Kramers’ escape problem

Secondly, Kramers took for WVfl a differential operator, which led to the celebrated Kramers escape problem, treated in XIII.6. 

This is Kramers escape problem. Since no analytic solution is known for any metastable potential of the shape in fig. 40 the quest is for suitable approximation methods. This problem has received an extraordinary amount of attention from physicists, chemists and mathematicians.5 0 We describe the main features - all present already in the seminal paper by Kramers. 

E. The Fractional Omstein-Uhlenbeck Process IV. The Fractional Kramers Escape Problem... 

B. The Fractional Generalization of the Kramers Escape Problem Mittag-Leffler Decay of the Survival Probability... 

The first paper that was devoted to the escape problem in the context of the kinetics of chemical reactions and that presented approximate, but complete, analytic results was the paper by Kramers [11]. Kramers considered the mechanism of the transition process as noise-assisted reaction and used the Fokker-Planck equation for the probability density of Brownian particles to obtain several approximate expressions for the desired transition rates. The main approach of the Kramers method is the assumption that the probability current over a potential barrier is small and thus constant. This condition is valid only if a potential barrier is sufficiently high in comparison with the noise intensity. For obtaining exact timescales and probability densities, it is necessary to solve the Fokker-Planck equation, which is the main difficulty of the problem of investigating diffusion transition processes. 

In the high barrier limit in this particular problem the inverse escape time is the Kramers escape rate. 

Coffey WT, Crothers DSF, Dormann JL, Geoghegan LJ, Kennedy EC, Wemsdorfer W (1998a) Range of validity of Kramers escape rates for non-axially symmetric problems in superparamagnetic relaxation. JPhys Condensed Matter 10 9093-9109... 

The original work of Kramers [11] stimulated research devoted to calculation of escape rates in different systems driven by noise. Now the problem of calculating escape rates is known as Kramers problem [1,47]. 

In the standard overdamped version of the Kramers problem, the escape of a particle subject to a Gaussian white noise over a potential barrier is considered in the limit of low diffusivity—that is, where the barrier height AV is large in comparison to the diffusion constant K [14] (compare Fig.6). Then, the probability current over the potential barrier top near xmax is small, and the time change of the pdf is equally small. In this quasi-stationary situation, the probability current is approximately position independent. The temporal decay of the probability to find the particle within the potential well is then given by the exponential function [14, 22]... 

While Kramers has considered escape of a particle from a single well, many processes involve transitions between locally stable states of a double well potential (Fig. 2). Dissociation and desorption are examples of single well problems unimolecular isomerization is a double well problem. Many-well problems are also of interest, such as diffusion of atoms or ions in solids. 

The conditions are such that the particle is originally in a potential hole, but it may escape in the course of time by passing over a potential barrier. The analytical problem is to calculate the escape probability as a function of the temperature and of the viscosity of the medium, and then to compare the values so found with the ones of the activated state method. For sake of simplicity, Kramers studied only the one-dimensional model, and the calculation rests on the equation of diffusion obeyed by a density distribution of particles in the. phase space. Definite results can be obtained in the limiting cases of small and large viscosity, and in both cases there is a close analogy with the Cristiansen treatment of chemical reactions as a diffusion problem. When the potential barrier corresponds to a rather smooth maximum, a reliable solution is obtained for any value of the viscosity, and, within a large range of values of the viscosity, the escape probability happens to be practically equal to that computed by the activated state method. 

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