Reaction, trapping, and multifractality in one-dimensional systems

Abstract

In the first part of this paper, we present two variants of the A + A --> A and A + A --> P reaction in one dimension that can be investigated analytically. In the first model, pairs of neighboring particles disappear reactively at a rate which is independent of their relative distance. It is shown that the probability density phi(x) for a nearest neighbor distance equal to x approaches the scaling form phi(x) approximately c exp(-cx/2)/(cx)1/2 in the long-time limit, with c being the concentration of particles. The second model is a ballistic analogue of the coagulation reaction A + A --> A. The model is solved by reducing it to a first-passage-time problem. The anomalous relaxation dynamics can be linked in a direct way to the fractal time properties of random walks. In the second part of this paper, we discuss the complications that arise in systems with disorder. We present a new approach that relates first-passage-time characteristics in a one-dimensional random walk to properties of random maps. In particular, we show that Sinai disorder is a borderline case for the appearance of multifractal properties. Finally, we apply a previously introduced renormalization technique to calculate the survival probability of particles moving on the line in the presence of a background of imperfect traps.