Fig. 2
Geometry control. Reaction time density H(r, t) for a reaction on an inner target of radius ρ/R = 0.01, for the different initial radii r indicated in the panels (r increasing from top to bottom). The values of the reactivity are a κ = ∞ (perfectly reactive) and b κR/D = 1 (partially reactive). The coloured vertical arrows indicate the mean reaction times for these cases (note that some arrows coincide). The vertical black dashed line indicates the crossover time tc = 2(R − ρ)2/(Dπ2) from the hump-like Lévy–Smirnov region to a plateau-like one. Thin coloured lines show the reaction time density H∞(r, t) from Eq. (6) for the unbounded case (R → ∞). The length and time units are fixed by setting R = 1 and R2/D = 1. Clearly the positions of the most probable reaction times are geometry-controlled by the initial distance to the target. Not surprisingly, for the largest initial distance the solution for the unbounded case underestimates the RTD hump. Note the extremely broad range of relevant reaction times (horizontal axis) spanning over 12 orders of magnitude in panel b. Coloured bar-codes c, d indicate the cumulative depths corresponding to four considered values of r/R in increasing order from top to bottom. Each bar-code is split into ten regions of alternating brightness, representing ten 10%-quantiles of the distribution. In spite of distinctions in the probability densities in panel b, the corresponding cumulative distributions are close to each other and result in very similar reaction depths
