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

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 reactivity k ( r ) on the interparticle distance r : square box, exponential, and inverse power type. The kinetics is obtained by analytical solution of the non-Markovian master equation and by direct numerical simulation of the process. The reaction kinetics is shown to be anomalous, namely, significantly different from that predicted for processes assisted by normal diffusion. The difference manifests itself both in static and time dependent characteristics of the kinetics depending on the form of k ( r ) and on the anomaly 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 ( r 0 ) dependence of reaction probability φ r ( r 0 ). For localized k ( r ) (square box and exponential) it is similar to that known for normal diffusion φ r ( r 0 )∼1/ r 0 . For k ( r )∼1/ r m , however, the behaviour of φ r ( r 0 ) depends on n = α m : at n >3 the function φ r ( r 0 ) is similar to that for localized k ( r ), but at 3> n >2 we found φ r ( r 0 )∼1/ r 0 n −2 . A similar change-over is also observed for the long time part n s ( t ) of the kinetics N ( t ) (the number of surviving pairs). For localized k ( r ) it is universal: n s ( t )∼1/ t α/2 . For k ( r )∼1/ r m and 3> n >2, the decay of n s ( t ) proves to be more dispersive: n s ( t )∼1/ t α(1−2/ n ) . Possible applications of the present results are discussed.