Defocusing of First-Reaction Times

In this chapter, we focus on the full probability density function \(H(\textbf{x}_0,t)\) of the first-reaction time (FRT) in a geometrically simple model, in which a diffusive particle, starting at a fixed position \(\textbf{x}_0\) , searches for and eventually binds to an immobile, partially reactive (that is, having a finite intrinsic reactivity \(\kappa \) ) spherical binding site, located in the center of a spherical domain. The entire domain is enclosed by an impermeable boundary. For such a model, we determine an exact expression for \(H(\textbf{x}_0,t)\) for arbitrary values of \(\kappa \) and \(\textbf{x}_0\) . Inspecting its functional form, we identify four distinct temporal regimes: a rise to the maximal value, a power-law descent, an extended in time plateau-like region and the terminal exponential tail, as well as the corresponding regime-delimiting time-scales. We show that the commonly studied mean FRT is associated with the large-t exponential tail and hence, its value is supported by anomalously long searching trajectories. In turn, the most probable FRTs, that should be observed for a substantial amount of realizations, can be several orders of magnitude shorter than the mean FRT, revealing a large defocusing of the first-reaction times. We also define the effective broadness of \(H(\textbf{x}_0,t)\) by calculating the corresponding coefficient of variation and specify the reaction depths at each of the temporal stages of the binding process. In particular, we demonstrate that a large fraction of trajectories react before the mean FRT. Finally, we present a simple approximate expression for \(H(\textbf{x}_0,t)\) which may be useful for fitting data garnered in experiments.