Branching annihilating random walks with long-range attraction in one dimension

Abstract

We study the annihilating random walk with long-range interaction in one dimension. Each particle performs random walks on a one-dimensional ring in such a way that the probability of hopping toward the nearest particle is W = [1 − ε(x + µ) −σ ]/2 (the probability of moving away from its nearest particle is 1 − W), where x is the distance from the hopping particle to its nearest particle and ε, µ, and σ are parameters. For positive (negative) ε, a particle is effectively repulsed (attracted) by its nearest particle and each hopping is generally biased. On encounter, two particles are immediately removed from the system. We first study the survival probability and the mean spreading behaves in the long-time limit if there are only two particles in the beginning. Then, we study how the density decays to zero if all sites are occupied at the outset. We find that the asymptotic behaviors are classified by seven categories; (i) σ > 1 or ε = 0, (ii) σ = 1 and 2ε > 1, (iii) σ = 1 and 2ε = 1, (iv) σ = 1 and 2ε < 1, (v) σ < 1 and ε > 0, (vi) σ = 0 and ε < 0, and (vii) 0 < σ < 1 and ε < 0. The asymptotic behaviors in each category are universal in the sense that µ (and sometimes ε) cannot affect the asymptotic behaviors.

Branching annihilating random walk with long-range repulsion: logarithmic scaling, reentrant phase transitions, and crossover behaviors

“…For σ < 1, the condition P s = 0 is applicable only to the BAWLA. Recall that the BAWLA with nonzero ε does not belong to the DI class [18,19]. Thus, there should be a crossover behavior for small |ε|, which is the topic of Sec.…”

“…If P s = Ps = 0 and q is small, then Eq. (19) shows that only pair annihilation governs the long-time behavior of the particle density. To confirm the scaling ansatz as well as the scenario for even ℓ, we performed Monte Carlo simulations for two cases, ℓ = 1 and ℓ = 2.…”

“…[21] (see also Refs. [19,22] for the AWL with attractive interaction). We summarize some results of Ref.…”

“…On the other hand, for even ℓ, the modified model in one dimension does not belong to the DI class if σ < 1. Rather, the critical exponents vary with σ [19]. In Ref.…”

“…Recently, a modified version of the BAW was suggested by introducing hopping bias in such a way that a particle is attracted by its closest particle [18,19]. The strength of the bias depends on the distance x between a particle and its closest particle in a power-law fashion x −σ .…”