DIFFUSION-CONTROLLED REACTIONS .2. AN APPROACH BASED ON A GENERALIZEDDIFFUSION EQUATION

In this work we present a new theoretical approach for diffusion-contr olled reactions which generalizes the classic theory of Smoluchowski a nd that of Collins and Kimball. In this approach, the non-Markovian ef fect is taken into account in the framework of a generalized diffusion equation with a time-dependent diffusion coefficient. When Smoluchows ki's absorbing boundary condition is considered, we have found the exa ct analytical solution of the generalized diffusion equation despite t he presence of a time-dependent diffusion coefficient. Our generalized Smoluchowski theory removes the unphysical singularity in reaction ra te, k(t), at t=0 in the classic Smoluchowski theory. This allows to ev aluate the initial value of reaction rate. The result shows that the i nitial reaction rate is overestimated for collision-induced reactions. This is due to the inappropriateness of the absorbing boundary condit ion for describing the collision-induced reactions. To take into accou nt more properly these kinds of reactions, Collins-Kimball's boundary condition is considered. In this case, a perturbation method and an ap proximation ansatz are developed to find an approximate solution, The approximation for reaction rate contains an adjustable parameter. A sc heme is prescribed for the optimal choice of this parameter in short a nd long time regions. With the help of this scheme, we obtain an appro ximation of the reaction rate which has the virtue to give simultaneou sly the exact asymptotic behavior at long time and the right initial v alue of the reaction rate.