Fractional Feynman-Kac Equation for non-Brownian Functionals

Introduction Results Applications

See also: L. Turgeman, S. Carmi, and E. Barkai, Phys. Rev. Lett. 103, 190201 (2009).

Lior Turgeman, Shai Carmi, Eli BarkaiDepartment of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel

Random walk functionals

A functional of a random walk:x(t) is the path, U(x) is some function.

t

dxUA0

)]([

Occupation time-How long is the particle at x>0 ?

Example: U(x)=Θ(x).

Functionals in nature:• Chemical reactions• NMR• Turbulent flow• Surface growth• Stock prices• Climate• Complexity of algorithms

Brownian functionals

G(x,A,t): the joint PDF of the particle to be at x and the functional to equal A.G(x,p,t): the Laplace transform of G(x,A,t) (A→p).For Brownian motion (normal diffusion: <x2>~t),

Feynman-Kac equation:

),,()(),,(),,(2

2

tpxGxpUtpxGx

tpxGt

Anomalous diffusion<x2>~tα

In many physical, biological, and other systems diffusion is anomalous.

What is the equation for the distribution of non-Brownian functionals?

Model: Continuous-time random-walk (CTRW)

• Lattice spacing a, jumps to nearest neighbors with equal probability.• Waiting times between jumps distributed according to ψ(t)~t.-

(1+α)

• For 0<α<1, sub-diffusion with <x2>~tα.

Fractional Feynman-Kac equation

Dt1-α is the fractional substantial derivative operator

In Laplace space (t→s), Dt1-α equals [s+pU(x)]1-α.

This is a non-Markovian operator- The evolution of G(x,p,t) depends on the entire history.

')'()'(

)()(

1)(

01

)()'(1 dttf

tt

expU

ttfD

t xpUtt

Variants

Backward equation:

In the presence of a force field F(x), replace Laplacian with Fokker-Planck operator:

Distribution of occupation times

• Consider the occupation time in half space , usually denoted with .• Boundary conditions:• For x→∞, G(A,t)=δ(A-t) G(p,s)=1/(s+p) (particle is always at x>0).• For x→-∞, G(A,t)=δ(A) G(p,s)=1/s (particle is never at x>0).• The distribution of f≡T+/t, the fraction of time spent at x>0:

)2/cos()1(2)1(

)1()2/sin()(

2/2/

12/12/

ffff

fffP

0

[ ( )]t

A x d

The particle trajectory is almost never symmetric:It usually sticks to one side.

Weak ergodicity breaking

• Consider the time average , where .• Assume harmonic potential .• For normal diffusion, the system is ergodic, that is for t→∞: .• For sub-diffusion, the time average is a random variable even in the long time limit - weak ergodicity breaking.

• Fluctuations in time average for t→∞ , .• What are the fluctuations of the time average for all t?• Use the Fractional Feynman-Kac equation:

0xx

No fluctuations for α=1.

α<1: Fluctuations exist- the system does not uniformly sample all available states.

Mittag-Leffler function

( ) ( )/x t A t t

T

2 2( ) / 2V x m x

• Qn(x,A,t)dxdA: the probability to arrive into [(x,x+dx),(A,A+dA)] after n jumps.• The time the particle performed the last jump in the sequence is (t-τ). • The particle is at (x,A) at time t if it was on [x,A- τ U(x)] at (t-τ) and did not move since.• The probability the particle did not move during (t- τ,t) is • Thus, G and Q are related via:

• To arrive into (x,A) at t the particle must have arrived into either [x+a,A- τ U(x+a)] or [x-a,A- τ U(x-a)] at (t-τ), and then jumped after waiting time τ.• Thus, a recursion relation exists for Qn:

• Solving in Laplace-Fourier space and taking the continuum limit, a→0, we get the

Analysis

0

')'(1)( dW

dtxUAxQWtAxGt

nn ]),(,[)(),,(

0 0

dtaxUAaxQtaxUAaxQttAxQ nn

t

n

]),(,[

2

1]),(,[

2

1)(),,(

0

1

21

2( , , ) ( , , ) ( ) ( , , )tG x p t G x p t pU x G x p t

t x

D

0 0 0

21

020

( , ) ( , ) ( ) ( , )x t x xG p t G p t pU x G p tt x

D

21

2

( )( , , ) ( , , ) ( ) ( , , )t

F xG x p t G x p t pU x G x p tt x x T

D

t

dxtA0

)()(

)2(E])[1(2

)(E]2[2

)1(

)(

3,22

0

3,20

2

2

2

txx

txx

x

tx

th

th

th

th

xx 22)1(

22

mT

xth