Strange kinetics: Levy flights and walks (Lecture 3 by J. Klafter)

Strange kinetics:
Levy flights and walks
(Lecture 3 by J. Klafter)
M. F. Shlesinger (ONR)
G. Zumofen (ETH)
A. Chechkin (Kharkov/TAU)
R. Metzler (Munich)

Brownian motion (simple random walk)
<
2
r ( t ) > ~
Kt
; K is the diffusion coefficient
Anomalous diffusion
(a)
<
2
r ( t ) > ~
α
t
α 1
Subdiffusion (dispersive)
α Superdiffusion
)
(b)
< r
2
( t)
> ~ log
β
( t
strong anomaly, ultraslow
Aim: creating framework for treating anomaly and strong anomaly in diffusion.

Classical scattering in
an egg crate potential.
J. Klafter et. al., Physics Today Feb (1996)
r
2
( )
7 / 4
t ≈ t
T. Geisel et al. Z. Phys. B71, 117 (1988)

Superdiffusion

Forms of stable distributions (Lévy)
P
α
( x → ∞)
~
0 < α < 2
x
1
1+
α
P
α
( x → ∞)
~
0 < α

Stable distributions and “heavy tails”
( )
1
ˆ
α
pX
( k; α, D) ≡ exp( ikX ) = exp − D | k | , 0 < α ≤ 2, D > 0 ⇒ pX
( x, α, D)
∝
0< α < 2
x→∞
1+
α
| x |
2
⇒ x =∞ , 0< α < 2
x , 0
q

Lévy stable probability laws
⎡ α α ⎛ k ⎞ ⎤
λ α, β ( k; µ , σ ) = exp ⎢−σ | k | ⎜1 − iβ ϖ ( k, α ) + iµ
k ,
| k |
⎟ ⎥
⎣ ⎝
⎠ ⎦
⎧ πα
tan , if α ≠ 1
⎪ 2
ϖ ( k, α)
= ⎨
⎪ 2
− ln | k | , if α = 1
⎪⎩ π
α = 2, β = 0 :
Examples:
( x − µ ) 2
1 ⎛ ⎞
L( x;2,0, µ , σ ) = exp −
,
2
2σ π ⎜ 4
σ
⎟
⎝ ⎠
α = 1, β = 0 :
σ 1
L( x;1,0, µ , σ ) =
,
π µ σ
α = 0.5, β = 0 :
( ) 2 2
x − +
σ
−
( ) 3/ 2 ⎛ σ ⎞
L( x;1/ 2,0, µ , σ ) = x − µ exp ⎜ − ⎟ .
2π ⎝ 2( x − µ ) ⎠
α = 0.5, β = −0.5 :
L x
1 1 ⎛ 1 ⎞
⎜ ⎟
2 π ⎝ 4 x ⎠
−3 / 2
( ;1 / 2, − 1 / 2, 0,1) = x exp − .

Lévy flights
1. Consider a random walker each of whose steps is an identically
distributed random variable with pdf p(x).
2. When can Pn(x), the pdf of being at x after n steps, be the same
as p(x)?
γ
3.
( k
)
=
exp[
−
n
k
] 0 < γ ≤ 2
P n
lim P n
x→∞
γ
( x)
−1−γ
4. = 2 , Gaussian;
2
< x ( n)
> ~ n (finite)
~
n x
0 < γ < 2
γ < 2
,
lim
k→
0
P n
( k)
~ 1−
c k
γ
<
x
2
( n)
>=
2 2
2
∫ dxPn
( x)
x = −∂ Pn
( k) / ∂k
k = 0
→
∞

Lévy walk
∞
~
5. Ψ(
r,
t)
~ δ ( r − vt)
∫ψ
( t')
dt'
t
ψ ( t)
~
t
−α
−1
,
0
< α <
2

Lévy walk

P
n
~ p
~
P
P
Comment: random walk on a lattice
( r) = P ( r − r'
) P ( r'
)
∑
r'
n−1
1
⎛ 2
k ⎞
∑
2 2
1
≈1−
r k ≈ exp⎜-
⎟ for k →
r=−∞
2
2
⎜ 2 r ⎟
⎝ ⎠
⎛ 2
nk ⎞
= −
⎜ 2
2 r
⎟
⎝ ⎠
∞
( k) = exp( ikr) P ( r)
n
n
1
n
( k) = [ ~
p
( k
)
]
≈
exp
⎜−
⎟
for
n
→ ∞
,
k
→
0
1
2π
π
−ikr
( r) ≈ e P ( k)
letting
letting
n
D
=
∫
−π
= t / τ ,
r
2
~
lim
n
→0,
τ →0
r
dk
2
=
/ τ
1
4πn
r
2
e
2
r
−
4n
r
2
=
1
4πDt
e
2
r
−
4Dt
,
0

Lévy walks
Lévy Walks is a space-time coupled random walk model characterized by
1
ψ ( x, t) ∝ δ ( x − t) ψ ( t)
2
1
Ψ( x, t) ∝ δ ( x − t) ψ ( t ') dt '
2
∫
∞
2
t
and
The mean squared displacement in Lévy walks follows the pattern
2
⎧ t < <
, 0 α 1
⎪ 2
t ln t α = 1
2 ⎪ 3−α
x ( t) ∝ ⎨ t 1< α < 2
⎪
t ln t α = 2
⎪
⎪ ⎩ t 2 < α
16

Diffusion of tracers in fluid flows.
Large scale structures (eddies, jets or convection rolls) dominate
the transport.
Example. Experiments in a rapidly rotating annulus (Swinney et al.).
Ordered flow:
Levy diffusion
(flights and traps)
Weakly turbulent flow:
Gaussian diffusion

V ( x,
y)
= A + B(cos
x + sin y)
+ C cos x cos y

Coupled model
Possibility to include within the velocity model, interruptions by spatial
localization (jump model)
~ ψ
(
t
)

Lévy walks

First Step: Towards a fractal
search pattern
reduce oversampling
J. E. Gillis, G.H. Weiss J. Mathematical Physics 11, 1307 (1970).
J. Klafter, I. M. Sokolov Physics World 18, 29 (2005).

Lévy flights/walks in foraging,
humans ??
“…..certain animals such as ants perform Levy walks when
searching for food in a new area. The above analysis may imply
that starving Levy walk ants possess a slight evolutionary
advantage over ants performing other walks, such as even the
SAW. Flying ants can be considered by the reader.”
Lévy-taxis?
M. F Shlesinger and J.Klafter (1986)
“On Growth and Form, Fractal and Non-Fractal Patterns in Physics.”

The flight of the albatross

Searching for food

Breakdown of segregation
(Mixing in A+B0)

Spatial derivatives
Riesz-Weil symmetric derivative:
d
d
α
x
α
⎧ 1
⎪
−
φ(
x)
=
2cos
⎨
⎪
⎩
( ) [ α α
+ ]
−∞
Dx
φ
xD∞φ
πα / 2
−
d
Hφ,
α = 1
dx
,
α ≠ 1
with H being a Hilbert-transform.
In the Fourier-representation
α
ˆ
⎛ d φ ⎞
Φ⎜ ⎟
α
⎝ d | x | ⎠
= −
|
k
|
α
ˆ φ

Space-fractional diffusion equation, U=0: free
Lévy flights
Random jumps
with Lévy PDF:
(n is time instant)
n
∑
j=
1
ikX
(
α
)
X ( n) = X j , pα
( x) ÷ pˆ
α ( k) = e = exp −D k
0 < α ≤ 2 f ( x, n) = ?
j
n
ik ∑ X j
n
( ) j 1 ikX
α
ikX n
j nD k
e e
= −
| |
= = = e = e
ˆ( , )
−tD k
α
n
fˆ( k , n )
⇐
f k t
= e
Equation for the characteristic
function of free Lévy flights ⇒
∂ fˆ
α
=− Dk fˆ
∂t
α
d φ
α
d x
α
÷ − k ˆ( φ k )
Equation for the PDF of free Lévy
flights
flights ⇒
∂f
∂t
=
α
∂ f
D
α
∂ x

Confined Lévy flights in bistable
potential: Kramers problem
dx dU Y α ( t )
dt dx
= − + ( )
2 4
⎛ 1 ⎞
TGauss
( D)
∝ exp ⎜ ⎟, D

Confined Lévy flights in bistable potential:
Kramers’ problem
lg T esc
5
4
3
2
1
1.0 1.5 2.0 2.5 3.0 3.5
-lg D
α=0.1
α=0.25
α=0.5
α=0.75
α=1.0
α=1.25
α=1.5
α=1.75
α=1.85
α=1.9
α=1.95
α=1.99
α=2.0
Escape time versus noise intensity
µ(α)
1.14
1.12
1.10
1.08
1.06
1.04
1.02
1.00
lg C(α)
T
=
2.0
1.5
1.0
0.5
0.0
0.0 0.5 1.0 1.5 2.0
α
C ( α )
D µ ⎛
α
⎞
⎜ ⎟
⎝ ⎠
0.98
0.5 1.0 1.5 2.0
α

Lévy flight in harmonic potential, 1≤ α

Confined Lévy flights. Quartic potential, U ∝ x 4 .
Cauchy case, α = 1
PDF:
f
( x)
=
π (1 −
x
1
2
+
x
4
)
Two properties of stationary PDF:
Bimodality: .1
local minimum at
two maxima at
x min = 0
x max = ± 1/ 2
2. Steep power law asymptotics with finite variance:
f ( x)
∝
4
x −
∞
x
2 2
= dx x f ( x)
= 1
∫
−∞
⇒
Stationary solution is out of the domain of attraction of stable laws

Two propositions
Proposition 1: Stationary PDF for the Lévy flights in external field
is not unimodal.
Proposition 2: Stationary PDF for the Lévy flights in external field
has power-law asymptotics,
f ( x)
≈
x
C
α
α + c−1
,
x
→ ∞
U
c
= x , c > 2
U
c
= x , c ≥ 2
−1
1 Cα = Γ α π sin πα 2
( ) ( )
Critical exponentc cr
= 4 −α
is a “universal constant”, i.e., it does not depend on c.
c
<
c
cr
⇒
2
x
= ∞
c
>
c
cr
⇒
2
x
< ∞
: “confined”

Leapover lengths and first passage
time statistics for Lévy flights
Question: What are the statistics of FPTs and FPLs ?

Trajectories: examples
T. Koren, A.V. Chechkin & J. Klafter Physica A 379, 10 (2007)

Results I - symmetric case
FPTs:
FPLs:
Sparre Andersen universality
(1953, 1954)
Decay slower than the step’s
PDF !
T. Koren et al. Phys. Rev. Lett. 99, 160602 (2007)

Results II – one-sided case
FPTs:
FPLs:

Anomalous is Normal...