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 )... 

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 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. 

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

We have already noted the difference between the Langevin description of stochastic processes in terms of the stochastic variables, and the master or Fokker-Planck equations that focus on their probabilities. Still, these descriptions are equivalent to each other when applied to the same process and variables. It should be possible to extract information on the dynamics of stochastic variables from the time evolution of their probabihty distribution, for example, the Fokker-Planck equation. Here we show that this is indeed so by addressing the passage time distribution associated with a given stochastic process. In particular we will see (problem 14.3) that the first moment of this distribution, the mean first passage time, is very useful for calculating rates. 

Equation (8.152) is an expression for the passage time distribution n(xi, Z xo)-The mean first passage time t(xi,xo) is its first moment... 

Passage time distributions and the mean first passage time operator. Applying it to (8.154) leads to... 

Passage time distributions and the mean first passage time provide a useful way for analyzing the time evolution of stochastic processes. An application to chemical reactions dominated by barrier crossing is given in Section 14.4.2 and Problem 14.3. 

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. 

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