Specific features of the kinetics of fractional-diffusion assisted geminate reactions

Abstract

Specific features of the kinetics of geminate reactions assisted by dispersive(subdiffusive) relative motion of reacting particles are analysed in detail.The analysis is made for different long-range dependences of the reactivityk(r) on the interparticledistance r: square box, exponential, and inverse power type. The kinetics is obtained by analyticalsolution of the non-Markovian master equation and by direct numerical simulation of theprocess. The reaction kinetics is shown to be anomalous, namely, significantly different fromthat predicted for processes assisted by normal diffusion. The difference manifests itselfboth in static and time dependent characteristics of the kinetics depending on the form ofk(r) and on theanomaly parameter α<1 (which determines the time dependence of the mean square displacement ). The anomalous kinetic peculiarities are most clearly demonstrated with the initial distance(r0) dependence ofreaction probability φr(r0). For localized k(r) (square box and exponential) it is similar to that known for normal diffusionφr(r0)∼1/r0. Fork(r)∼1/rm, however, thebehaviour of φr(r0) depends on n = αm: atn>3 the functionφr(r0) is similar tothat for localized k(r), but at 3>n>2 we found φr(r0)∼1/r0n−2. A similar change-over is also observed for the long time partns(t) of thekinetics N(t) (the number of surviving pairs). For localizedk(r) it isuniversal: ns(t)∼1/tα/2. For k(r)∼1/rm and3>n>2, the decay ofns(t) proves to bemore dispersive: ns(t)∼1/tα(1−2/n). Possible applications of the present results are discussed.