Escape time calculations, probability

Probably the best way of assessing fire hazard is by calculations via mathematical fire growth and transport models, such as HAZARD I [58], FAST [59], HARVARD [60] or OSU [61]. These models predict times to reach untenable situations. They are often combined with fire escape models and will, then, yield times to escape. 

In the preceding part of this section, we have concentrated on the electron escape probability, which is an important quantity in the geminate phase of recombination, and can be experimentally observed. However, modern experimental techniques also give us a possibility to observe the time-resolved kinetics of geminate recombination in some systems. Theoretically, the decay of the geminate ion pairs can be described by the pair survival probability, W t), defined by Eq. (4). One method of calculating W t) is to solve the Smoluchowski equation [Eq. (2)] for w r,t) and, then, to integrate the solution over the space variable. Another method [4] is to directly solve Eq. (7) under relevant conditions. 

Figure 2 Survival probability of geminate ion pairs as a function of time. The two solid lines correspond to two different values of the initial electron-cation distance. The broken lines show the asymptotic kinetics calculated from Eq. (25). The value of the escape probability for Tq = O.Sr is indicated by

Figure 3 Escape probability as a function of the initial electron-cation distance. The lower broken curve is calculated from the Onsager equation [Eq. (15)]. The numerical results for different mean free times x were taken from Ref. [22]. The unit of x is rJ(ksTlmf, where m is the electron mass. The upper broken curve was calculated using the energy diffusion model. (From Ref. 23.)...

Computer simulation has also been used to calculate the external electric field effect on the geminate recombination in high-mobility systems [22]. For the mean free time x exceeding -0.05, the field dependence of the escape probability was found to significantly deviate from that obtained from the diffusion theory. Furthermore, the slope-to-intercept ratio of the field dependence of the escape probability was found to decrease with increasing x. Unlike in the diffusion-controlled geminate recombination, this ratio is no longer independent of the initial electron-ion separation [cf. Eq. (24)]. 

With this realization we may evaluate the rate by considering an artificial situation in which A is maintained strictly constant (so the quasi-steady-state is replaced by a true one) by imposing a source inside the well and a sink outside it. This source does not have to be described in detail We simply impose the condition that the population (or probability) inside the well, far from the barrier region, is fixed, while outside the well we impose the condition that it is zero. Under such conditions the system will approach, at long time, a steady state in which dP/dt = 0 but J 0. The desired rate k is then given by J/A. We will use this strategy to calculate the escape rate associated with Eq. (14.44). 

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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