Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion...
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Transcript of Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion...
![Page 1: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/1.jpg)
Lecture 10: Anomalous diffusion
Outline:• generalized diffusion equation• subdiffusion• superdiffusion• fractional Wiener process
![Page 2: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/2.jpg)
anomalous diffusionRecall derivation of Fokker-Planck equation:
![Page 3: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/3.jpg)
anomalous diffusionRecall derivation of Fokker-Planck equation:
€
∂P(x, t)
∂t= ds r(x − s;s)P(x − s, t) − r(x;−s)P(x, t)[ ]∫
= −∂
∂xsr(x,s)ds∫( )P(x, t)[ ] +
∂ 2
∂x 212 s2r(x,s)ds∫( )P(x, t)[ ] +L
= −∂
∂xr1(x)P(x, t)( ) +
1
2
∂ 2
∂x 2r2(x)P(x, t)( ) +L
![Page 4: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/4.jpg)
anomalous diffusionRecall derivation of Fokker-Planck equation:
€
∂P(x, t)
∂t= ds r(x − s;s)P(x − s, t) − r(x;−s)P(x, t)[ ]∫
= −∂
∂xsr(x,s)ds∫( )P(x, t)[ ] +
∂ 2
∂x 212 s2r(x,s)ds∫( )P(x, t)[ ] +L
= −∂
∂xr1(x)P(x, t)( ) +
1
2
∂ 2
∂x 2r2(x)P(x, t)( ) +L
But what if ?
€
s2r(x,s)ds∫ = ∞
![Page 5: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/5.jpg)
anomalous diffusionRecall derivation of Fokker-Planck equation:
€
∂P(x, t)
∂t= ds r(x − s;s)P(x − s, t) − r(x;−s)P(x, t)[ ]∫
= −∂
∂xsr(x,s)ds∫( )P(x, t)[ ] +
∂ 2
∂x 212 s2r(x,s)ds∫( )P(x, t)[ ] +L
= −∂
∂xr1(x)P(x, t)( ) +
1
2
∂ 2
∂x 2r2(x)P(x, t)( ) +L
But what if ?
And what if the distribution of time steps has infinite mean?
€
s2r(x,s)ds∫ = ∞
![Page 6: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/6.jpg)
anomalous diffusionRecall derivation of Fokker-Planck equation:
€
∂P(x, t)
∂t= ds r(x − s;s)P(x − s, t) − r(x;−s)P(x, t)[ ]∫
= −∂
∂xsr(x,s)ds∫( )P(x, t)[ ] +
∂ 2
∂x 212 s2r(x,s)ds∫( )P(x, t)[ ] +L
= −∂
∂xr1(x)P(x, t)( ) +
1
2
∂ 2
∂x 2r2(x)P(x, t)( ) +L
But what if ?
And what if the distribution of time steps has infinite mean?
Go back and reformulate the problem: €
s2r(x,s)ds∫ = ∞
![Page 7: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/7.jpg)
continuous-time random walkdistribution of jump sizes r(x), distribution of waiting times w(t)
![Page 8: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/8.jpg)
continuous-time random walkdistribution of jump sizes r(x), distribution of waiting times w(t)
can have joint distribution ψ(x,t); here, ψ(x,t) = r(x)w(t)
![Page 9: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/9.jpg)
continuous-time random walkdistribution of jump sizes r(x), distribution of waiting times w(t)
can have joint distribution ψ(x,t); here, ψ(x,t) = r(x)w(t)
Let η(x,t) = probability density of x at a t right after a jump
![Page 10: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/10.jpg)
continuous-time random walkdistribution of jump sizes r(x), distribution of waiting times w(t)
can have joint distribution ψ(x,t); here, ψ(x,t) = r(x)w(t)
Let η(x,t) = probability density of x at a t right after a jump
€
η(x, t) = d ′ x d ′ t ψ (x − ′ x , t − ′ t )η ( ′ x , ′ t ) + δ(x)δ(t)∫∫
![Page 11: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/11.jpg)
continuous-time random walkdistribution of jump sizes r(x), distribution of waiting times w(t)
can have joint distribution ψ(x,t); here, ψ(x,t) = r(x)w(t)
Let η(x,t) = probability density of x at a t right after a jump
€
η(x, t) = d ′ x d ′ t ψ (x − ′ x , t − ′ t )η ( ′ x , ′ t ) + δ(x)δ(t)∫∫= d ′ x r(x − ′ x ) d ′ t w(t − ′ t )η ( ′ x , ′ t ) + δ(x)δ(t)∫∫
![Page 12: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/12.jpg)
continuous-time random walkdistribution of jump sizes r(x), distribution of waiting times w(t)
can have joint distribution ψ(x,t); here, ψ(x,t) = r(x)w(t)
Let η(x,t) = probability density of x at a t right after a jump
€
η(x, t) = d ′ x d ′ t ψ (x − ′ x , t − ′ t )η ( ′ x , ′ t ) + δ(x)δ(t)∫∫= d ′ x r(x − ′ x ) d ′ t w(t − ′ t )η ( ′ x , ′ t ) + δ(x)δ(t)∫∫
€
P(x, t) = d ′ t 1− d ′ ′ t w( ′ ′ t )0
t− ′ t
∫[ ]0
t
∫ η ( ′ x , ′ t )
Then
![Page 13: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/13.jpg)
continuous-time random walkdistribution of jump sizes r(x), distribution of waiting times w(t)
can have joint distribution ψ(x,t); here, ψ(x,t) = r(x)w(t)
Let η(x,t) = probability density of x at a t right after a jump
€
η(x, t) = d ′ x d ′ t ψ (x − ′ x , t − ′ t )η ( ′ x , ′ t ) + δ(x)δ(t)∫∫= d ′ x r(x − ′ x ) d ′ t w(t − ′ t )η ( ′ x , ′ t ) + δ(x)δ(t)∫∫
€
P(x, t) = d ′ t 1− d ′ ′ t w( ′ ′ t )0
t− ′ t
∫[ ]0
t
∫ η ( ′ x , ′ t )
Then
______________ prob to survive from t’ to t without a jump
![Page 14: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/14.jpg)
Fourier & Laplace transform:Fourier transform in space, Laplace transform in time:
![Page 15: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/15.jpg)
Fourier & Laplace transform:Fourier transform in space, Laplace transform in time:
€
P(k,s) =1− w(s)
s
1
1− r(k)w(s)
![Page 16: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/16.jpg)
Fourier & Laplace transform:Fourier transform in space, Laplace transform in time:
€
P(k,s) =1− w(s)
s
1
1− r(k)w(s)
The conventional case:
![Page 17: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/17.jpg)
Fourier & Laplace transform:Fourier transform in space, Laplace transform in time:
€
P(k,s) =1− w(s)
s
1
1− r(k)w(s)
The conventional case:
€
w(t) =1
τexp − t τ( ) ⇒ w(s) =
1
1+ sτ≈1− sτ
![Page 18: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/18.jpg)
Fourier & Laplace transform:Fourier transform in space, Laplace transform in time:
€
P(k,s) =1− w(s)
s
1
1− r(k)w(s)
The conventional case:
€
w(t) =1
τexp − t τ( ) ⇒ w(s) =
1
1+ sτ≈1− sτ
r(x) =1
4πξ 2exp −
x 2
ξ 2
⎛
⎝ ⎜
⎞
⎠ ⎟ ⇒ r(k) = exp −k 2ξ 2
( ) ≈1− k 2ξ 2
![Page 19: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/19.jpg)
Fourier & Laplace transform:Fourier transform in space, Laplace transform in time:
€
P(k,s) =1− w(s)
s
1
1− r(k)w(s)
The conventional case:
€
w(t) =1
τexp − t τ( ) ⇒ w(s) =
1
1+ sτ≈1− sτ
r(x) =1
4πξ 2exp −
x 2
ξ 2
⎛
⎝ ⎜
⎞
⎠ ⎟ ⇒ r(k) = exp −k 2ξ 2
( ) ≈1− k 2ξ 2
€
P(k,s) ≈sτ
s
1
1− 1− (kξ )2( ) 1− sτ( )
![Page 20: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/20.jpg)
Fourier & Laplace transform:Fourier transform in space, Laplace transform in time:
€
P(k,s) =1− w(s)
s
1
1− r(k)w(s)
The conventional case:
€
w(t) =1
τexp − t τ( ) ⇒ w(s) =
1
1+ sτ≈1− sτ
r(x) =1
4πξ 2exp −
x 2
ξ 2
⎛
⎝ ⎜
⎞
⎠ ⎟ ⇒ r(k) = exp −k 2ξ 2
( ) ≈1− k 2ξ 2
€
P(k,s) ≈sτ
s
1
1− 1− (kξ )2( ) 1− sτ( )
≈1
s + ξ 2 τ( )k 2
![Page 21: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/21.jpg)
Fourier & Laplace transform:Fourier transform in space, Laplace transform in time:
€
P(k,s) =1− w(s)
s
1
1− r(k)w(s)
The conventional case:
€
w(t) =1
τexp − t τ( ) ⇒ w(s) =
1
1+ sτ≈1− sτ
r(x) =1
4πξ 2exp −
x 2
ξ 2
⎛
⎝ ⎜
⎞
⎠ ⎟ ⇒ r(k) = exp −k 2ξ 2
( ) ≈1− k 2ξ 2
€
P(k,s) ≈sτ
s
1
1− 1− (kξ )2( ) 1− sτ( )
≈1
s + ξ 2 τ( )k 2=
1
s + Dk 2
![Page 22: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/22.jpg)
Fourier-Laplace inversion
2 ways: (D = 1)
![Page 23: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/23.jpg)
Fourier-Laplace inversion
2 ways:1. Invert the Laplace transform first:
(D = 1)
![Page 24: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/24.jpg)
Fourier-Laplace inversion
2 ways:1. Invert the Laplace transform first:
€
P(k,s) =1
s + k 2⇒ P(k, t) = exp −k 2t( )
(D = 1)
![Page 25: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/25.jpg)
Fourier-Laplace inversion
2 ways:1. Invert the Laplace transform first:
€
P(k,s) =1
s + k 2⇒ P(k, t) = exp −k 2t( )
P(x, t) =ds
2π∫ exp −ikx − k 2t( )
(D = 1)
![Page 26: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/26.jpg)
Fourier-Laplace inversion
2 ways:1. Invert the Laplace transform first:
€
P(k,s) =1
s + k 2⇒ P(k, t) = exp −k 2t( )
P(x, t) =ds
2π∫ exp −ikx − k 2t( )
= exp −x 2
4 t
⎛
⎝ ⎜
⎞
⎠ ⎟
ds
2π∫ exp −ikx − k 2t +
x 2
4t
⎛
⎝ ⎜
⎞
⎠ ⎟
(D = 1)
![Page 27: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/27.jpg)
Fourier-Laplace inversion
2 ways:1. Invert the Laplace transform first:
€
P(k,s) =1
s + k 2 ⇒ P(k, t) = exp −k 2t( )
P(x, t) =ds
2π∫ exp −ikx − k 2t( )
= exp −x 2
4 t
⎛
⎝ ⎜
⎞
⎠ ⎟
ds
2π∫ exp −ikx − k 2t +
x 2
4t
⎛
⎝ ⎜
⎞
⎠ ⎟
=1
4πtexp −
x 2
4 t
⎛
⎝ ⎜
⎞
⎠ ⎟
(D = 1)
![Page 28: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/28.jpg)
other way:
2. Invert the Fourier transform first:
![Page 29: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/29.jpg)
other way:
2. Invert the Fourier transform first:
€
P(k,s) =1
s + k 2⇒ P(x,s) =
1
2 sexp − x s( )
![Page 30: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/30.jpg)
other way:
2. Invert the Fourier transform first:
€
P(k,s) =1
s + k 2⇒ P(x,s) =
1
2 sexp − x s( )
P(x, t) =ds
2πi∫ 1
2 sexp − x s + st( )
![Page 31: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/31.jpg)
other way:
2. Invert the Fourier transform first:
€
P(k,s) =1
s + k 2⇒ P(x,s) =
1
2 sexp − x s( )
P(x, t) =ds
2πi∫ 1
2 sexp − x s + st( )
=du
2πexp −i x u − u2t( )∫ iu = s( )
![Page 32: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/32.jpg)
other way:
2. Invert the Fourier transform first:
€
P(k,s) =1
s + k 2⇒ P(x,s) =
1
2 sexp − x s( )
P(x, t) =ds
2πi∫ 1
2 sexp − x s + st( )
=du
2πexp −i x u − u2t( )∫ iu = s( )
=1
4πtexp −
x 2
4 t
⎛
⎝ ⎜
⎞
⎠ ⎟
![Page 33: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/33.jpg)
anomalous diffusion:
€
w(t)∝τ α
tα +1⇒ w(s) ≈1− (sτ )α (α <1)long waiting times:
![Page 34: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/34.jpg)
anomalous diffusion:
€
w(t)∝τ α
tα +1⇒ w(s) ≈1− (sτ )α (α <1)
r(x)∝ξ σ
x1+σ⇒ r(k) ≈1− (kξ )σ (σ < 2)
long waiting times:
long jumps:
![Page 35: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/35.jpg)
anomalous diffusion:
€
P(k,s) =1− w(s)
s
1
1− r(k)w(s)€
w(t)∝τ α
tα +1⇒ w(s) ≈1− (sτ )α (α <1)
r(x)∝ξ σ
x1+σ⇒ r(k) ≈1− (kξ )σ (σ < 2)
long waiting times:
long jumps:
=>
![Page 36: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/36.jpg)
anomalous diffusion:
€
P(k,s) =1− w(s)
s
1
1− r(k)w(s)
≈(sτ )α
s
1
1− 1− (kξ )σ( ) 1− (sτ )α
( )
€
w(t)∝τ α
tα +1⇒ w(s) ≈1− (sτ )α (α <1)
r(x)∝ξ σ
x1+σ⇒ r(k) ≈1− (kξ )σ (σ < 2)
long waiting times:
long jumps:
=>
![Page 37: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/37.jpg)
anomalous diffusion:
€
P(k,s) =1− w(s)
s
1
1− r(k)w(s)
≈(sτ )α
s
1
1− 1− (kξ )σ( ) 1− (sτ )α
( )
≈1
s1−α
1
sα + ξ σ τ α( )kσ
€
w(t)∝τ α
tα +1⇒ w(s) ≈1− (sτ )α (α <1)
r(x)∝ξ σ
x1+σ⇒ r(k) ≈1− (kξ )σ (σ < 2)
long waiting times:
long jumps:
=>
![Page 38: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/38.jpg)
anomalous diffusion:
€
P(k,s) =1− w(s)
s
1
1− r(k)w(s)
≈(sτ )α
s
1
1− 1− (kξ )σ( ) 1− (sτ )α
( )
≈1
s1−α
1
sα + ξ σ τ α( )kσ
=1
s1−α
1
sα + ˜ D kσ
€
w(t)∝τ α
tα +1⇒ w(s) ≈1− (sτ )α (α <1)
r(x)∝ξ σ
x1+σ⇒ r(k) ≈1− (kξ )σ (σ < 2)
long waiting times:
long jumps:
=>
![Page 39: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/39.jpg)
Subdiffusion: long wait time distribution
€
P(k,s) =1
s1−α
1
sα + k 2
![Page 40: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/40.jpg)
Subdiffusion: long wait time distribution
Invert Fourier transform first:
€
P(k,s) =1
s1−α
1
sα + k 2
![Page 41: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/41.jpg)
Subdiffusion: long wait time distribution
Invert Fourier transform first:
€
P(k,s) =1
s1−α
1
sα + k 2
€
P(k,s) =1
s1−α
1
sα + k 2
![Page 42: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/42.jpg)
Subdiffusion: long wait time distribution
Invert Fourier transform first:
€
P(k,s) =1
s1−α
1
sα + k 2
€
P(k,s) =1
s1−α
1
sα + k 2⇒ P(x,s) =
1
2s1−α / 2exp − x sα / 2
( )
![Page 43: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/43.jpg)
Subdiffusion: long wait time distribution
Invert Fourier transform first:
€
P(k,s) =1
s1−α
1
sα + k 2
€
P(k,s) =1
s1−α
1
sα + k 2⇒ P(x,s) =
1
2s1−α / 2exp − x sα / 2
( )
P(x, t) =ds
2πi∫ 1
2s1−α / 2exp − x sα / 2 + st( )
![Page 44: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/44.jpg)
Subdiffusion: long wait time distribution
Invert Fourier transform first:
€
P(k,s) =1
s1−α
1
sα + k 2
€
P(k,s) =1
s1−α
1
sα + k 2⇒ P(x,s) =
1
2s1−α / 2exp − x sα / 2
( )
P(x, t) =ds
2πi∫ 1
2s1−α / 2exp − x sα / 2 + st( )
=1
2tα / 2
du
2πiexp −i x / tα / 2
( )uα / 2 + u( )∫ u = st( )
![Page 45: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/45.jpg)
Subdiffusion: long wait time distribution
Invert Fourier transform first:
€
P(k,s) =1
s1−α
1
sα + k 2
€
P(k,s) =1
s1−α
1
sα + k 2⇒ P(x,s) =
1
2s1−α / 2exp − x sα / 2
( )
P(x, t) =ds
2πi∫ 1
2s1−α / 2exp − x sα / 2 + st( )
=1
2tα / 2
du
2πiexp −i x / tα / 2
( )uα / 2 + u( )∫ u = st( )
=1
tα / 2f x / tα / 2( )
![Page 46: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/46.jpg)
Subdiffusion: long wait time distribution
Invert Fourier transform first:
€
P(k,s) =1
s1−α
1
sα + k 2
€
P(k,s) =1
s1−α
1
sα + k 2⇒ P(x,s) =
1
2s1−α / 2exp − x sα / 2
( )
P(x, t) =ds
2πi∫ 1
2s1−α / 2exp − x sα / 2 + st( )
=1
2tα / 2
du
2πiexp −i x / tα / 2
( )uα / 2 + u( )∫ u = st( )
=1
tα / 2f x / tα / 2( )
€
x 2(t) = x 2P(x)dx = tα∫ y 2 f (y)dy∫
![Page 47: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/47.jpg)
Subdiffusion: long wait time distribution
Invert Fourier transform first:
α < 1: subdiffusion
€
P(k,s) =1
s1−α
1
sα + k 2
€
P(k,s) =1
s1−α
1
sα + k 2⇒ P(x,s) =
1
2s1−α / 2exp − x sα / 2
( )
P(x, t) =ds
2πi∫ 1
2s1−α / 2exp − x sα / 2 + st( )
=1
2tα / 2
du
2πiexp −i x / tα / 2
( )uα / 2 + u( )∫ u = st( )
=1
tα / 2f x / tα / 2( )
€
x 2(t) = x 2P(x)dx = tα∫ y 2 f (y)dy∫
![Page 48: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/48.jpg)
long-tailed jump distribution:(α = 1, σ < 2)
€
P(k,s) =1
s + kσ
![Page 49: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/49.jpg)
long-tailed jump distribution:(α = 1, σ < 2)
First invert Laplace transform:
€
P(k,s) =1
s + kσ
![Page 50: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/50.jpg)
long-tailed jump distribution:(α = 1, σ < 2)
First invert Laplace transform:
€
P(k,s) =1
s + kσ
€
P(k, t) = exp −kσ t( )
![Page 51: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/51.jpg)
long-tailed jump distribution:(α = 1, σ < 2)
First invert Laplace transform:
€
P(k,s) =1
s + kσ
€
P(k, t) = exp −kσ t( )
P(x, t) =1
t1/σSσ (x / t1/σ )
![Page 52: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/52.jpg)
long-tailed jump distribution:(α = 1, σ < 2)
First invert Laplace transform:
€
P(k,s) =1
s + kσ
€
P(k, t) = exp −kσ t( )
P(x, t) =1
t1/σSσ (x / t1/σ ) Sσ = stable distribution
of order σ
![Page 53: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/53.jpg)
long-tailed jump distribution:(α = 1, σ < 2)
First invert Laplace transform:
€
P(k,s) =1
s + kσ
€
P(k, t) = exp −kσ t( )
P(x, t) =1
t1/σSσ (x / t1/σ )
x →∞ ⏐ → ⏐ ⏐ ∝
t
x1+σ
Sσ = stable distributionof order σ
![Page 54: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/54.jpg)
long-tailed jump distribution:(α = 1, σ < 2)
First invert Laplace transform:
€
P(k,s) =1
s + kσ
€
P(k, t) = exp −kσ t( )
P(x, t) =1
t1/σSσ (x / t1/σ )
x →∞ ⏐ → ⏐ ⏐ ∝
t
x1+σ
Sσ = stable distributionof order σ
x scales like t1/σ
![Page 55: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/55.jpg)
long-tailed jump distribution:(α = 1, σ < 2)
First invert Laplace transform:
€
P(k,s) =1
s + kσ
€
P(k, t) = exp −kσ t( )
P(x, t) =1
t1/σSσ (x / t1/σ )
x →∞ ⏐ → ⏐ ⏐ ∝
t
x1+σ
Sσ = stable distributionof order σ
x scales like t1/σ (superdiffusion: faster than √t ),
![Page 56: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/56.jpg)
long-tailed jump distribution:(α = 1, σ < 2)
First invert Laplace transform:
€
P(k,s) =1
s + kσ
€
P(k, t) = exp −kσ t( )
P(x, t) =1
t1/σSσ (x / t1/σ )
x →∞ ⏐ → ⏐ ⏐ ∝
t
x1+σ
Sσ = stable distributionof order σ
x scales like t1/σ (superdiffusion: faster than √t ), but <x2(t)> = ∞.
![Page 57: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/57.jpg)
long-tailed jump distribution:(α = 1, σ < 2)
First invert Laplace transform:
€
P(k,s) =1
s + kσ
€
P(k, t) = exp −kσ t( )
P(x, t) =1
t1/σSσ (x / t1/σ )
x →∞ ⏐ → ⏐ ⏐ ∝
t
x1+σ
Sσ = stable distributionof order σ
x scales like t1/σ (superdiffusion: faster than √t ), but <x2(t)> = ∞.fractional moments:
![Page 58: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/58.jpg)
long-tailed jump distribution:(α = 1, σ < 2)
First invert Laplace transform:
€
P(k,s) =1
s + kσ
€
P(k, t) = exp −kσ t( )
P(x, t) =1
t1/σSσ (x / t1/σ )
x →∞ ⏐ → ⏐ ⏐ ∝
t
x1+σ
Sσ = stable distributionof order σ
x scales like t1/σ (superdiffusion: faster than √t ), but <x2(t)> = ∞.
€
x λ (t) = x λ P(x)dx = t λ /σ∫ y λ f (y)dy∫ < ∞, λ < σ
fractional moments:
![Page 59: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/59.jpg)
Fractional Wiener process
For an ordinary Wiener process,
€
x 2(t) = σ 2t
![Page 60: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/60.jpg)
Fractional Wiener process
For an ordinary Wiener process,
How can we get ?
€
x 2(t) = σ 2t
€
x 2(t) ∝σ 2t H
![Page 61: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/61.jpg)
Fractional Wiener process
For an ordinary Wiener process,
How can we get ?
Consider
€
x 2(t) = σ 2t
€
x 2(t) ∝σ 2t H
€
x(t) = C d ′ t 1
(t − ′ t )a0
t
∫ ξ ( ′ t ); ξ (t)ξ ( ′ t ) = δ(t − ′ t ), a < 12
![Page 62: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/62.jpg)
Fractional Wiener process
For an ordinary Wiener process,
How can we get ?
Consider
Then
€
x 2(t) = σ 2t
€
x 2(t) ∝σ 2t H
€
x(t) = C d ′ t 1
(t − ′ t )a0
t
∫ ξ ( ′ t ); ξ (t)ξ ( ′ t ) = δ(t − ′ t ), a < 12
€
x 2(t) = σ 2C2 d ′ t 1
(t − ′ t )2a0
t
∫ =σ 2C2
1− 2at1−2a ⇒ a = 1
2 (1− H)
![Page 63: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/63.jpg)
Fractional Wiener process
For an ordinary Wiener process,
How can we get ?
Consider
Then
Laplace-transformed:
€
x 2(t) = σ 2t
€
x 2(t) ∝σ 2t H
€
x(t) = C d ′ t 1
(t − ′ t )a0
t
∫ ξ ( ′ t ); ξ (t)ξ ( ′ t ) = δ(t − ′ t ), a < 12
€
x 2(t) = σ 2C2 d ′ t 1
(t − ′ t )2a0
t
∫ =σ 2C2
1− 2at1−2a ⇒ a = 1
2 (1− H)
€
x(s) = C dt t−ae−st
0
∞
∫[ ]ξ (s) = C ⋅Γ(1− a)
s1−a⋅ξ (s)
![Page 64: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/64.jpg)
Fractional Wiener process
For an ordinary Wiener process,
How can we get ?
Consider
Then
Laplace-transformed:
so choose
€
x 2(t) = σ 2t
€
x 2(t) ∝σ 2t H
€
x(t) = C d ′ t 1
(t − ′ t )a0
t
∫ ξ ( ′ t ); ξ (t)ξ ( ′ t ) = δ(t − ′ t ), a < 12
€
x 2(t) = σ 2C2 d ′ t 1
(t − ′ t )2a0
t
∫ =σ 2C2
1− 2at1−2a ⇒ a = 1
2 (1− H)
€
x(s) = C dt t−ae−st
0
∞
∫[ ]ξ (s) = C ⋅Γ(1− a)
s1−a⋅ξ (s)
€
C =1
Γ(1− a)
![Page 65: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/65.jpg)
fractional derivatives
€
x(s) =1
s1−aξ (s)
![Page 66: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/66.jpg)
fractional derivatives
€
x(s) =1
s1−aξ (s) ⇒ x(t) =
d
dt
⎛
⎝ ⎜
⎞
⎠ ⎟−(1−a )
ξ (t)
![Page 67: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/67.jpg)
fractional derivatives
€
x(s) =1
s1−aξ (s) ⇒ x(t) =
d
dt
⎛
⎝ ⎜
⎞
⎠ ⎟−(1−a )
ξ (t) ≡1
Γ(1− a)d ′ t
1
(t − ′ t )a0
t
∫ ξ ( ′ t )
![Page 68: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/68.jpg)
fractional derivatives
€
x(s) =1
s1−aξ (s) ⇒ x(t) =
d
dt
⎛
⎝ ⎜
⎞
⎠ ⎟−(1−a )
ξ (t) ≡1
Γ(1− a)d ′ t
1
(t − ′ t )a0
t
∫ ξ ( ′ t )
€
d
dt
⎛
⎝ ⎜
⎞
⎠ ⎟(1−a )
x(t) = ξ (t)or
![Page 69: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/69.jpg)
fractional derivatives
€
x(s) =1
s1−aξ (s) ⇒ x(t) =
d
dt
⎛
⎝ ⎜
⎞
⎠ ⎟−(1−a )
ξ (t) ≡1
Γ(1− a)d ′ t
1
(t − ′ t )a0
t
∫ ξ ( ′ t )
€
d
dt
⎛
⎝ ⎜
⎞
⎠ ⎟(1−a )
x(t) = ξ (t)
d
dt
⎛
⎝ ⎜
⎞
⎠ ⎟−a
˙ x (t) = ξ (t)
or
or
![Page 70: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/70.jpg)
fractional derivatives
€
x(s) =1
s1−aξ (s) ⇒ x(t) =
d
dt
⎛
⎝ ⎜
⎞
⎠ ⎟−(1−a )
ξ (t) ≡1
Γ(1− a)d ′ t
1
(t − ′ t )a0
t
∫ ξ ( ′ t )
€
d
dt
⎛
⎝ ⎜
⎞
⎠ ⎟(1−a )
x(t) = ξ (t)
d
dt
⎛
⎝ ⎜
⎞
⎠ ⎟−a
˙ x (t) = ξ (t)
1
Γ(a)d ′ t
1
(t − ′ t )1−a0
t
∫ ˙ x ( ′ t ) = ξ (t)
or
i.e.,
or
![Page 71: Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process.](https://reader036.fdocuments.us/reader036/viewer/2022062322/5697bfd11a28abf838cab3b7/html5/thumbnails/71.jpg)
fractional derivatives
€
x(s) =1
s1−aξ (s) ⇒ x(t) =
d
dt
⎛
⎝ ⎜
⎞
⎠ ⎟−(1−a )
ξ (t) ≡1
Γ(1− a)d ′ t
1
(t − ′ t )a0
t
∫ ξ ( ′ t )
€
d
dt
⎛
⎝ ⎜
⎞
⎠ ⎟(1−a )
x(t) = ξ (t)
d
dt
⎛
⎝ ⎜
⎞
⎠ ⎟−a
˙ x (t) = ξ (t)
1
Γ(a)d ′ t
1
(t − ′ t )1−a0
t
∫ ˙ x ( ′ t ) = ξ (t)
or
i.e.,
or
nonlocal!