Kramers Problem in
anomalous
dynamics
Sliusarenko O.Yu.
Akhiezer Institute for Theoretical Physics
NSC KIPT, Kharkiv, Ukraine
Classical Kramers Problem
 Calculating the mean escape rate of a particle from a
potential well due to the influence of an external random
force with Gaussian probability distribution law


Pontryagin L.S., Andronov A.A., Witt A.A. JETP 3, 165 (1933).
H. A. Kramers, Physica Amsterdam 7, 284 (1940).
A
xmax
B
Modeling of some chemical
reactions
The electroconductivity theory of
crystals
Nucleation theories
Climatic dynamics etc.
Classical Kramers Problem
continued.
Assumptions:
 One-dimensional motion, for simplicity;
 All the particles are concentrated in one point at the initial
moment of time;
 The particles do not interact;
 The potential’s height is much larger, than the heat motion
energy;
H kT
 All this leads to the problem with
quasi-stationary conditions
Н
A
xmax
B
Classical Kramers Problem
continued..

Chandrasekhar Stochasic Processes in Physics and Astronomy
 An integral representation of Fokker-Planck equation
A
U / kT A,
j    eU / kT ds  kT
we
m
B
B
 where the integral is taken through an arbitrary path, from the
point A to the point B, j is the current’s density, β is some
constatnt.
 Considering a one-dimensional problem,
weU ( x)/ kT
j  kT B
m
A
B
U ( x)/ kT dx
e

A
.
Classical Kramers Problem
continued...
 In the point A we have a Maxwell-Bolzmann disrtibution. Then,
the number of particles near the point A will be
d A  wAeU / kT dx.
 Expanding the potential
U (x)  A2 x2,
x ~0, A  const
2
U (x)  H  1max2(x  xmax )2,
x xmax , max  const
2
 We can calculate the integral approximately
A
B

 eH / kT .
2

Tesc   
  eU (x)/ kT dx   2
j
A kT A
A max
PART I
astable Levy Motion
Levy Probability Distribution Law
0
a
a
a
a
a
-2
0.30
a
a
a
-6
-8
ln p(x)
0.25
-4
-10
-12
-14
0.20
-16
p(x)
-18
-20
0.15
-4
-2
0
2
4
6
ln x
0.10
0.05
0.00
-10
-5
0
x
5
10


 
x
 a
  1a dx  x
, a
 x 
x
x




x
 x  
  1a dx  ln x ,   a ,
x

x

Kramers Problem for Levy Statistics
Why the straight analytical approach is not
possible?
 The Fokker-Planck equation now has fractional
derivatives
 => a complicated integro-differential equation in
partial derivatives
 The Levy PDF does not have an analytical
representation in real space;
 The infinite variance of the noise.
One of the ideas:
 Langevin numerical simulations.
Numerical Simulations
 Langevin equations









dx t 
 v t  ,
dt
dv t 
dU  x 
m
  mv t  
 ma t  ,
,D
dt
dx
x is the particle’s coordinate, v is its
velocity, m is its mass, γ is a friction
constant, U(x) is an external potential,
ξα(t) is a random force, D is its intensity,
α is the Levy index.
 Let us examine the strong friction case, when
dx  t 
dU  x 
 
   1


t
a ,D  
m dx
dt
dv t 
  v t  :
dt
 Or, in dimensionless variables after the time quantization:
Motion in the Potential
 Let us study the double-well potential
4
2
x
x
U1 x   a  b , a,b  0.
4
2
0.4
 The time-discrete Langevin equation:




a ,1 
0.2
n
0.1
U1(x)
xn1  xn
1/a


3
 ( x  x)t  Dt 
0.3
2
0.0
-0.1
-0.2
1
-0.3
-2.0
-1.5
-1.0
-0.5
0.0
x(t)
x
0
-1
-2
0
500
1000
1500
t
2000
2500
3000
0.5
1.0
1.5
2.0
The First Passage Time
 We place the particle to the left potential’s minimum (x0= -



0.4
0.3
0.2
0.1
U1(x)

1);
The iterations of the time-discrete Langevin equation begin;
When the particle reaches the point x=0, we regard it as
the escaped one
We stop the timer;
The algorithm is re-executed for 100000 times to gain the
statistics, then the time is averaged.
0.0
-0.1
-0.2
-0.3
-2.0
-1.5
-1.0
-0.5
0.0
x
0.5
1.0
1.5
2.0
The First Passage Time
continued.
T a,D 



5
log10 (Tesc)
4



C a 
D
 a 
 a   d lgT .
d lg1 D
3
2
1
1.0
1.5
2.0
2.5
-log10 (D)
3.0
a
a
a
a
a
a
a
a
a
3.5 a
,
The First Passage Time
continued..
T a , D  
1.14
2.0
1.10
1.5
1.06
log10 C(a)
1.12
1.08
a

D
 a 
 a   d lgT .
1.0
d lg1 D
0.5
0.0
1.04

C a 
0.0
0.5
1.0
1.5
2.0
a
1.02
1.00
0.98
0.5
1.0
1.5
a
2.0
,
U(x)
U(x)
The Mean First Passage Time (other
potentials)
x
x
4.5
3.5
lg Tesc
3.0
2.5
2.0
1.5
1.0
0.5
4
3
lg Tesc
a
a
a
a
a
a
a
a
a
a
a
a
a
4.0
2
1
0
-1
0.0
0.0
0.5
1.0
1.5
-lg D
2.0
2.5
3.0
-2
-4
a
a
a
a
a
a
a
a
a
a
a
a
a
-3
-2
-1
0
-lg D
1
2
3
Simulating the FP time PDF
 We place the particle to the left potential’s minimum




(x0= -1);
The iterations of the time-discrete Langevin equation begin;
When the particle reaches the point x=0, we regard it as
the escaped one
We stop the timer;
The algorithm is re-executed for 1000000 times to gain the
statistics, then the times are treated with a procedure that
extracts the PDF of the data.
The FP time PDF
-5
a
a
a
a
-6
ln p(t)
-7
-8
-9
D 102.0
-10
-11
-12
0
500
1000
t
1500
p t   1 exp  t / T .
T


2000
T1  1 ;
p(0)





T2   d ln p(t )
dt





1
.
The FP time PDFs
(other potentials)
Cubic potential
-5
-5
a
a
a
a
-6
-8
-9
-7
-8
-9
-10
-10
-11
-12
a
a
a
a
-6
ln p(t)
-7
ln p(t)
Harmonic potential
0
500
1000
t
1500
2000
-11
0
250
500
750
t
1000
1250
Analytical Approach. The Constant
Flux Approximation
 FFPE in dimensionless variables
 In terms of probability flux
 After the Fourier transformation of both equations
Analytical Approach. The Constant
Flux Approximation continued.
 Consider a constant probability flux
 Solving for f(k), executing an inverse Fourier transformation
Analytical Approach of Imkeller and
Pavlyukevich
P.Imkeller, I. Pavlyukevich J. Phys. A: Math. Gen. 39 (2006) L237–L246
Levy noise
Gaussian-like noise
Between the large jumps
makes the particles
relax to the potential’s bottom
Large “outliers”
The escapes are done
during a single jump
If one jump is not enough
Analytical Approach of Imkeller and
Pavlyukevich continued.
2.0
Numerical simulation
Imkeller-Pavlyukevich' theory
log10 C(a)
1.5
1.0
0.5
0.0
0.0
0.5
1.0
a
1.5
2.0
application
A Problem from Climatic Dynamics
Peter D. Ditlevsen, Geophysical Research Letters, 26, 1441 (1999)
The fluctuations of Calcium concentration inside the ice core
was studied
1. The times between the two
states of the system are nicely
described with the Poisson
process;
2. The PDF is bimodal => the
double-well “potential” is
possible;
3. The noise is white but with
strongly non-Gaussian PDF
A Problem from Climatic Dynamics
continued.
dy    dU / dy  dt 1dx  2dL,


dx   xdt  1 x2 dB
Gaussian noise,
year fluctuations
Levy noise,
α=1,75
1000-2000year fluctuations
Publications:
1. A.V. Chechkin, O.Yu. Sliusarenko On Lévy flights in potential well. Ukr. J.
Phys., 2007, v. 52, № 3, p. 295–300
2. Aleksei V. Chechkin, O.Yu. Sliusarenko, Ralf Metzler, and Joseph Klafter
Barrier crossing driven by Lévy noise: Universality and the role of noise
intensity. Physical Review E, 2007, v. 75, 041101, p. 041101-1–041101-11
3. A.V. Chechkin, O.Yu. Sliusarenko Generalized Kramers’ problem for Lévy
particle. Problems of Atomic Science and Technology, 2007, № 3(2), p. 293–
296
Conferences:
1. 373th Wilhelm und Else Heraeus-Seminar, Anomalous Transport:
Experimental Results and Theoretical Challenges, Bad Honnef, Germany, July
12-16, 2006
2. 2-nd International conference on Quantum electrodynamics and statistical
physics (QEDSP2006), Kharkiv, Ukraine, September 19-23, 2006
3. Physics of Fluctuations far from Equilibrium, Dresden, Germany, July 02-06,
2007
PART II
Fractional Brownian Motion.
Fractional Gaussian Noise
1
exp   2 / 4 
4
PDF   
1000
2
<x >
noise
H=0.3 100H=0.7
H=0.1
H=0.2
H=0.3
H=0.4
H=0.5
10
1
1
10
100
j
free motion
H=0.3
H=0.7
x j 2  2D j
2H
1000
Simulation Procedure
dU  xn 
H
xn 1  xn  
 t  D1/ 2  t   H  n  .
dx
U  xn   xn 2 / 2
1.00
0
0.96
2
-2
Simulation
Analytics
0.98
<x >st
2
ln <x >
-1
D=1
D=0.5
D=0.25
D=0.1
2H
t
-3
0.94
0.92
-4
0.90
-5
0.88
-8 -7 -6 -5 -4 -3 -2 -1
ln t
0
1
2
3
0.0
0.1
0.2
0.3
H
0.4
0.5
Mean Escape Time
200 200
150 150
6
1.0
5
100
2
0
0.5
1
H=0.1
H=0.2
H=0.3
H=0.4
H=0.5
1000
1.5
100 100
50 50
0
2.0
1 2
2 3
3 4 4 5
0.0
1/D 1/D
-2
5 6
6
4
ln Tesc
Tesc
Tesc
300 300
250 250
H=0.5
H=0.5
H=0.45
H=0.45
H=0.4
H=0.4
H=0.35
H=0.35
H=0.3
H=0.3
H=0.25
H=0.25
H=0.2
H=0.2
H=0.1
H=0.1
The escape time of the particle from the
truncated harmonic potential well as the
function of an inverse noise intensity 1/D.
<x >
400 400
350 350
10
-1
0
1
3
2
2
1
x =1.4142
0
The same, but in a logarithmic scale. Now,
the exponential behaviour is clearly
noticeable.
1 0
1
10
-1
-2
100
j
1
2
3
4
1/D
H=0.5
H=0.4
1000H=0.3
H=0.2
H=0.1
5
6
MET vs Hurst Exponent
6
Dependence of mean escape time on
Hurst exponent (anti-persistent case),
four noise intensity values, a
logarithmic plot.
5
3
2
H
1
4.5
D=1
D=0.5
D=0.25
D=0.167
0
-1
Simulation D=0.25
Linear Fit
Polynomial Fit
4.0
3.5
0.1
0.2
0.3
0.4
0.5
H
Dependence of mean escape time on Hurst
exponent (anti-persistent case), a
logarithmic plot. Solid line is a linear fitting
dashed line is a parabolic fitting. It is clear,
that the linear fitting of the data is not correct.
3.0
ln Tesc
ln Tesc
4
2.5
2.0
1.5
1.0
0.5
0.1
0.2
0.3
H
0.4
0.5
Exponential Behavior of MET
0.96
1.0
A1
Parabolic fit
0.94
0.5
0.0
-0.5
A2
A1
0.92
0.90
-1.0
-1.5
0.88
-2.0
0.86
0.84
A2
Linear fit
-2.5
0.1
0.2
0.3
0.4
-3.0
0.5
0.1
H
0.2
0.3
0.4
0.5
H
ln Tesc
1
 A1  H   A2  H 
D
A1  0.70516  1.48947 H  2.28094 H 2
A2  3.01894  7.29582 H
PDF of ETs
H=0.1
H=0.2
H=0.3
Exponential fit of H=0.1
Exponential fit of H=0.2
Exponential fit of H=0.3
p(t)
0.1
Probability density function of
mean escape times as the
function of walking time,
logarithmic plot. The exponential
behaviour is observed.
0.01
p t  
1E-3
0
10
20
30
40
50
t
60
70
80
90
1
exp  t / Tesc 
Tesc