Theory of non-Markovian reversible dissociation reactions

We consider a reversible dissociation-recombination reaction in solution which is described by a distribution of waiting times rather than a single dissociation rate constant. This is a non-Markovian generalization of the backreaction boundary condition. We formulate the new boundary condition in terms of the residence time in the bound state and illustrate the theory by assuming a stable-law density for the residence time. Explicit expressions are found for the Laplace transform of the survival probability in one and three dimensions, which can be inverted analytically for special values of the stable-law parameter α and numerically for other values of a. We derive the long-time behavior of the survival probability for arbitrary α, and note that the survival probability undergoes a first-order phase transition in one dimension, in which its asymptotic value changes abruptly at α = 1/2. In three dimensions it undergoes a second-order phase transition at α = 1, in which only the asymptotic slope of the survival probability changes discontinuously.