First passage time analyses

A more tractable theory based on the probability that a reactant pair will react at a time t (pass from reactants to products) is that due to Szabo et al. [282]. If the survival probability of a geminate pair of reactants initially formed with separation r0 is p (r0, t) at time t, the average lifetime of the pair is /dr0 p(r0, t)t and this is longer for larger initial separation distances. It provides a convenient and approximate description of the rate at which a reactant pair can disappear, but it does so without the need of a full time-dependent solution of the appropriate equations. Nevertheless, as a means of comparing time-dependent theory and experiment in order to measure the value of unknown parameters, it cannot be regarded as satisfactory. 

By now the opinions of the author on the direction of future work should begin to have become apparent. Above all else, more detailed and careful experimental work is needed. Much progress has been made since the time of Noyes early studies on iodine atom recombination. There are so many holes in our understanding of experimental systems that many of the articles, and especially those on nanosecond or picosecond time-resolved studies of reaction rates which have been published so far, are important events for a kineticist It is to be hoped that the increasing interest in this experimental field will continue to grow. 

There is a great danger in making recommendations for future work on many counts, not least because a new experimental discovery or technique may completely invalidate or supercede the more conventional and current approaches in use. Nevertheless, rather than recapitulate the discussion of previous studies, it is probably of more interest and help to make a few suggestions about which directions on research into diffusion-limited reaction rates might be more fruitful. 

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I. Goychuk and P. Hanggi, Ion channel gating A first-passage time analysis of the Kramers type. Proc. Natl. Acad. Sci. USA 99(6) 3552-3556 (2002). 

Dohnal G. First-passage time analysis for Markovian deteriorating model. In Safety, Reliability and Risk Analysis, CRC Press (2008). 

The master equation is solved numerically from which the mean-first passage time is extracted. Analysis of the mean-first passage time indicates that even a moderate increase in the critical number Nc of the beta process leads to entropic slowdown of the dynamics [96]. Furthermore, the fragility of the system is controlled by the ratio of the critical number Nc to total number Ap of beta process [96]. 

We defer the proof to the appendix. The reason for presenting two methods for evaluating the mean first passage time is based on their diflierent scopes of applicability. If we were only interested in T, then equation (11.30) would be preferable because it requires less computation. However, we are limited to the first moment [8]. The advantage of the first approach is that we obtain any moment by one integration. Moreover, we have access to the time evolution of the escape process which allows for a more detailed analysis. 

We describe some models of the disastrous events spreading. This is a very important part of risk analysis. Despite the strong dependency of successive events, after some arrangements we can use Markov models for the description. It allows us to compute several characteristics of such system. When we consider a system of objects among them a disastrous event could spread, we can compute a probabdity distribution of absorbing states, first passage times for any of the objects and many others. This modeling can help us to make some preventive decision or to prepare disaster recovery plans. In the paper, the model is described and some computations are outlined. Keywords risk, safety, successive event, disastrous event, markov chain. 

Information flow is related to the mean first passage time (MFPT) concept used in Markov chain analysis. MFPT represents the mean time it takes for a signal released at a node to reach another node in the Markov chain. 

Passage times and distribution of passage times in recirculating systems were first considered by Shinnar et al. (64) in their analysis of RTD in closed-loop systems. The most important such system is that of blood circulation, but the analysis cited is also relevant to engineering systems such as fluidized-bed reactors. The main objective of this work was the analysis of tracer experiments in recirculating systems. The renewal theory discussed by Cox (65) served as the theoretical framework for their analysis. Both Shinnar et al. (64), and later Mann and Crosby (66) and Mann et al. (67) have shown that the NPD functions can be evaluated from the passage time distribution function, which in turn can be obtained from the renewal theory. 

A first approximation to the description of the full process is to assume that, by some mechanism independent of the fluctuations, the system remains uniform in space. In this "zero-dimensional" description, considerable attention was focussed on the decay from an initial unstable state, and on the passage times between simultaneously stable states [25-29]. Two characteristic scales emerge from this analysis. For the evolution around the unstable state, the time needed for the probability distribution to forget the initial condition and begin to develop peaks toward the stable attractors is... 

In a 500-mL. round-bottomed, four-necked glass reactor, fitted with an efllcient mechanical stirrer, thermometer. sintered gas inlet, and a condenser (cooled to — 40 C). was charged the benzenethiol 7 (0.2 mol), finely crushed NaOH (20 g, 0.5 mol). tris[2-(2-methoxyethoxy)ethyl]amine (TDA-I 3.2 g, 0.01 mol), and solvent (TCB or toluene. 200 mL). This mixture was vigorously stirred while chlorodifluoromethane (8) was bubbled through the solution. A noticeable exothermy was observed during the first few min. Then, the medium was heated up to the desired temperature (vide supra) and maintained there for the time indicated, whilst passage ofS was continued. The progress of the reaction was monitored by GC analysis. 

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