t
t<τ
t>τ
h∆X 2 i ∼ t
h∆X 2 i ∼ tβ
1<β<2
transient subdiffusion
biological media
standard diffusion
homogeneous media
superdiffusion
turbulent plasmas
Levy flights
transport in polymers
h∆ℓ2 i ∼ t3
Richardson diffusion
atmospheric turbulence
Fractional Diffusion – Theory and Applications – Part I – p. 29/3
SCIENTIA
Fractional Diffusion
MA
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Over the past decade a new theoretical framework has been
developed to model anomalous diffusion. The new framework
is based around the physics of continuous time random walks
and the mathematics of fractional calculus.
One can ask what would be a differential
having as its exponent a fraction. Although
this seems removed from Geometry . . . it appears that one day these paradoxes will yield
useful consequences.
Gottfried Leibniz (1695)
Fractional Diffusion – Theory and Applications – Part I – p. 30/3