First passage-time distribution

Obtain explicitly the first-passage time distribution. Compare with (II.6.1). 

All quantities are functions of X. These two equations determine the first-passage time distributions fRm and fL m. 

Exercise. A particle moves in (L, R) with a velocity that at random moments jumps between two possible values c and —c. Show that its first-passage time distribution requires the solution of )... 

The effect of the mobile phase dispersion on the retention time can be handled in the same manner as the effect of the nonconstant number of adsorption-desorption events on the stationary phase time [8,9,98]. Felinger et at. modeled the mobile phase dispersion by a one-dimensional random walk and by the first passage time distribution arising from the random walk. When combined with the stochastic process of adsorption-desorption, this approach leads to a rather general stochastic representation of the chromatographic process. They obtained the following solution via the characteristic function method [9] ... 

Equation 6.120 gives the characteristic fimction of the band profile recorded with a destructive detector, such as the flame ionization detector because the mobile phase process is modeled with the first passage time distribution. In destructive detectors, the molecule is destroyed as soon as it enters the detector cell, therefore it is not possible that one molecule is detected twice due to backward diffusion. On the other hand, with UV detection in HPLC, a molecule might diffuse back to the detector cell, just after it has left the cell. This distinction, in theory, gives different band profiles for destructive and nondestructive detectors. In practice, however, the difference between the band profiles calculated by the two approaches is minuscule and experimentally carmot be measured. The band profile in the case of a nondestructive detector is obtained if not the first-passage distribution but the probability distribution of a diffusing molecule is used to describe the mobile phase process. 

ShaUoway, D. and A.K. Faradjian, Efficient computation of the first passage time distribution of the generalized master equation by steady-state relaxation. Journal of Chemical Physics, 2006,124(5) 054112. 

It should be noted that CARPT experiments in the gas—soHd riser produced for the first time the definitive soHds residence time distribution in the riser itself (Fig. 1.12). Precise monitoring of the time when the tracer particle enters the system across the inlet plane, and the time when it exits across either inlet or exit plane, provides its actual residence time in the riser. Ensemble averaging for several thousand particle visits yields the solids RTD. The first passage time distribution is also readily be obtained. This information cannot be obtained by measuring the response at the top of the riser to an impulse injection of tracer at the bottom. By using CARPT, true descriptions ofsoHds residence time distributions can be obtained in the riser (Bhusarapu et al., 2004, 2006). One task of CFD modelers is to develop codes that can predict the experimental observations of CARPT. Here, it... 

Sherif YS. Smith L.M. 1980. First-Passage Time Distribution of Brownian Motion as a Reliability Model. IEEE Transaction on Reliability 5(2) 425-426. 

For the solution of real tasks, depending on the concrete setup of the problem, either the forward or the backward Kolmogorov equation may be used. If the one-dimensional probability density with known initial distribution deserves needs to be determined, then it is natural to use the forward Kolmogorov equation. Contrariwise, if it is necessary to calculate the distribution of the mean first passage time as a function of initial state xo, then one should use the backward Kolmogorov equation. Let us now focus at the time on Eq. (2.6) as much widely used than (2.7) and discuss boundary conditions and methods of solution of this equation. 

When the initial probability distribution is not a delta function, but some arbitrary function Wo(xo) where xq C (c,d), then it is possible to calculate moments of the first passage time, averaged over initial probability distribution ... 

Probably, a similar procedure was previously used (see Refs. 1 and 93-95) for summation of the set of moments of the first passage time, when exponential distribution of the first passage time probability density was demonstrated for the case of a high potential barrier in comparison with noise intensity. 

The moments of transition time of a dynamical system driven by noise, described by arbitrary potential cp(x) such that cp( oo) = oo, symmetric relatively to some point x = d, with initial delta-shaped distribution, located at the point xo < d [Fig. A 1(a)], coincides with the corresponding moments of the first passage time for the same potential, having an absorbing boundary at the point of symmetry of the original potential profile [Fig. A 1(b)]. 

In a realistic simulation, one initiates trajectories from the reactant well, which are thermally distributed and follows the evolution in time of the population. If the phenomenological master equations are correct, then one may readily extract the rate constants from this time evolution. This procedure has been implemented successfully for example, in Refs. 93,94. Alternatively, one can compute the mean first passage time for all trajectories initiated at reactants and thus obtain the rate, cf. Ref 95. 

If P(t) is the distribution of first passage times for transitions past level N, the number of molecules which pass N in the interval (t, t + 6t) is P(t) 8t. Then... 

The distribution of first passage times can be found by summing these equations to give... 

The standard theories of chemical kinetics are equilibrium theories in which a Maxwell-Boltzmann distribution of reactants is postulated to persist during a reaction.68 The equilibrium theory first passage time is the TV -> oo limit in Eq. (6), Corrections to it then are to be expected when the second term in this equation is no longer negligible, i.e., when N is not much greater than e — e- )-1. The mean first passage time and rate of activation deviate from their equilibrium value by more than 10% when... 

One may ask the following question. Suppose the random walker starts out at site m at t = 0 how long does it take him to reach a given site R for the first time This first-passage time is, of course, different for the different realizations of his walk and is therefore a random quantity. Our purpose is to find its probability distribution, and in particular the average or mean first-passage time ]... 

The quantities 7r, t, considered so far are the first moments of the probability distribution /(t) of the first-passage time. Specifically, for a one-step process there are two distributions fRttn(t) and /L>m(t) for the probabilities to arrive at R and L at a time t after starting out at site m. We derive an equation for them. By a similar argument as used above one obtains... 

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. 

The relative Brownian motion between the constituents of doublets consisting of sufficiently small equal-size aerosol particles is described by a one-dimensional Fokker-Planck equation in the particle energy space. A first passage time approach is employed for the calculation of the average lifetime of the doublets. This calculation is based on the assumption that the initial distribution of tire energy of the relative motion of the constituent particles is Maxwellian. The average dissociation time of doublets, in air at 1 atm and 298 K, for a Hamaker constant of 10 12 erg has been calculated for different sizes of the constituent particles. The calculations are found to be consistent with the assumption that the... 

Passage time distributions and the mean first passage time... 

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