Probability distribution decay time results

In the case when either the barrier intensity Q [150] or temperature T [151] are slowly modulated, the resulting waiting time distribution becomes a superposition of infinitely many exponentials. At least since the important work of Shlesinger and Hughes [152], and probably earlier, it is known that a superposition of infinitely many exponentially decaying functions can generate an inverse power law. 

In spite of its prevalence in the fluorescence decay literature, we were not universally successful with this fitting method. Most reports of hi- or multiexponential decay analysis that use a time-domain technique (as opposed to a frequency-domain technique) use time-correlated photon counting, not the impulse-response method described in Section 2.1. In time-correlated photon-counting, noise in the data is assumed to have a normal distribution. Noise in data collected with our instrument is probably dominated by the pulse-to-pulse variation of the laser used for excitation this variation can be as large as 10-20%. Perhaps the distribution or the level of noise or the combination of the two accounts for our inconsistent results with Marquardt fitting. 

From such single trajectories we determine the individual barrier crossing times as the time interval between a jump into one well across the zero line x = 0 and the escape across x = 0 back to the other well. In Fig. 18, we demonstrate that on average, the crossing times are distributed exponentially, and thus follow the same law (112) already known from the Brownian case. Such a result has been reported in a previous study of Kramers escape driven by Levy noise [91]. In fact, the exponential decay of the survival probability... 

In real systems, typical decay functions are far from exponential, and loss curves, s (log y), appear much broader than the ones found by the Debye model. A phenomenological approach to deal with this observation is to consider that this enlargement in the response is the result of the existence of a distribution of relaxation times, with a probability density function of p(ln x) ... 

---