Kramers Problem in Kramers Problem in anomalous anomalous dynamicsdynamics

Sliusarenko O.Yu.Akhiezer Institute for Theoretical Physics

NSC KIPT, Kharkiv, Ukraine

Classical Kramers ProblemClassical 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).

Bxmax

A

Modeling of some chemical Modeling of some chemical reactionsreactions The electroconductivity theory of The electroconductivity theory of crystalscrystalsNucleation theoriesNucleation theoriesClimatic dynamics etc.Climatic dynamics etc.

Classical Kramers Problem continued.

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; All this leads to the problem with

quasi-stationary conditions

Bxmax

A Bxmax

A

Н

H kT

Classical Kramers Problem continued..

Classical Kramers Problem continued..

Chandrasekhar Stochasic Processes in Physics and Astronomy

An integral representation of Fokker-Planck equation

where the integral is taken through an arbitrary path, from the point A to the point B, is the current’s density, β is some constatnt.

Considering a one-dimensional problem,

/ / ,A AU kT U kT

BB

kTj e ds wem

j

( )/

( )/.

AU x kTB

BU x kT

A

wekTjm

e dx

In the point A we have a Maxwell-Bolzmann disrtibution. Then, the number of particles near the point A will be

Expanding the potential

We can calculate the integral approximately

Classical Kramers Problem continued...

Classical Kramers Problem continued...

/ .U kTA A

d w e dx

2 2( ) , ~0,2 A A

U x x x const

,2 2max max max max

1( ) ( ) ,2

U x H x x x x const

esc( )/ /

max.22 B

U x kT H kTAA AA

T e dx ej kT

PART Istable Levy Motion

PART Istable Levy Motion

Levy Probability Distribution LawLevy Probability Distribution Law

-10 -5 0 5 10

0.00

0.05

0.10

0.15

0.20

0.25

0.30

p(x

)

x

-10 -5 0 5 10

0.00

0.05

0.10

0.15

0.20

0.25

0.30

p(x

)

x

-4 -2 0 2 4 6

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

ln p

(x)

ln x

-4 -2 0 2 4 6

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

ln p

(x)

ln x

,,ln

,

1

1

xdxx

xx

xdxx

xx

x

x

,,ln

,

1

1

xdxx

xx

xdxx

xx

x

x

Kramers Problem for Levy StatisticsKramers 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 SimulationsNumerical Simulations

Langevin equations

Let us examine the strong friction case, when :

Or, in dimensionless variables after the time quantization:

,

,

,D

dx tv t

dtdv t dU x

m mv t m tdt dx

,

,

,D

dx tv t

dtdv t dU x

m mv t m tdt dx

dv t

v tdt

,

1D

dx t dU xtmdt dx

,1

D

dx t dU xtmdt 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.

Motion in the PotentialMotion in the Potential

Let us study the double-well potential

The time-discrete Langevin equation:

4 21

, , 0.4 2x xU x a b a b

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

U1(x

)

x

0 500 1000 1500 2000 2500 3000-2

-1

0

1

2

x(t

)

t

1/

31 ,1( )nnx x x x t D t n

1/3

1 ,1( )nnx x x x t D t n

The First Passage TimeThe First Passage Time

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 100000 times to gain the

statistics, then the time is averaged.

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

U1(x

)

x

The First Passage Timecontinued.The First Passage Timecontinued.

1.0 1.5 2.0 2.5 3.0 3.5

1

2

3

4

5

log 10

(T

esc)

-log10

(D)

.

, ,

lglg1

CT D

D

d Td D

The First Passage Time continued..

The First Passage Time continued..

.

, ,

lglg1

CT D

D

d Td D

0.5 1.0 1.5 2.0

0.98

1.00

1.02

1.04

1.06

1.08

1.10

1.12

1.14

0.0 0.5 1.0 1.5 2.0

0.0

0.5

1.0

1.5

2.0

log

10 C

()

0.5 1.0 1.5 2.0

0.98

1.00

1.02

1.04

1.06

1.08

1.10

1.12

1.14

0.0 0.5 1.0 1.5 2.0

0.0

0.5

1.0

1.5

2.0

log

10 C

()

U(x

)

x

U

(x)

x

The Mean First Passage Time (other potentials)The Mean First Passage Time (other potentials)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

lg T

esc

-lg D

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

lg T

esc

-lg D

-4 -3 -2 -1 0 1 2 3-2

-1

0

1

2

3

4

lg T

esc

-lg D

-4 -3 -2 -1 0 1 2 3-2

-1

0

1

2

3

4

lg T

esc

-lg D

Simulating the FP time PDFSimulating 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 PDFThe FP time PDF

0 500 1000 1500 2000-12

-11

-10

-9

-8

-7

-6

-5

ln p

(t)

t

2.010D

1

1

2

1 ;(0)

ln ( ) .

Tp

d p tTdt

1 exp / .p t t TT

The FP time PDFs(other potentials)The FP time PDFs(other potentials)

0 500 1000 1500 2000-12

-11

-10

-9

-8

-7

-6

-5

ln p

(t)

t

0 500 1000 1500 2000-12

-11

-10

-9

-8

-7

-6

-5

ln p

(t)

t

0 250 500 750 1000 1250-11

-10

-9

-8

-7

-6

-5

ln p

(t)

t

0 250 500 750 1000 1250-11

-10

-9

-8

-7

-6

-5

ln p

(t)

t

Cubic potential Harmonic potential

Analytical Approach. The Constant Flux ApproximationAnalytical 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.

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 PavlyukevichAnalytical Approach of Imkeller and Pavlyukevich

Levy noise

Gaussian-like noise Large “outliers”

The escapes are done during a single jump

Between the large jumps makes the particles

relax to the potential’s bottom

If one jump is not enough

P.Imkeller, I. Pavlyukevich J. Phys. A: Math. Gen. 39 (2006) L237–L246

Analytical Approach of Imkeller and Pavlyukevich continued.

0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

1.5

2.0 Numerical simulation Imkeller-Pavlyukevich' theory

log 1

0 C(

)

A Problem from Climatic DynamicsA 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

application

A Problem from Climatic Dynamics continued.

A Problem from Climatic Dynamics continued.

1 2

2

/ ,

1

dy dU dy dt dx dL

dx xdt x dB

1 2

2

/ ,

1

dy dU dy dt dx dL

dx xdt x dB

Levy noise,α=1,75

1000-2000year fluctuations

Levy noise,α=1,75

1000-2000year fluctuations

Gaussian noise,year fluctuations

Gaussian noise,year fluctuations

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

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, 20063. Physics of Fluctuations far from Equilibrium, Dresden, Germany, July 02-06, 2007

Publications:

Conferences:

PART IIFractional Brownian Motion.

PART IIFractional Brownian Motion.

Fractional Gaussian NoiseFractional Gaussian Noise

21exp / 4

4PDF

21exp / 4

4PDF

noiseH=0.3 H=0.7

free motionH=0.3 H=0.7

22 2H

jx D j

1 10 100 10001

10

100

1000

<x2 >

j

H=0.1 H=0.2 H=0.3 H=0.4 H=0.5

Simulation ProcedureSimulation Procedure

1/ 21 .

Hnn n H

dU xx x t D t n

dx

1/ 21 .

Hnn n H

dU xx x t D t n

dx

0.0 0.1 0.2 0.3 0.4 0.5

0.88

0.90

0.92

0.94

0.96

0.98

1.00 Simulation Analytics

<x2 >

st

H

0.0 0.1 0.2 0.3 0.4 0.5

0.88

0.90

0.92

0.94

0.96

0.98

1.00 Simulation Analytics

<x2 >

st

H-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3

-5

-4

-3

-2

-1

0

ln <

x2 >

ln t

D=1 D=0.5 D=0.25 D=0.1

t2H

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3

-5

-4

-3

-2

-1

0

ln <

x2 >

ln t

D=1 D=0.5 D=0.25 D=0.1

t2H

2 / 2n nU x x 2 / 2n nU x x

Mean Escape TimeMean Escape Time

1 2 3 4 5 6

0

50

100

150

200

250

300

350

400

Tes

c

1/D

H=0.5 H=0.45 H=0.4 H=0.35 H=0.3 H=0.25 H=0.2 H=0.1

1 2 3 4 5 6-2

-1

0

1

2

3

4

5

6

ln T

esc

1/D

H=0.5 H=0.4 H=0.3 H=0.2 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.

The same, but in a logarithmic scale. Now, the exponential behaviour is clearly

noticeable.

-2 -1 0 1 2

0.0

0.5

1.0

1.5

2.0

x0=1.4142

1 10 100 10001

10

100

1000

<x2 >

j

H=0.1 H=0.2 H=0.3 H=0.4 H=0.5

1 2 3 4 5 6

0

50

100

150

200

250

300

350

400

Tes

c

1/D

H=0.5 H=0.45 H=0.4 H=0.35 H=0.3 H=0.25 H=0.2 H=0.1

MET vs Hurst ExponentMET vs Hurst Exponent

0.1 0.2 0.3 0.4 0.5

-1

0

1

2

3

4

5

6

ln T

esc

H

D=1 D=0.5 D=0.25 D=0.167

0.1 0.2 0.3 0.4 0.50.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Simulation D=0.25 Linear Fit Polynomial Fit

ln T

esc

H

Dependence of mean escape time on Hurst exponent (anti-persistent case), four noise intensity values, a logarithmic plot.

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.

H

Exponential Behavior of METExponential Behavior of MET

0.1 0.2 0.3 0.4 0.50.84

0.86

0.88

0.90

0.92

0.94

0.96 A1 Parabolic fit

A1

H

esc 1 2

1lnT A H A H

D

0.1 0.2 0.3 0.4 0.5-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

A2

H

A2 Linear fit

2 3.01894 7.29582A H 21 0.70516 1.48947 2.28094A H H

PDF of ETsPDF of ETs

0 10 20 30 40 50 60 70 80 90

1E-3

0.01

0.1

p(t)

t

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

Probability density function of mean escape times as the function of walking time, logarithmic plot. The exponential behaviour is observed.

escesc

1exp /p t t T

T