Time-dependent survival probability

Yet unless very detailed information is available to describe the initial distribution of separations, p(r, 0), it will not be possible to use measured time-dependent survival probabilities to probe details of dynamic liquid structure. Currently, experimental uncertainties at 30% are so large that such a probe is not possible, since the effects of the short-range caging region are only 30%, at the most, of the rate coefficient or escape probability. 

Figure 28. Simulated time-dependent survival probability of the LE form calculated from Eq, (42) and p(z,t) shown in Figure 27.

Figure 6.7 Time-dependent survival probability, Pij(t), calculated without (solid line) and with (dotted line) the intrashell interaction matrix elements, within the EDA. The initial hydrogen state. Is, interacts with a laser pulse of = (1 — l/n gJ/2, ripes = 85, of F = 5 x 10 a.u. and of duration 50 ps (FWHM = 10 ps).

The dangerous cuts of a random sequence of particular members correspond to extreme loading situations ofstructures. Instantaneous survival probabilities at these sequence cuts form series systems. The time-dependant survival probability of particular members as series systems may be assessed by Monte Carlo simulation and numerical integration methods. However, it is more reasonable to use imsophisticated, simplified but quite exact method of transformed conditional probabilities (Kudzys 2007). The resistance of non deteriorating members may be treated as stationary process. Thus, their long-term survival probability may be calculated by the Equation ... 

ABSTRACT The implantation of simple engineering techniques in time-dependent survival probability predictions of structural and technical systems as stochastic systems of events is discussed. A possibility to avoid complicated multidimensional integrations in a probabilistic safety analysis of stochastic systems with perfectly ductile components is based on the approaches of Transformed Conditional Probabilities (TCP) and Conventional Correlation Vectors (CCV). Dynamic (time-dependent) autosystems of extreme events of particular single and mixed ductile elements are treated as correlated components of static stochastic systems with m random failure modes. The unsophisticated and fairly exact prediction of the probability-based reliability of stochastic systems of events is demonstrated by numerical examples and histograms of their reliability indices. 

Fig. 5 Time-dependent survival probability for an electronic wavepacket initially occupying the LUMO of ligand BPl. Time-propagations performed by the Chebyshev (solid line) and combined AO/MO (circles) methods. The inset shows the [Ru(bpy)3] " chemical structure.

From Sect. 2.4.1 the time dependent survival probability for a neutral pair is known to be... 

A complete solution of eqn. (151) for the time-dependent density distribution does not appear feasible, but Hong and Noolandi [323—325] have found the long-time behaviour, as well as the steady-state solution. The mathematics are very complex since the complications encountered in the analysis of the Debye—Smoluchowski equation (Appendix A) are compounded by the applied electric field. For small electric fields and long times, the survival probability is approximately... 

The result of a molecular dynamics simulation is a time dependent wavefunction (quantum dynamics) or a swarm of trajectories in a phase space (classical dynamics). To analyze what are the processes taking the place during photodissoeiation one can directly look at these. This analysis is, however, impractical for systems with a high dimensionality. We can calculate either (juantities in the time domain or in the energy domain, fn the time domain survival probabilities can be measured by pump-probe experiments [44], in the energy domain the distribution of the hydrogen kinetic energy can be experimentally obtained [8]. 

In addition to the survival probability of the initial state, we will also be interested in time-dependent transition probabilities. The transition probability at time t between the time-evolving state tj/(r)) an( another zero-order state ) is the absolute square of the transition amplitude... 

Much of the research on reaction rates between ion-pairs has been specifically aimed at probing the initial separation of ions after they are formed. As discussed in Chap. 7, Sect. 3.2 and Sect. 2.2 of this chapter, the initial distribution of ions pairs (etc.) is probably the major uncertainty in any analysis of the reaction rate. Indeed, the cart is put before the horse and reaction rates, survival probabilities, etc. are used to calculate the initial distribution Despite very detailed and careful work (see Chap. 7, Sect. 3), there is extensive disagreement amongst various authors about which functional form best fits experimental data this difference of opinion often applies to similar experiments on the same solvents. What is probably safe to say is that the initial distribution deduced from an analysis of time or electric field-dependent survival probabilities is very sensitive to the experimental conditions and purity of the solvent. Despite the fact that an applied electric field may quite typically vary from 0.1 to... 

Taking into account that neutralization means tunneling of a target conduction-band electron to the ion, the time integral can easily be replaced by integration over the distance from the surface, s, by use of the identity dt = ds/v , where Vj is the component of the ion velocity perpendicular to the surface. Prom this, the velocity-dependence of the survival probability, P , is obtained ... 

The long-time survival probability for the entire track approaches the limit P(t)/ = 1 + 0.6x-0 6, where T is a normalized time, much earlier than the P(r)/Pesc = 1 + (7TT)-0-5 predicted by the free diffusion theory (Bartczak and Hummel,1997). Notice that the T 06 dependence of the existence probability had been established eariler in the experiment of van den Ende et al.(1984). 

Recently, experiments have been reported where the time dependence of the radical survival probability has been measured. Not only is the (long time) escape or recombination probability measured, but also the time scale over which the initial concentration of radicals decays to the final radical concentration has been noted [266—68]. Such studies provide extremely valuable additional information, because the time scale for reaction is the time scale it takes for the radicals to diffuse together again and hence these experiments give some insight into the distribution of initial separation distances. For instance, radicals separated by r0 1 nm take rl/6D 0.16 ns to diffuse together in a solvent of diffusion coefficient 10 9 m2 s-1. Once the theory of radical recombination has been discussed in the remainder of this section, these time-dependent studies will be reconsidered in Sect. 3. 

Figure 18 shows the time dependence of this probability for reaction between spherical reactants with R = 0.5 nm for two cases, i.e. where D = 2 x 10-9 m2 s-1, r0 = 0.65 nm and D = 10-9 m2 s 1, r0 = 0.525 nm. In the former case, not only is escape more probable, but also the time scale over which the survival probability relaxes to a nearly constant value is longer. 

The observed survival probability depends not only on the survival probability of an ion-pair formed with an initial separation, r0, but also on the distribution of initial separations, w(r0). As will be discussed in Sects. 3 and 4, the rate of loss of energy of ions after heterolytic bond fission (or, alternatively, the range of electrons formed by ionisation of a molecule) depends sensitively on the energetics of the solvent molecules. These separation distances can range up to 10 nm or more and are specifically discussed in Sect. 3. The survival probability of a collection of isolated ion-pairs, P(t), formed at a time t0 — 0, and with a distribution of initial distances w(r0) is... 

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. 

The N terms in the diffusion operator (Sano and Baird s [504] M operator) can be retained because Piv-i.jw-i does not retain any dependence on rfe, so the term in Vft, etc. is zero. In the reactive sink term, the overlap of the kv pair, S/ v must be removed. However, it could have been taken to the right-hand side as a source term and represented as feact/d r0 9 vnbii M. There are (N — 1) (M — 1) such equations and N — 2) (M — 2) equations involving the survival probability Pat-2,m-2> when derived from an initial cluster of N anions and M cations. The sum total of all the probabilities that a particular distribution exists at a time f0, is... 

Many years ago Polya [20] formulated the key problem of random walks on lattices does a particle always return to the starting point after long enough time If not, how its probability to leave for infinity depends on a particular kind of lattice His answer was a particle returns for sure, if it walks in one or two dimensions non-zero survival probability arises only for the f/iree-dimensional case. Similar result is coming from the Smoluchowski theory particle A will be definitely trapped by B, irrespectively on their mutual distance, if A walks on lattices with d = 1 or d = 2 but it survives for d = 3 (that is, in three dimensions there exist some regions which are never visited by Brownian particles). This illustrates importance in chemical kinetics of a new parameter d which role will be discussed below in detail. 

In other words, the peculiarity of the two-dimensional motion which has led to the zero survival probability of correlated pairs, equation (3.2.26), for randomly distributed particles consists of the complete zerofication of the reaction rate at a great time, K(oo) = 0. The logarithmic dependence of the reaction rate on time does not considerably affect the asymptotic behaviour of macroscopic concentrations. Introducing the critical exponent a... 

Fig. 5.14. The time dependence of the survival probability for one-dimensional diffusion with mobile, noninteracting traps for various values of k. Curve 1 is exact for static particles (k = 0) and identical to the Smoluchowski result curve 4 is exact for static traps (n = 1), curve 2 for equal diffusion coefficients (k = 1/2), and curve 3 for k = 1 are obtained from...

Time-Dependent Escape Rate (Reaction Rate).186 To explain the meaning of P (t)—the probability of finding the particle still in the well at time t, recalling that it was generated within it with uniform probability—let the random variable T be the time at which the particle crosses the barrier (suffers a reaction). It is obvious that the survival probability defined as the probability of the particle being in the well after time t satisfies... 

Use of the Condon approximation for the active and inactive modes causes the matrix element (2.59) to break up into a product of overlap integrals (for the inactive modes) and a constant factor V responsible for interaction of the potential energy terms (for the active modes). In this approximation the time dependence of the survival probability of A1 is given by... 

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