On CTRWs and Fractional Fokker-Planck-Equations

Thomas Staudt

Seminar on the Maths of Randomness in Biology, Georg-August University Göttingen

2017/01/18

Appetizer: Anomal Diffusion

MMMMMMMMMMMMMMMM\(\langle x^2\rangle \sim {D}_\alpha t^\alpha\)

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Overview

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MMMMMMMMMM Fractional Integrals and Derivatives

Fractional Integrals and Derivatives

Who Would Think of "Fractional Derivatives"?!

• 1695: Birth of fractional calculus:

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  Leibniz	"Can derivatives be generalized with non-integer orders?"
  L'Hospital	"What if the order is 1/2?"
  Leibniz	"An apparent paradox, from which one day useful consequences will be drawn."

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• A wealth of approaches: Leibniz, Euler, Liouville, Riemann, Fourier, Weyl, Laplace, Lagrange, Riesz ...

Fractional Integrals and Derivatives

Iterated Integration is the Key

Cauchy's formula for iterated integration

MMMM\(~ {}_aI^n f(x)\)	\(= \int_a^x dx_1 \int_a^{x_1} dx_2 \cdots \int_a^{x_{n-1}} dx_n \,f(x_n)\)
\(~\)	\(= \frac{1}{(n-1)!}\,\int_a^x (x - y)^{n-1} f(y) \, dy\)
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Replace \(n \in \mathbb{N}\) with \(\alpha > 0\)

MMMM\(~ {}_aI^\alpha f(x)\)

\(:= \frac{1}{\Gamma(\alpha)}\int_a^x (x - y)^{\alpha-1} f(y) \, dy\)

with the Gamma function \(\quad\Gamma(\alpha) = \int_0^\infty e^{-x}x^{\alpha - 1} \,dx\)

Fractional Integrals and Derivatives

Fractional Derivatives

Define \(~ {}^{}_aD^{n-\alpha}_x f(x) = \frac{d^n}{dx^n}\, {}_aI^\alpha f(x)\), \(~ 0 \le \alpha < 1 ~\)

• Weyl (\(a = -\infty\))
• Riemann-Liouville (\(a = 0\))

\(~ {}^{}_0D^{n-\alpha}_x f(x)\)	\(= \frac{d^n}{dx^n}\, \int_0^x (x - y)^{\alpha-1} f(y)\, dy\)
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Grünwald-Letnikov representation

\({}^{}_aD^{n-\alpha}_x f(x)\)

\(= \lim_{N\to\infty}\left[\left(\frac{N}{x -a}\right)^\alpha\sum_{j=0}^N \frac{(-1)^j\Gamma(\alpha + 1)}{\Gamma(\alpha - j + 1)j!}f\!\left(x - j \frac{x - a}{N}\right)\right]\)

Fractional Integrals and Derivatives

"Fractional Exponentials": Mittag-Leffler-Functions

Fractional analogon to exponentials: Mittag-Leffler functions

MMMMMMMMM\({E}_\alpha(y)\)	\(:= \sum_{k=0}^{\infty} \frac{y^k}{\Gamma(k\alpha + 1)}\)
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The fractional differential equation

MMMMMMMMM\(-b\,\frac{d}{dx} f(x)\)

\(= \,{}^{}_0D^{1-\alpha}_x f(x)\)

is solved by

MMMMMMMMM\(f(x)\)

\(= {E}_\alpha(-x^\alpha / b)\)

Fractional Integrals and Derivatives

Nonlocal Dynamics -- How is this Physics?

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Fractional time derivatives:

• dynamics with memory
• respects causality

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Fractional spatial derivatives:

• fundamental problem: action at distance
• decoupling system from environment hard or impossible
• phenomenological models / non-inertial dynamics

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MMMMMMMM Continuous-Time Random Walks (CTRWs)

Continuous-Time Random Walks

Definition

Look at random walker

Position	\( x(t) = \sum_{i}^{N(t)} y_i \)
Number of steps MM	\( N(t) ~\) until time \(~t\)
Single Jumps	\( y_i ~\) with pdf \(~\lambda(y)\)
Waiting times	\( \tau_i ~\) with pdf \(~\omega(\tau)\)

For CTRWs: \(~y_i, \tau_i ~\) independent

# Simulate CTRW with N steps
waiting_times = draw_from_lambda(N)
jumps = draw_from_omega(N)

t = cumsum(waiting_times)
x = cumsum(jumps)

Continuous-Time Random Walks

Should I Wait or Rather Fly?

Basic classification of CTRWs

MMM	\(~\langle y_i^2 \rangle\) MM	\(~\langle \tau_i \rangle\)	MMMMMMM e.g.
Normal diffusion M	\(< \infty\)	\(< \infty\)	MMM\(~\omega(t) \sim \exp(-t)~\,\) (exponential)
Subdiffusion	\(< \infty\)	\(= \infty\)	MMM\(~\omega(t) \sim t^{-1-\alpha}~\) Mi (power-law)
Levy Flights	\(= \infty\)	\(< \infty\)	MMM\(~\lambda(y) \sim \frac{1}{1 + y^2}~\) Mxl (Cauchy)

Continuous-Time Random Walks

Memory and Aging

CTRWs are usually not Markovian

• Markovian only if \(\tau_i\) exponentially distributed
• Then CTRW is a MCCT (Markov Chain in Continuous Time)

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Non-exponential CTRWs memorize when they last jumped

• \(x(t)\) alone does not condition \(x(t+s)\), but \(x(t)\) + time waited already (renewal process)
• \( P(\tau_i > a + b \,|\, \tau_i > b) \neq P(\tau_i > a) \) \(~\Leftrightarrow~\) \(\tau_i\) not memoryless
• For subdiffusion: The jumping rates decline with time, as particles that already waited long are expected to wait even longer (aging)

Continuous-Time Random Walks

Properties

Properties for large \(t\) in the normal and subdiffusive case

M		normal MM	subdiffusive
Average jumps MM	\(\langle N(t)\rangle\)	\(t\)	\(t^\alpha\)
Jump rate	\(k(t) = \frac{d}{dt}\langle N(t)\rangle\)	const	\(t^{\alpha-1}\)
Spatial MSD	\(\langle x^2 \rangle = \langle y^2 \rangle \langle N(t) \rangle\) MM	\(t\)	\(t^\alpha\)
M
Temporal MSD	\(\overline{x^2}\)	\(t\)	\(t~ {}^{*}\)
Stationarity		yes	no
Ergodicity		yes	no

\({}^{*}\) Different constants of proportionality of different realizations

Continuous-Time Random Walks

Propagator

The general solution \(P(x, t)\) is best expressed in Fourier-Laplace space (\(k,s)\)

\(\qquad P(k, s) = \frac{1 - \omega(s)}{s} \frac{1}{1-\lambda(k)\omega(s)}\qquad\) Montroll-Weiss-Formula

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This is equivalent to

\( \!\!\!\!\!\partial_t P(x, t) = \partial_t\int_0^t dt'\, M(t - t') \int dx'\, P(x', t')\, [\lambda(x - x') - \delta(x - x')] \)

with memory kernel

\( \quad M(s) = \frac{\omega(s)}{1 - \omega(s)} \quad \Longleftrightarrow \quad M(t) \sim \begin{cases} t^{\alpha-1} \quad \text{if}~\omega(t) \sim t^{-\alpha - 1}\\ 1 ~~\,~~\quad \,\text{if}~\omega(t) \sim \exp(-t) \end{cases}\)

Continuous-Time Random Walks

Remarks

CTRWs are models to describe

• normal diffusion if \(\langle \tau_i \rangle < \infty\), \(\langle y_i^2 \rangle < \infty~~~\) (weak memory, CLT)
• subdiffusive behaviour if \(\langle \tau_i \rangle = \infty~~~\) (trapping mechanism, aging)
• Levi-flights if \(\langle y_i^2 \rangle = \infty~~~\) (spontaneous large jumps)

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CTRWs are not well suited to

• model interaction with external fields
• boundary value problems

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MMMMMMMMM Fractional Fokker-Planck Equations (FFPEs)

Fractional Fokker-Planck Equations

From CTRWs to FFPEs

Approximate the density P(x', t') to second order, then the Montroll-Weiss formula in \((x, t)\) space becomes

\( \partial_t P(x, t) = \partial_t \,\int_0^t \,M(t - t')\,[A\,\partial_x P(x, t') + B\, \partial_x^2 P(x, t')] \)

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If \(\alpha = 1\) then \(M(t) = \text{const}\) and

\( \partial_t P(x, t) = A\,\partial_x P(x, t) + B\, \partial_x^2 P(x, t) \quad\qquad\qquad\) (FPE)

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If \(\alpha < 1\) then \(M(t) \sim t^{\alpha - 1}\) and

\( \partial_t P(x, t) = {}_0D^{1-\alpha}_t [A\,\partial_x P(x, t) + B\, \partial_x^2 P(x, t)] ,\qquad\) (FFPE)

Fractional Fokker-Planck Equations

General FFPEs

Under general physical conditions, the FFPE is given by

\( \qquad \qquad\partial_t P = {}_0D_t^{1 - \alpha}\,\left( \partial_x \frac{V'(x)}{m\eta_\alpha} + {K}_\alpha\partial^2_x\right)\,P \)

where

\(m\)	Particle mass
\(\eta_\alpha\)	Generalized friction constant
\({K}_\alpha\)	Generalized diffusion constant

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Moments:

\( \qquad \frac{d}{dt} \langle x^n\rangle = {}_0{D}^\alpha_t \left( -\frac{n}{m\eta_\alpha}\,\langle x^{n-1}\,V'\rangle + n(n-1)\,{K}_\alpha\,\langle x^{n-2} \rangle \right) \)

Fractional Fokker-Planck Equations

Free and Harmonic Fractional Particles

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Free particle (leads to Fractional Diffusion Equation, FDE)

\( \qquad \qquad\partial_t P = {K}_\alpha~ {}_0D_t^{1 - \alpha}\,\partial^2_x\,P \quad \Longrightarrow \quad \langle x^2 \rangle \sim {K}_\alpha t^\alpha \)

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Harmonic Potential \( V(x) = m\omega^2x^2/2\)

\( \qquad \qquad \langle x \rangle = {E}_\alpha(- \omega^2 t^\alpha/\eta_\alpha) \)

\( \qquad \qquad \langle x^2 \rangle = {x}_\text{th}^2 + ( {x}_0^2 - {x}_\text{th}^2)\, {E}_\alpha(- 2 \omega^2 / \eta_\alpha t^\alpha) \)

with thermal equilibrium \( x^2_\text{th} = {k}_\text{B}T/ m\omega^2 \)

Fractional Fokker-Planck Equations

Stationary State and Generalized Einstein Relations

Similar to standard FPE, the stationary solution is

\(\qquad \qquad \qquad \qquad {P}_\text{st}(x) \sim \exp\left( - \frac{V(x)}{m \eta_\alpha {K}_\alpha} \right) \)

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Comparing with Bolzmann distribution reveals

\(\quad~~ \qquad {K}_\alpha = {k}_\text{B} T / m \eta_\alpha \qquad\) (generalized Einstein Relation)

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By a linear response approach, one can also show

\(\quad \qquad \langle x \rangle_F = \frac{1}{2} \frac{F}{{k}_\text{B}T} \langle x^2 \rangle\qquad\) (generalized second Einstein Relation)

for a constant force \(F\)

Conclusion

This week:

• CTRWs are versatile mathematical models to describe anomal diffusion
• Remarkable features: Aging, non-stationary, non-ergodic for \(\alpha < 1\)
• There is a close relation between CTRWs and FFPEs
• For the FFPE similar properties to the original FPE can be derived

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Next week:

• Other models to describe subdiffusion, with different properties
• Methods of how to distinguish between different Mechanism driving anomal diffusion