Kramers escape rate

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... 

Therefore, in the present context, the Kramers escape rate can be best understood as playing the role of a decay parameter in the Mittag-Leffler functions governing the highly nonexponential relaxation behavior of the system. 

Equation (156) illustrates how anomalous relaxation influences the complex susceptibility arising from the slowest relaxation mode in the noninertial limit. Therefore, in anomalous relaxation the Kramers escape rate can be best understood as playing the role of a decay parameter in the Mittag-Leffler function governing the relaxation behavior of the system. We remark that the slowest decay mode, the parameters of which are determined by Eq. (156), will... 

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]. 

It should be stressed that for the double-well reaction model in the non-Markovian case a general result similar to the Kramers expression (4.160) cannot be found. To evaluate the thermally activated escape rate, the motion within the barrier region is described by means of a GLE in which the potential near the barrier is linearized, that is,... 

The Need for Generalization of the Kramers Theory The Generalized Kramers Model Non-Markovian Effects in the One-Dimensional Case The Escape Rate of a Non-Markov Multidimensional Process... 

The objective is to find the steady-state escape rate k out of the potential well. Before presenting the Kramers solution it is important to note that for such a (quasi) steady state to be established, a clear separation of time scales has to exist, whereupon the escape occurs on a time scale much longer than all time scales associated with the motion inside the well. In particular this implies that the well should be deep enough (see below). 

When y is very small, the thermal relaxation in the well is not fast relative to the escape rate, and the assumption that the distribution within the well can be represented by the equilibrium Boltzmann distribution no longer holds. On the other hand we can make use of the fact that the total energy E varies on a time scale much longer than either x or u (it is conserved for y = 0). Thus changing variables from (x, v) to ( , ) and eliminating the fast phase variable (f> leads to a Smoluchowski (diffusion) equation for E. [Kramers gave the equivalent equation in terms of the action variable J( ).]... 

The same equation can be derived within a simple Kramers picture [55,80,86] for the escape from a well (locked state), assuming that the pulling force produces a small constant potential bias that reduces a height of a potential barrier. The progressive increase of the force results in a corresponding increase of the escape rate that leads to a creep motion of the atom. However, this behavior is different from what occurs when the atom (or an AFM tip) is driven across the surface and the potential bias is continuously ramped up as the support is moved [87,88]. The consequences of this effect will be discussed in more detail in Section III.B. 2. 

This is the Kramers low friction result for the escape rate k. As anticipated, the rate in this limit is proportional to the friction y which determines the efficiency of energy accumulation and loss in the well. 

At t=T the escape rate begins to depart from the above reported law, tending to a temperature-independent quantum transition rate (Fig. G.l), which has the same form, in the usual WKB (Wentzel-Kramers-Brillouin) approximation for quantum tunneling, as in the case of switching of magnetization via thermal activation ... 

Rg is proportional to y, and this implies that Rg has a maximum at some intermediate y With some simplifying harmonic assumptions about V(q), Kramers calculates the low friction escape rate... 

Regard the energy landscape pictured in Fig. 3.3 and suppose that the protein is initially trapped in the basin of attraction surrounding state A. According to Kramers theory (Van Kampen 1992 Risken 1996), the protein state will eventually escape from this basin due to thermal perturbations, and the escape rate... 

It should be noted that dynamical equations of the trap model do not lead to a Boltzmann distribution at T since the equations do not obey detailed balance and, specifically, the traps are sampled from a quenched distribution. The escape rate from a trap is, however, determined by the Kramers process [29]. 

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. 

It is also important to note that for a system coupled linearly to a finite discrete set of harmonic oscillators, the rate of escape over the barrier is described by the TST [150] and the TST rate is exactly given by Eq. (320). In the limit of continuum of oscillators, the Kramers-Grote-Hynes result is regained [Eq. (320)] [150]. 

Instead of the quasi-stationary state assumption of Kramers, he assumed only that the density of particles in the vicinity of the top of the barrier was essentially constant. Visscher included in the Foldcer-Planck equation a source term to accoimt for the injection of particles so as to compensate those escaping and evaluated the rate constant in the extreme low-friction limit. Blomberg considered a symmetric, piecewise parabolic bistable potratial and obtained a partial solution of the Fokker-Hanck equation in terms of tabulated functions by requiring this piecewise analytical solution to be continuous, the rate constant is obtained. The result differs from that of Kramers only when the potential has a sharp, nonharmonic barrier. 

Note that Eq. (4.24) provides the same stationary second moment as Eq. (4.19). Equation (4.24), via the well-known Kramers approacl " (see also Fonseca et al., Chapter IX) leads us to the rate of escape from the well ... 

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