Escape time

Without implying anything about the shape of the tube, it is clear that it has the same length as the polymer chain itself, that is, nlo. This is the displacement of interest in calculating the escape time from the tube. Therefore Eq. (2.63) becomes... 

Figure 7-76. Maximum radiation intensity vs. escape time based on 5 seconds reaction time. By permission, Kent, Hydrocarbon Processing, V. 43, No. 8 (1964), p. 121 [60].

In that pioneering study28 it was not possible to obtain a time resolved distribution of the backspillover oxygen species despite the fast, 40 ms, time resolution of the video-frames. If the spillover distance is 100 pm, this implies surface spillover oxygen diffusivities as high as 10 3 cm2/s. If, however, microcracks exist in the film, which is very likely, then the spillover distance is much shorter and thus much lower diffusivities would suffice to escape time-dependent detection. 

On the other hand, both activity ratios decrease for increasing values of the escape time 0, (2i0Bi/2i0pb)G ratios being more affected than ( °Po/ °Pb)G ratios because of the very short half-life of i (T = 5.01 d). 

Initially, an overdamped Brownian particle is located in the potential minimum, say somewhere between x and X2- Subjected to noise perturbations, the Brownian particle will, after some time, escape over the potential barrier of the height AT. It is necessary to obtain the mean decay time of metastable state [inverse of the mean decay time (escape time) is called the escape rate]. 

To calculate the mean escape time over a potential barrier, let us apply the Fokker-Planck equation, which, for a constant diffusion coefficient D = 2kT/h, may be also presented in the form... 

Let us consider the case when the diffusion coefficient is small, or, more precisely, when the barrier height A is much larger than kT. As it turns out, one can obtain an analytic expression for the mean escape time in this limiting case, since then the probability current G over the barrier top near xmax is very small, so the probability density W(x,t) almost does not vary in time, representing quasi-stationary distribution. For this quasi-stationary state the small probability current G must be approximately independent of coordinate x and can be presented in the form... 

The escape time is introduced as the probability P divided by the probability current G. Then, using (3.4) and (3.6), we can obtain the following expression for the escape time ... 

We may then extend the integration boundaries in both integrals to oo and thus obtain the well-known Kramers escape time ... 

Influence of the shape of potential well and barrier on escape times was studied in detail in paper by Agudov and Malakhov [49]. 

So, if one will compare the temperature dependence of the experimentally obtained escape times of some unknown system with the temperature dependence of Kramers time presented in Table I, one can make conclusions about potential profile that describes the system. 

Let us calculate the relaxation time of particles in this potential (escape time over a barrier) which agrees with inverse of the lowest nonvanishing eigenvalue Yj. Using the method of eigenfunction analysis as presented in detail in Refs. 2, 15, 17, and 18 we search for the solution of the Fokker-Planck equation in the... 

For a small noise intensity, the double integral may be evaluated analytically and finally we get the following expression for the escape time (inverse of the eigenvalue yj) of the considered bistable potential ... 

The obtained escape time iy for the bistable potential is two times smaller than the Kramers time (3.10) Because we have considered transition over the barrier top x = 0, we have obtained only a half. 

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

Hence, according to (5.107), taking into account that c < x < d, we finally arrive at the exact expression of the escape time of the Brownian particles from decision interval [c, d] for an arbitrary potential profile [Pg.404]

It is very interesting to note that in this case the factor e2P arises in the escape time instead of the Kramers factor associated with the good possibility for the Brownian particles to diffuse back to the potential well from a flat part of the potential profile, resulting in strong increasing of the escape time from the well (see, e.g., Ref. 83). 

Use the NIOSH web site to determine an escape time period for a person subjected to an IDLH concentration. 

Charge neutralization in media of low viscosity. Secondary reactions including intertrack reactions. Electron escape time in low-viscosity media. Intratrack reactions completed. Secondary radical formation and reaction. 

Type (iv). Consider a particle starting at y e (M, R). If we impose the reflecting boundary condition at R the probability for it to leave through M is unity and the mean escape time is, according to (3.7),... 

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