Fig. 1: Imperfect reaction kinetics in confinement.

(a) In the case of imperfect reactions, multiple random interaction events between reactants are typically required before reaction occurs. The reaction time T_r for a random walker starting from r with a target (red dot) can then be written \(T_{{{{{{\bf{r}}}}}}}= \tau_{{{{{{{\bf{r}}}}}}}} + \mathop{\sum}\nolimits_{k=2}^{n} \tau^{(k)}_1\), where τ_r is the first passage time (FPT) to the target, n is the total number of visits to the target before reaction, and \({\tau }_{1}^{(k)}\) is a first return time to the target. (b) In the case of discrete space models, imperfect reactivity is parametrized by the probability p that reaction occurs at each visit of the random walker to the target. In the case of continuous space models, imperfect reactivity is modeled either by (c) a reaction rate k(r) when the random walker is within the reactive volume that defines the target, or (d) partially absorbing boundary conditions (parametrized by κ) at the target boundary.