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). Modeling of some chemical reactions Modeling of some chemical reactions The electroconductivity theory of crystals The electroconductivity theory of crystals Nucleation theories Nucleation theories Climatic dynamics etc. 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; All this leads to the problem with quasi-stationary conditions Н

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,

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...

PART I  stable Levy Motion

Levy Probability Distribution Law

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 Let us examine the strong friction case, when : Or, in dimensionless variables after the time quantization: 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, ξ α (t) is a random force, D is its intensity, α is the Levy index.

Motion in the Potential Let us study the double-well potential The time-discrete Langevin equation:

The First Passage Time We place the particle to the left potential’s minimum (x 0 = - 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 times to gain the statistics, then the time is averaged.

The First Passage Time continued.

The First Passage Time continued..

The Mean First Passage Time (other potentials)

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 times to gain the statistics, then the times are treated with a procedure that extracts the PDF of the data.

The FP time PDF

The FP time PDFs (other potentials) Cubic potentialHarmonic potential

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 Levy noise Gaussian-like noiseLarge “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.

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 application

A Problem from Climatic Dynamics continued. Levy noise, α=1, year fluctuations Levy noise, α=1, year 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– 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, , p ‑ 1– ‑ A.V. Chechkin, O.Yu. Sliusarenko Generalized Kramers’ problem for Lévy particle. Problems of Atomic Science and Technology, 2007, № 3(2), p. 293– th Wilhelm und Else Heraeus-Seminar, Anomalous Transport: Experimental Results and Theoretical Challenges, Bad Honnef, Germany, July 12-16, nd International conference on Quantum electrodynamics and statistical physics (QEDSP2006), Kharkiv, Ukraine, September 19-23, Physics of Fluctuations far from Equilibrium, Dresden, Germany, July 02-06, 2007 Publications: Conferences:

PART II Fractional Brownian Motion.

Fractional Gaussian Noise noise H=0.3 H=0.7 free motion H=0.3 H=0.7

Simulation Procedure

Mean Escape Time 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.

MET vs Hurst Exponent 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.

Exponential Behavior of MET

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