lecture 10: anomalous diffusion outline: generalized diffusion equation subdiffusion superdiffusion...

Lecture 10: Anomalous diffusion

Outline:• generalized diffusion equation• subdiffusion• superdiffusion• fractional Wiener process

anomalous diffusionRecall derivation of Fokker-Planck equation:

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

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∫ = ∞

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∫ = ∞

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∫ = ∞

continuous-time random walkdistribution of jump sizes r(x), distribution of waiting times w(t)

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)

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

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)∫∫

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)∫∫

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

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

Fourier & Laplace transform:Fourier transform in space, Laplace transform in time:

Fourier & Laplace transform:Fourier transform in space, Laplace transform in time:

€

P(k,s) =1− w(s)

s

1

1− r(k)w(s)

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:

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

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

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( )

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

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

Fourier-Laplace inversion

2 ways: (D = 1)

Fourier-Laplace inversion

2 ways:1. Invert the Laplace transform first:

(D = 1)

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)

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)

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)

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)

other way:

2. Invert the Fourier transform first:

other way:

2. Invert the Fourier transform first:

€

P(k,s) =1

s + k 2⇒ P(x,s) =

1

2 sexp − x s( )

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( )

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( )

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

⎛

⎝ ⎜

⎞

⎠ ⎟

anomalous diffusion:

€

w(t)∝τ α

tα +1⇒ w(s) ≈1− (sτ )α (α <1)long waiting times:

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:

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:

=>

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:

=>

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:

=>

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:

=>

Subdiffusion: long wait time distribution

€

P(k,s) =1

s1−α

1

sα + k 2

Subdiffusion: long wait time distribution

Invert Fourier transform first:

€

P(k,s) =1

s1−α

1

sα + k 2

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

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

( )

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( )

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( )

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( )

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∫

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∫

long-tailed jump distribution:(α = 1, σ < 2)

€

P(k,s) =1

s + kσ

long-tailed jump distribution:(α = 1, σ < 2)

First invert Laplace transform:

€

P(k,s) =1

s + kσ

long-tailed jump distribution:(α = 1, σ < 2)

First invert Laplace transform:

€

P(k,s) =1

s + kσ

€

P(k, t) = exp −kσ t( )

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 )

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

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

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

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 ),

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)> = ∞.

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:

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:

Fractional Wiener process

For an ordinary Wiener process,

€

x 2(t) = σ 2t

Fractional Wiener process

For an ordinary Wiener process,

How can we get ?

€

x 2(t) = σ 2t

€

x 2(t) ∝σ 2t H

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

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)

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)

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)

fractional derivatives

€

x(s) =1

s1−aξ (s)

fractional derivatives

€

x(s) =1

s1−aξ (s) ⇒ x(t) =

d

dt

⎛

⎝ ⎜

⎞

⎠ ⎟−(1−a )

ξ (t)

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 )

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

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

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

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!