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Reaction, trapping, and multifractality in one-dimensional systems

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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 ϕ(x) for a nearest neighbor distance equal tox approaches the scaling formϕ(x) ∼c exp(−cx/2)/(cx)1/2 in the long-time limit, withc 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-passagetime 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.

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Author notes

  1. C. Van den Broeck

    Present address: L.U.C., B-3590, Diepenbeek, Belgium

Authors and Affiliations

  1. Department of Chemistry, University of California at San Diego, 92093-0340, La Jolla, California

    C. Van den Broeck

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Van den Broeck, C. Reaction, trapping, and multifractality in one-dimensional systems. J Stat Phys 65, 971–990 (1991). https://doi.org/10.1007/BF01049593

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