Fractional Feynman-Kac Equation for non-Brownian Functionals
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Transcript of Fractional Feynman-Kac Equation for non-Brownian Functionals
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
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')'(1)( dW
dtxUAxQWtAxGt
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0 0
dtaxUAaxQtaxUAaxQttAxQ nn
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]),(,[
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0
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