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

Abstract

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 < p ≤ 1 is developed for the coagulation reaction S + S → S in a one-dimensional lattice gas. This solution is exact for the case p = 1 . It reproduces the analytical form derived by Privman [V. Privman, Phys. Rev. E 50 (1) (1994) 50–53. Also available as arXiv.org preprint cond-mat/9310079v1], and gives excellent agreement with computer simulations for p < 1 . Kinetics for the annihilation reaction S + S → n o t h i n g are derived from the kinetics of the coagulation reaction S + S → S using Privman’s transformation in the above cited work. Using the recursive solution we are able to demonstrate data collapse for p < 1 . A quantitative measure for the effect of fluctuations caused by reaction using pair correlations of gap frequencies was developed and studied.