Fig. 1
Reaction control. Reaction time density H(r, t) for a reaction on an inner target of radius ρ/R = 0.01, with starting point a r/R = 0.2 and b r/R = 0.02 for four progressively decreasing (from top to bottom) values of the dimensionless reactivity κ′ = κR/D indicated in the plot. Note that κ′ includes R and D such that smaller values of κ′ can also be achieved at a fixed κ upon lowering R or by increasing the values of D. The coloured vertical arrows indicate the mean reaction times for these cases. The vertical black dashed line indicates the crossover time tc = 2(R − ρ)2/(Dπ2) above which the contribution of higher order Laplacian eigenmodes become negligible. This characteristic time marks the end of the hump-like region (Lévy–Smirnov region specific to an unbounded system, see below and the Methods section for more details) and indicates the crossover to a plateau region with equiprobable realisations of the reaction times. This plateau region spans a considerable window of reaction times, especially for lower reactivity values. Thin coloured lines show the reaction time density H∞(r, t) from Eq. (6) for the unbounded case (R → ∞). Length and time units are fixed by setting R = 1 and R2/D = 1. Note the extremely broad range of relevant reaction times (the horizontal axis) spanning over 12 orders of magnitude for the panel b. Coloured bar-codes c, d indicate the cumulative depths corresponding to four considered values of κ′ in decreasing order from top to bottom. Each bar-code is split into ten regions of alternating brightness, representing ten 10%-quantiles of the distribution (e.g., the first dark blue region of the top bar-code in panel c indicates that 10% of reaction events occur till \(Dt/R^2 \simeq 1\))
