First passage times, theory

There have been many other theoretical analyses of geminate radical recombination probabilities, some of which are considered further below. They can be divided into three types (a) diffusion equation treatments, (b) first passage time methods, and (c) kinetic theory applications. 

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... 

Exercise. The second factor is dominated by the terms near v = b. They are of the order (p ) 1, which is the Arrhenius factor. In the theory of molecular dissociation this factor has suggested the idea of a transition complex , i.e., an imagined intermediate molecule corresponding to the unstable state b. The reaction from a to c is then visualized as two successive steps a -> b and, subsequently, b -> c. The first step determines the overall rate. This has led some authors to identify the reaction rate with the first-passage time xba. Show that this is only half the correct amount. 

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. 

V. General Theory of Mean First Passage Time.377... 

The discussion of the last section can be generalized to include the possibility of a chemical reaction.18 Consider the case in which the achievement of the (N + l)st level represents the completion of the reaction and in which the reaction occurs only by a molecule passing into the (N+l)st level. Any molecule which reaches this level is absorbed or "dies. The reaction rate is determined by the rate at which molecules in their "random walk from level to level reach the (IV-f-l)st level for the first time. In the language of the theory of stochastic processes the mean time for level (IV 4-1) to be reached is the mean first passage time for the IVth level (the time required to pass N for the first time). 

Fig. 4. The ratio of the equilibrium theory first passage time to exact first passage time as a function of number of levels for various values 6t hv/kT.

Kramers approach to rate theory in the underdamped and spatial-diffusion-limited regimes spurred extensions which were applicable to the much more complex STGLE. Grote and Hynes (23) used a parabolic barrier approximation to derive the rate expression for the GLE in the spatial diffusion limit. Carmeli and Nitzan derived expressions for the rate of the GLE (24) and the STGLE (25) in the underdamped limit. The overdamped limit for the rate in the presence of delta correlated friction was solved using the mean first passage time expression (26,27). A turnover theory, valid for space- and time-dependent friction, has only been recently presented by Haynes, Voth, and Poliak... 

The equilibrium theory first passage time is applicable in limit as N ao. Hence corrections to it are to be expected when the second term in the product above is not negligible, i.e., when N is not much greater than The mean first passage time and... 

Vega, J.L., Guantes, R., Miret-Artes, S. Mean first passage time and the Kramers turnover theory in activated atom-surface diffusion, Phys. Chem. Chem. Phys. 2002,4,4985. 

The theory of first passage times enables one to take a master equation like eq. (15.12) that describes the evolution of a probability distribution P t) for a population, and derive from it an equation for the probability distribution of times T t) for reaching a population m at time t when we start out with n at time zero. For the Szabo model of eq. (15.12), we find... 

The theory of first passage times has also been employed by us recently in order to introduce a new variant of the method of photobleaching to measure diffusion in membranes. This variant [R. Peters et al., Proc. Natl. Acad. Sci. USA 78, 62 (1981)] employs continuous radiation at an intermediate light level and allows one to monitor in living cells the lateral transport of membrane constituents of very small concentrations, e.g. cell surface receptors. 

The organization of this article is as follows in the next section we first introduce physical principles of excitation transfer based on Fdrster theory. In section 3 the average excitation lifetime and quantum yield are defined in terms of excitation transfer rates. In section 4, representative pathways of excitation migration are described in terms of mean first passage times to a reaction center. In section 5, an expansion method for excitation migration in terms repeated trapping and detrapping events is introduced. In section 6, some measures of robustness and optimality of a... 

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. 

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