Time-dependent probability analysis

Time-dependent probability analysis of fiber-reinforced polymer rehabilitated pipes... 

For the present analysis, it suffices to adopt a model for internal motion that contains the motion as an equilibrium between two states—the conformational parameters of which can be specified. It is required, as usual, that this motion be independent from overall motion. If these states are labeled 1 and 2 and the conformational change is assumed to follow first-order kinetics, time-dependent probabilities are... 

The last two results are rather similar to the quadratic forms given by Fox and Uhlenbeck for the transition probability for a stationary Gaussian-Markov process, their Eqs. (20) and (22) [82]. Although they did not identify the parity relationships of the matrices or obtain their time dependence explicitly, the Langevin equation that emerges from their analysis and the Doob formula, their Eq. (25), is essentially equivalent to the most likely terminal position in the intermediate regime obtained next. 

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 diffusion equation analysis is discussed in Sect. 2. It has been used very much more frequently in studies of diffusion-limited reactions rates than the analysis based on molecular pair behaviour, which is discussed in Sect. 3. This is probably because the diffusion equation approach is rather more direct, clear and versatile than the molecular pair analysis (furthermore, time-dependent Green s functions are required for the molecular pair approach). Besides, the probability that a molecular pair will reencounter one another is often derived from a diffusion equation analysis in any case and under these circumstances the two approaches are identical. 

B. Peters, K. Janek, U. Kuckelkorn, and H. G. Holzhiitter. Assessment of protea-somal cleavage probabilities from kinetic analysis of time-dependent product formation./. Mol. Biol, 318 847-862, 2002. 

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