Diffusion controlled reactions with interacting potentials: Comparison of boundary conditions and sink term methods

The two particle diffusion controlled reaction rate with Smoluchowski boundary conditions is derived by direct integration of the diffusion equation in the long time limit, together with an integral equation for the spatial part of the long time probability function. In a first order approximation this rate constant takes the value of the exact inverse mean first passage time. It is also shown how to obtain from the two particle result the rate constant for N concurrent independent two particle reactions between unbound particles. A new derivation is given of the mean first passage time in the Wilemski–Fixman closure approximation which clarifies the differences between the boundary conditions and sink term methods. Our previous statement that the exact and closure approximation results coincide to the third term of the expansion in the reaction sink radius is confirmed.