Exact algorithm for<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>d</mml:mi></mml:math>-dimensional walks on finite and infinite lattices with traps. II. General formulation and application to diffusion-controlled reactions

Abstract

A method for calculating exactly the expected walk length $〈n〉$ for random walks on $d$-dimensional lattices with traps, reported recently by the authors [Phys. Rev. Lett. 47, 1500 (1981)], is elaborated in some detail in order to exhibit the underlying structure of the theory and to demonstrate the generality of the approach. Formulated as a problem in matrix transformation theory, the properties of a certain linear operator $A$ and its inverse ${A}^{\ensuremath{-}1}$ are explored in $d=1, 2, 3$. In $d=1$, the analytic result $〈n〉=\frac{N(N+1)}{6}$ derived by Montroll for trapping on a (periodic) chain with a single, deep trap is recovered. In the higher dimensions $d=2, 3$, extensive new data are reported on the results of exact calculations of $〈n〉$ for two types of reaction-diffusion processes. The first is that of a reactant migrating toward a target molecule in a volume of $d$ dimensions, and reacting there irreversibly upon first encounter. Then, it is assumed that the $N\ensuremath{-}1$ sites surrounding the target molecule are not passive (nonabsorbing, neutral) but may react with the diffusing molecule to form an excited-state complex which may, with nonzero probability $s$, result in the irreversible removal of reactant from the system. In both models, the efficiency of reaction is studied as a function of the spatial extent of the reaction volume and of the boundary conditions imposed on the underlying lattice.