Defocusing of First-Reaction Times

In this chapter, we focus on the full probability density functionupper H left parenthesis bold x 0 comma t right parenthesisH(x0, t) of the first-reaction time (FRT) in a geometrically simple model, in which a diffusive particle, starting at a fixed positionbold 0x0, searches for and eventually binds to an immo-bile, partially reactive (that is, having a finite intrinsic reactivitykappa) 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 upper parenthesisH(x0, t) for arbitrary values ofkappa andbold 0x0. 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-tt 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 parenthesisH(x0, t) by calculat-ing 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 parenthesisH(x0, t) which may be useful for fitting data garnered in experiments.