Kramers’ method

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 foregoing treatment it is implicitly assumed that the system relaxes to the quasi-stationary distribution Pqs(x, t) on a time scale Tqs that is much shorter than the time scale ts = 1/ s characterizing the evolution to Ps(x). The following remarks on the Kramers method are of importance ... 

Numerical results have been given,173 174 based on the original model of Kramers, which reproduce the high-friction limit accurately. Recently, the Kramers method has been reformulated135,137,175 in order to emphasize the underlying assumptions, namely, the quasi-stationary (long-time) behavior of the system and the concept of a two-state system... 

This section is based mostly on the results presented in Ref. 78 and is arranged in the following way. In Section III.B.l we note mentioned the problem of superparamagnetic relaxation, which has been already tackled by means of the Kramers method, in the in Section II.A), and show how to obtain the analytical solution (in the form of asymptotic series) for the micromagnetic Fokker-Planck equation in the uniaxial case. In Section III.B.l the perturbative... 

B. The Bistable Model and Other Generalizations of the Kramers Method.397... 

Brinkman, Landauer and Swanson, and Doimelly and Roberts made important progress in extending Kramers method to models with several spatial dimensions. For the relatively simple models that were worked out, the major conclusions attained by Kramers do hold well. (A more detailed discussion of this point is given in the next section.)... 

Johnson, M. W., Curtiss, C.F. Potential flows of dilute polymer solutions by Kramers method. J. Chem. Phys. 51, 3023-3026 (1969). 

For an arbitrary continuous potential barrier, an approximate general expression for transition probability (barrier permeability) can be derived using the approach of ZWAAN-KEMBIE /60/, which is a generalization of the familiar BWK (BRILLOUIN-WENTZEL-KRAMERS) method/6l/. 

Mossoba, M.E.R. Dugan, and J.K.G. Kramer. Methods for Analysis of Conjugated Linoleic Acids and trans- %-. Isomers in Dairy Fats by Using a Combination of Gas Chromatography, Silver-Ion Thin-layer Chromatography/Gas Chromatography, and Silver-Ion Liquid Chromatography, ]. AOAClnt.. 87, 545—562 (2004). 

Kramers foimd the Eyiing-Polanyi equation (16) as a particular case of medium-small viscosity. It may seem impressive to see that the results of TST, based on quantum mechanics, come out as a p>articular case of Kramers pure classical method. But there are precise limitations to the use of Kramers method. 

Kramers method can be also extended to calculation of the probabilities, bnt not the times of exit into mnltiple absorbing states. For instance, the probability to exit through state starting from state S. is... 

The simplest situation is found when the bulk concentrations are kept constant by an appropriate stirring device (usually RDE), hence AC7 = 0. The AC are given by the solution of the set of ordinary linear first-order differential equations obtained by linearization of Eq. (2) under a sine wave potential perturbation IsE = A exp jtat. Resolution by the Kramers method immediately shows that AC /AE is expressed by a rational function of the imaginary angular frequency jar. 

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