Lecture 13
Drunk Man Walks
H. Risken, The Fokker-Planck Equation (Springer-Verlag Berlin 1989)
C. W. Gardiner, Handbook of Stochastic Methods (Springer Berlin 2004)
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Great White Shark swims 12,400 miles, shocks scientists
WCS release
October 6, 2005
Random Walks and Levy flights
Random Walk
The term “random walk” was first used by Karl Pearson in 1905. He proposed a
simple model for mosquito infestation in a forest: at each time step, a single
mosquito moves a fixed length at a randomly chosen angle. Pearson wanted to
know the mosquitos’ distribution after many steps.
The paper (a letter to Nature ) was answered by Lord Rayleigh, who had already
solved the problem in a more general form in the context of sound waves in
heterogeneous materials.
Actually, the the theory of random walks was developed a few years before
(1900) in the PhD thesis of a young economist: Louis Bachelier. He proposed
the random walk as the fundamental model for financial time series. Bachelier
was also the first to draw a connection between discrete random walks and the
continuous diffusion equation.
Curiously, in the same year of the paper of Pearson (1905) Albert Einstein
published his paper on Brownian motion which he modeled as a random walk,
driven by collisions with gas molecules. Einstein did not seems to be aware of
the related work of Rayleigh and Bachelier.
Smoluchowski in 1906 also published very similar ideas.
The simplest possible problem: 1 dimensional Random Walk
0 1 2 3-4 -3 -2 -1
The “walker” starts at position x = 0 at the step t = 0; at each
time-step the walker can go either forward or backward of one
position with equal probabilities 1/2.
We ask the probability P(x,t) to find the walker at the position x
a the time step t.
!
p(x, t) =t!
(t + x) /2( )! (t " x) /2( )!1
2
#
$ % &
' (
t
~t>>1
2
) te
"x 2
2t
probability of any given sequence of t steps
Number of sequences that take to x in t steps
(t+x)/2 steps taken in the positive
direction
(t-x)/2 steps taken in the negative
direction
Stochastic process - normal noise (finite variance) 1 dimension
!
x(t +1) = x(t) +"(t)
Central limit theorem (lecture 2)
The sum of independent identically-distributed variables with
finite variance will tend to be normally distributed. [lecture 2]
!
x(t) ="=1
t
# $(" )
!
p(x, t) ~t>>1
1
2" t #2exp $
x2
2t #2
%
&
' '
(
)
* *
Average traveled
distance:
!
xt= " = 0
!
x2
t
" xt
2
= #2 t $ t!
" = 0( )
Stochastic process - normal noise (finite variance) D dimensions
!
r r (t +1) =
r r (t) +
r " (t)
iid D-dimesional vectors
isotropic (uniformly distributed in the direction)
with moduli distributed accordingly to a givenprobability density function p(|!|).
!
p(r r ,t +1) =
0
"
# p(r $ )p(
r r %
r $ ,t)dD r
$
!
p(r r ,t +1) =
0
"
# p(r $ ) p(
r r ,t) %
r $
r & p(
r r ,t) +
1
2
r $
r &
r & p(
r r ,t)
r $ + ....
'
( ) *
+ , d
D r $
!
p(r r ,t +1) " p(
r r ,t) =
#2
2D$2
p(r r ,t) + ....
!
"p
"t=#2
2D$2p
!
p(",t) =2D
D / 2"D#1
$(D/2) 2 % 2 t( )D / 2exp
#D"2
2 %2 t
&
'
( (
)
*
+ +
!
" =r r
Random Walk - any dimension
D =1 D =2
D =3 D =10
D =100P(") at t = 0,1,..,9
for <!> =0 and <!2> =1
p(",t)
p(",t)
p(",t)
P(",t)
P(",t)
"
!
" =# (D+1)
2( )# D
2( )2 $2
Dt
!
"2 = #2 t"
""
"
This is not a sphere.
Is this a sphere?
...further jumps…
Stochastic process with large noise fluctuations (non-finite
variance)
!
x(t +1) = x(t) +"(t)
We already discussed the 1 dimensional case in the context
of the central limit theorem and stable distributions [Lecture 3].
!
p(x, t)"a(t)
x#
!
p(")#1
"$!
f (x,",#,c,µ) = dk$%
+%
& eikxexp ikµ $ ck
"$11$ i#
k
k'(" $1,k)
(
) *
+
, -
(
) * *
+
, - -
3.0
2.5
2.0
1.5
Super diffusive behavior
Probability of a jump larger or equal to Lmax
in t steps I have a finite probability of a jump equal to Lmax
if:
!
Pr(L " Lmax ) = p(#)d#Lmax
$
% ~ 1L
max
&'1
!
t Pr(L " L
max) ~ 1
!
Lmax~ t
1/"#1
Mean Square Displacement:
!
x2~ t
1/ 0.9= t
1.11
!
x ~ t1/ 0.9
= t1.11!
" =1.9 Large jumps
dominate the
behavior!
x(t +1) = x(t) +"(t)
!
p(")#1
"$
!
x2
= t "2 = t "2p(")d"Lmin
Lmax
# ~ tLmax3$%
~ t2 /(%$1)
!
" >1
!
1<" < 3
!
x ~ Lmax
!
x2~ L
max
2
!
x " t = t0.5
!
x2
= "2 t ~ t 0.5
Diffusive behavior (‘jumps’ with finite variance)
Super-diffusive behavior
!
x2~ t
1/1.5= t
0.66
!
x ~ t1/1.5
= t0.66
!
x2~ t
1/ 2= t
0.5
!
x ~ t1/ 2
= t0.5
!
x2~ t
1/ 0.9= t
1.11
!
x ~ t1/ 0.9
= t1.11
!
" = 3
!
" = 2.5
!
" =1.9
Scale free and Self similarity
1010
105
108
104
106
103
104
102
!
"x ~ "t( )2
!
"x ~ "t( )1/#
!
" = 0.5
Higher dimensions: Levi Flights
!
r r (t +1) =
r r (t) +
r " (t)
!
" =1.1
!
p(r " ) ~
1
"d +#
!
ˆ p (r k ) ~ exp("c k
#)
Sub-diffusive behaviors
!
x(t + ") = x(t) +#(t)
!
"2with finite, but
with power law distributed waiting times:
!
p(") ~1
"#
In n steps the mean square displacement will grow as
the total time elapsed is
!
x2
= n "2
!
x2~ t
"#1sub-diffusive!
!
t ~ n "
!
" > 2
!
t ~ n
!
1<" < 2
!
t ~ "max~ n
1/(#$1)
!
x2~ t diffusive
(same reasoning as for Lmax in previous slide)
Case
Case
Random walk on graphs
!
P( j, t +1) = pstay ( j)P( j, t) + pout ( j "1)P( j "1,t) + pin ( j +1)P( j +1,t)
The probability to be at time t+1 at a given geodesic distance j form the
starting point is given by the probability that at time t the walker is
probability to be at distance j and stay there
probability to be at distance j-1 and move forward probability to be at distance j+1 and move inward
!
P(0,t +1) = pin (1)P(1,t)
!
P( j,0) = " j ,0
with
and
Which is the probability to find the walker at distance j after t steps?
!
P( j, t +1) = pstay ( j)P( j, t) + pout ( j "1)P( j "1,t) + pin ( j +1)P( j +1,t)
Random walk on graphs - continuous limit
the equation for P(j,t) can be re-written as
!
P( j, t +1) " P( j, t) = 1
2pin ( j +1) + pout ( j +1)[ ]P( j +1,t)
+1
2pin ( j "1) + pout ( j "1)[ ]P( j "1,t)
+ pin ( j) + pout ( j)[ ]P( j, t)
+1
2pin ( j +1) " pout ( j +1)[ ]P( j +1,t)
-1
2pin ( j "1) " pout ( j "1)[ ]P( j "1,t)
the continuous limit (j #"; t # $) gives (Fokker-Plank Equation)
!
"P(#,$ )
"t=" 2
"#2pout (#) + pin (#)
2
%
& ' (
) * P(#,$) +
"
"#pout (#) + pin (#)[ ]P(#,$)
Random walk on graphs - shell map
Each one of the probabilities
!
pin ( j)
!
pout ( j)
!
pstay ( j)
is proportional to the relative number of paths that take the
walker inwards, outwards or within the “shell” j
j
J+1
J-1
square lattice (j > 1): - inward 4(j - 1);
- outward 4(j+1);
- stay 0
For a broad class of graphs holds (j>>1):
!
pout ( j) + pin ( j) ~ Const
!
pout ( j) " pin ( j)#V ( j +1) "V ( j)
V ( j)
Random walk on graphs - continuous asymptotic solutions
!
pout (") + pin (") ~ Const =# 2
D
!
pout (") # pin (") =$ 2
DV (")
%V (")
%"
Finite dimensions:
!
V (") ~ "D#1
!
P(",t) =2D
D / 2"D#1
$(D/2) 2% 2t( )D / 2exp
#D"2
2% 2t
&
' (
)
* +
Hyperbolic spaces:
!
V (") ~ exp(")
!
pout (") # pin (") ~ Const
which leads to an equation of the form
!
"P(#,$ )
"t= %D
1
"
"#P(#,$ ) + D
2
" 2
"#2P(#,$ )
and the solution for large times is a density wave which
moves ballistically outwards:<"> ~ t , < "2> ~ t2 and < "2> - < ">2 ~ t
T. Aste, "Random walks on disordered networks'', Phys. Rev. E 55 (1997), p.6233-6236.
!
pout (") # pin (") ~(D#1)$ 2
D"D#1"D#2
~1
"
Will the walker ever return to the origin?
Will the walker ever return to the origin?
The mean time spent in the origin is:
!
P(0,t)t= 0
"
#
!
" =1#1
P(0,t)t= 0
$
%
The probability to return to the origin is (Polya theorem)
!
P(0,t) =1
2"( )D
ˆ P (r k ,t)d
Dr k
#$
+$
%
!
ˆ P (r k ,t) = ˆ p (
r k )[ ]
t
!
t= 0
"
# P(0,t) =1
2$( )D
ˆ p (r k )[ ]
t
t= 0
"
# dDr k
%"
+"
& =1
2$( )D
1
1% ˆ p (r k )
dDr k
%"
+"
&
Which can be computed using the characteristic function
!
1
1" p(r k )
dDr k
"#
+#
$ ~1
k%
dDr k ~
"#
+#
$k
D"1
k%
d k"#
+#
$
!
p(r k ) ~ exp("c k
#) ~ 1" c k
#
!
" =
1 for D # 2
<1 for D > 2
$ % &
1 for D # '
<1 for D > '
$ % &
$
%
( (
&
( (
Finite
variance
Power Law,
Non defined
variance
! < 2
He might never get back home…
Recurrent " = 1(he always get back)
Transient " < 1(he might never get back)
D =1P(",t)
"
D =2P(",t)
"
D =100P(") at t =
0,1,..,9for <!> =0
and <!2> =1
P(",t)
"
Are sharks intelligent mathematicians?
G. M. Viswanathan, V. Afanasyev, S. V. Buldyrev, E. J. Murphy, P. A. Prince, H. E. Stanley, Levy flight search
patterns of wandering albatrosses, Nature 381 (1996) 413 - 415.A.M. Reynolds, Cooperative random Levy flight searches and the flight patterns of honeybees, Physics Letters A 354 (2005)
384-388.
“maximally searchable graphs” (lecture 9)
!
ps(i, j)" di, j#$
We want to be able to navigate efficiently with
local information only (greedy algorithm).
Let us put the vertices on a lattice and choose
shortcuts between lattice vertices with
probability
with di,j the distance between the vertices on
the lattice.
It results that any target vertex from any
random starting vertex is found
within a time ~ (Log V )2 only when:
!
" = space dimension
J. M. Kleinberg, Navigation in asmall world, Nature, 406 (2000),
p. 845.
Shortfin Mako Shark 53802 1507012 26
Jul 2007 to 23 Mar 2008
Shortfin Mako Shark 68509 1507007
13 Jul 2007 to 9 Mar 2008
Shortfin Mako Shark 53802
1507012 26 Jul 2007 to 23 Mar
2008
Two dimensions:
!
ps(r)" r#2
http://topp.org/species/mako_shark
Supplemetary material
Weierstrass Random WalkThe “walker” starts at position x = 0 at step
t = 0; at each time-step the walker can go
either forward or backward bn positions
with probability ~ pn.
p
pn
!=b
!=bn
!
p(n) ~ pn
!
ˆ p (k) = p(")#$
+$
% eik"
= (1# p) pm
cos(kbm
)m= 0
$
&k'$~ exp(#c(() k
()
!
" = ±bn
!
n =log(" )
b
!
p(") ~1
"1+#
!
p(") =(1# p)
2m= 0
$
% pn &(" # bn ) + &(" + bn )[ ]
!
" = #log p( )log(b)
!
P(x, t) ~1
x1+"
!
p(") ~1
"1+#
Levy stable
Characteristic function
!
p(x, t) = d"1...
#$
+$
% d"t
#$
+$
% p("1)...p("t )&("1 + ...+"t # x)
!
ˆ p (k, t) = dxeikx
"#
+#
$ d%1...
"#
+#
$ d%t
"#
+#
$ p(%1)...p(%t )&(%
1+ ...+%t " x)
!
ˆ p (k, t) = ˆ p (k)[ ]t
~ exp "Deff t k#( )
!
" ˆ P (k, t)
"t= #Deff k
$ ˆ P (k, t)
Diffusive behavior % = 2
!
"P(x, t)
"t= Deff
" 2 ˆ P (k,t)
"x2
Fractional diffusion behavior
!
"P(x, t)
"t= Deff
"# ˆ P (k,t)
"x#
!
"#
"x#:= $
1
2%dk
$&
+&
' k#e$ikx Fractional calculus
Non-equally distributed time-steps can lead to the same
kind of fat-tailed jump probabilities.
solution
Probabulity to return to the origin, on graphs…
!
P( j, t +1) = pstay ( j)P( j, t) + pout ( j "1)P( j "1,t) + pin ( j +1)P( j +1,t)
From the above relation one can calculate the mean time
spent in the origin:
!
t= 0
"
# P( j, t) =1
# of paths between shell j and shell j +1j= 0
"
#
T. Aste, "Random walks on disordered networks'', Phys. Rev. E 55 (1997), p.6233-6236.
!
(# of paths between shell j and shell j +1 ~k
2
j
kj
V ( j) ~
V ( j) for k 2 finite
V ( j)[ ]2
"#1 scale free with 1 <" < 2
V ( j)[ ]"
"#1 scale free with 2 $" < 3
%
&
' '
(
' '
!
" =1#1
P(0,t)t= 0
$
%~
1 for D & 2 and finite k2
1 for scale free with 2 &' < 3 and D < 2 - (3 -')2
1 for scale free with 1<' < 2 and D < 2 - 1'
< 1 otherwhise
(
)
* *
+
* * !
V ( j) ~ jD"1
Brownian motion is a fractal with
dimension 4/3
G.F. Lawler, W. Werner (1999), Intersection exponents for planar Brownian motion, Ann. Probab. 27, 1601-1642.