Diffusion-controlled reaction on a one dimensional lattice: Dependence on jump and reaction probabilities

In an earlier study [J.C. Lewis, H. Wheeler, Physica A 271 (1999) 63–86] of the dependence on jump probability p of the rates of diffusion-controlled reactions on simple cubic lattices in dimension 2≤d≤4 we found that the dependence was non-linear, which is not in accord with what would be expected on the basis of theories of such reactions in continua [M. von Smoluchowski, Wien. Ber. 124 (1915) 263; Phys. Zeit. 17 (1916) 557–585; Z. Phys. Chem. 92 (1917) 129; S. Chandrasekhar, Revs. Modern Phys. 15 (1) (1943) 1–89]. In the present work we examine the d=1 case. Jump probabilities less than one are of particular importance in that the p=1 case, in which all particles move simultaneously, is not physical. A recursive solution for the concentration of reactant S as a function of time and of jump probability 0