Kramers approximation

Fig. 2.6. Schematic representation of a reaction treated in the Kramers approximation. The shape of the probability density distribution is assumed to have reached equilibrium (i.e., time independence) at the bottom of the reactant valley. Only the weight of P x. t) (total number of reactants) diminishes by activated diffusion across the barrier.

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. 

Note that with respect to the usual adiabatic analysis we have ad hoc substituted the approximate Kramers time by the exact one, Eq. (6.16), and found a surprisingly good agreement of this approximate expression with the computer simulation results in a rather broad range of parameters.]... 

This is a harmonic mean because it is really the mean free path that is relevant. At high temperatures (> 105 K), the main sources of opacity and approximate formulae for them (the first two originally due to H. Kramers) are ... 

Stacey, J.S. and Kramers, J.D. (1975). Approximation of terrestrial lead isotope evolution by a two-stage model. Earth and Planetary Science Letters 26 207-221. 

The process of formation of a bubble having a critical radius, can be computed using a semiclassical approximation. The procedure is rather straightforward. First one computes, using the well known Wentzel-Kramers-Brillouin (WKB) approximation, the ground state energy Eq and the oscillation frequency //() of the virtual QM drop in the potential well U JV). Then it is possible to calculate in a relativistic framework the probability of tunneling as (Iida Sato 1997)... 

The problem of the electron spin relaxation in the early work from Sharp and co-workers (109 114) (and in some of its more recent continuation (115,116)) was treated only approximately. They basically assume that, for integer spin systems, there is a single decay time constant for the electron spin components, while two such time constants are required for the S = 3/2 with two Kramers doublets (116). We shall return to some new ideas presented in the more recent work from Sharp s group below. 

Aiba (A3), Fox and Gex (F8), Kramers, Baars and Knoll (K15), Metzner and Taylor (MIO), Norwood and Metzner (N3), Van de Vusse (V5) and Wood et al. (W12) have studied flow patterns and mixing times. In addition, Brothman et al. (B22), Gutoff (G9), Sinclair (S16) and Weber (W3) analyzed flow in a stirred tank in terms of the recycle flow model of Fig. 23F. This model corresponds to the draft-tube reactor, and with sufficiently large recycle rate the performance prediction of this model approximates backmix flow. 

Field emission is a tunneling phenomenon in solids and is quantitatively explained by quantum mechanics. Also, field emission is often used as an auxiliary technique in STM experiments (see Part II). Furthermore, field-emission spectroscopy, as a vacuum-tunneling spectroscopy method (Plummer et al., 1975a), provides information about the electronic states of the tunneling tip. Details will be discussed in Chapter 4. For an understanding of the field-emission phenomenon, the article of Good and Muller (1956) in Handhuch der Physik is still useful. The following is a simplified analysis of the field-emission phenomenon based on a semiclassical method, or the Wentzel-Kramers-Brillouin (WKB) approximation (see Landau and Lifshitz, 1977). 

Kramers, Grote and Hynes and Hanggi and Mojtabai showed that if one assumes that the spatial diffusion across the top of the barrier is the rate limiting step, then by approximating the barrier as being parabolic with frequency co one finds (see also Eq. 7) that the rate is given by the expression... 

Wentzel Kramer-Brillouin approximation, tunneling, 1492 Whewefi, Reverend, 1050 Whiskers, 1327, 1336 White, and organic adsorption. 979 Wieckowski, A., 1146 Will, 1205... 

This is called the Kramers-Moyal expansion. ) Formally (2.6) is identical with the master equation itself and is therefore not easier to deal with, but it suggests that one may break off after a suitable number of terms. The Fokker-Planck approximation assumes that all terms after v = 2 are negligible. Kolmogorov s proof is based on the assumption that av = 0 for v>2. This, however, is never true in physical systems In the next chapter we shall therefore expand the M-equation systematically in powers of a small parameter and find that the successive orders do not simply correspond to the successive terms in the Kramers-Moyal expansion. 

Exercise. Write the Kramers-Moyal expansion for one-step processes using (5.2). Exercise. Construct the Fokker-Planck approximation for the M-equation (VI.9.12) and use it to find [Pg.209]

We shall now solve the Kramers equation (7.4) approximately for large y by means of a systematic expansion in powers of y-1. Straightforward perturbation theory is not possible because the time derivative occurs among the small terms. This makes it a problem of singular perturbation theory, but the way to handle it can be learned from the solution method invented by Hilbert and by Chapman and Enskog for the Boltzmann equation.To simplify the writing I eliminate the coefficient kT/M by rescaling the variables,... 

Incidentally, suppose one replaces the M-equation (3.4) by the naive Fokker-Planck approximation (VIII.5.3), obtained by breaking off the Kramers-Moyal expansion after the second term rather than by the systematic expansion of chapter X. This cannot be correct for small n and cannot therefore reproduce the evolution starting from small initial m. It is therefore not paradoxical that the absorbing site n = 0 does not translate into an absorbing boundary condition of the Fokker-Planck equation - as remarked in an Exercise of XII.5. 

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. 

Thus we have derived the Kramers equation (VIII.7.4) as an approximation for short tc. It becomes exact in the white noise limit (3.12). The coefficient of the fluctuation term is the integrated autocorrelation function of the fluctuating force, in agreement with (IX.3.5) and (IX.3.6).110... 

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