Lecture 10: Anomalous diffusion Outline: generalized diffusion equation subdiffusion superdiffusion fractional Wiener process

anomalous diffusion Recall derivation of Fokker-Planck equation:

anomalous diffusion Recall derivation of Fokker-Planck equation:

anomalous diffusion Recall derivation of Fokker-Planck equation: But what if ?

anomalous diffusion Recall derivation of Fokker-Planck equation: But what if ? And what if the distribution of time steps has infinite mean?

anomalous diffusion Recall derivation of Fokker-Planck equation: But what if ? And what if the distribution of time steps has infinite mean? Go back and reformulate the problem:

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

continuous-time random walk distribution 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 walk distribution 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 walk distribution 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 walk distribution 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 walk distribution 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 Then

continuous-time random walk distribution 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 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:

Fourier & Laplace transform: Fourier transform in space, Laplace transform in time: The conventional case:

Fourier & Laplace transform: Fourier transform in space, Laplace transform in time: The conventional case:

Fourier & Laplace transform: Fourier transform in space, Laplace transform in time: The conventional case:

Fourier & Laplace transform: Fourier transform in space, Laplace transform in time: The conventional case:

Fourier & Laplace transform: Fourier transform in space, Laplace transform in time: The conventional case:

Fourier & Laplace transform: Fourier transform in space, Laplace transform in time: The conventional case:

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: ( 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: ( D = 1 )

Fourier-Laplace inversion 2 ways: 1. Invert the Laplace transform first: ( D = 1 )

other way: 2. Invert the Fourier transform first:

other way: 2. Invert the Fourier transform first:

other way: 2. Invert the Fourier transform first:

other way: 2. Invert the Fourier transform first:

other way: 2. Invert the Fourier transform first:

anomalous diffusion: long waiting times:

anomalous diffusion: long waiting times: long jumps:

anomalous diffusion: long waiting times: long jumps: =>

anomalous diffusion: long waiting times: long jumps: =>

anomalous diffusion: long waiting times: long jumps: =>

anomalous diffusion: long waiting times: long jumps: =>

Subdiffusion: long wait time distribution

Invert Fourier transform first:

Subdiffusion: long wait time distribution Invert Fourier transform first:

Subdiffusion: long wait time distribution Invert Fourier transform first:

Subdiffusion: long wait time distribution Invert Fourier transform first:

Subdiffusion: long wait time distribution Invert Fourier transform first:

Subdiffusion: long wait time distribution Invert Fourier transform first:

Subdiffusion: long wait time distribution Invert Fourier transform first:

Subdiffusion: long wait time distribution Invert Fourier transform first: α < 1 : subdiffusion

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

long-tailed jump distribution: ( α = 1, σ < 2 ) First invert Laplace transform:

long-tailed jump distribution: ( α = 1, σ < 2 ) First invert Laplace transform:

long-tailed jump distribution: ( α = 1, σ < 2 ) First invert Laplace transform:

long-tailed jump distribution: ( α = 1, σ < 2 ) First invert Laplace transform: S σ = stable distribution of order σ

long-tailed jump distribution: ( α = 1, σ < 2 ) First invert Laplace transform: S σ = stable distribution of order σ

long-tailed jump distribution: ( α = 1, σ < 2 ) First invert Laplace transform: S σ = stable distribution of order σ x scales like t 1/σ

long-tailed jump distribution: ( α = 1, σ < 2 ) First invert Laplace transform: S σ = stable distribution of order σ x scales like t 1/σ (superdiffusion: faster than √t ),

long-tailed jump distribution: ( α = 1, σ < 2 ) First invert Laplace transform: S σ = stable distribution of order σ x scales like t 1/σ (superdiffusion: faster than √t ), but = ∞.

long-tailed jump distribution: ( α = 1, σ < 2 ) First invert Laplace transform: S σ = stable distribution of order σ x scales like t 1/σ (superdiffusion: faster than √t ), but = ∞. fractional moments:

long-tailed jump distribution: ( α = 1, σ < 2 ) First invert Laplace transform: S σ = stable distribution of order σ x scales like t 1/σ (superdiffusion: faster than √t ), but = ∞. fractional moments:

Fractional Wiener process For an ordinary Wiener process,

Fractional Wiener process For an ordinary Wiener process, How can we get ?

Fractional Wiener process For an ordinary Wiener process, How can we get ? Consider

Fractional Wiener process For an ordinary Wiener process, How can we get ? Consider Then

Fractional Wiener process For an ordinary Wiener process, How can we get ? Consider Then Laplace-transformed:

Fractional Wiener process For an ordinary Wiener process, How can we get ? Consider Then Laplace-transformed: so choose

fractional derivatives

or

fractional derivatives or

fractional derivatives or i.e., or

fractional derivatives or i.e., or nonlocal!