Universality classes of absorbing phase transitions in generic branching-annihilating particle systems with nearest-neighbor bias

One-dimensional annihilating random walk with long-range interaction

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-ɛ(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 ɛ, μ, and σ are parameters. For positive (negative) ɛ, 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 ɛ=0, (ii) σ=1 and 2ɛ>1, (iii) σ=1 and 2ɛ=1, (iv) σ=1 and 2ɛ<1, (v) σ<1 and ɛ>0, (vi) σ=0 and ɛ<0, and (vii) 0<σ<1 and ɛ<0. The asymptotic behaviors in each category are universal in the sense that μ (and sometimes ɛ) cannot affect the asymptotic behaviors.

Crossover behaviors in branching annihilating attracting walk

We introduce branching annihilating attracting walk (BAAW) in one dimension. The attracting walk is implemented by a biased hopping in such a way that a particle prefers hopping to a nearest neighbor located on the side where the nearest particle is found within the range of attraction. We study the BAAW with four offspring by extensive Monte Carlo simulation. At first, we find the critical exponents of the BAAW with infinite range of attraction, which are different from those of the directed Ising (DI) universality class. Our results are consistent with the recent observation [Daga and Ray, Phys. Rev. E 99, 032104 (2019)2470-004510.1103/PhysRevE.99.032104]. Then, by studying crossover behaviors, we show that as far as the range of attraction is finite the BAAW belongs to the DI class. We conclude that the origin of non-DI critical behavior of the BAAW with infinite range of attraction is the long-range nature of the attraction.

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

We introduce and numerically study the branching annihilating random walks with long-range attraction (BAWL). The long-range attraction makes hopping biased in such a manner that particle's hopping along the direction to the nearest particle has larger transition rate than hopping against the direction. Still, unlike the Lévy flight, a particle only hops to one of its nearest-neighbor sites. The strength of bias takes the form x^{-σ} with non-negative σ, where x is the distance to the nearest particle from a particle to hop. By extensive Monte Carlo simulations, we show that the critical decay exponent δ varies continuously with σ up to σ=1 and δ is the same as the critical decay exponent of the directed Ising (DI) universality class for σ≥1. Investigating the behavior of the density in the absorbing phase, we argue that σ=1 is indeed the threshold that separates the DI and non-DI critical behavior. We also show by Monte Carlo simulations that branching bias with symmetric hopping exhibits the same critical behavior as the BAWL.

Effect of bias in a reaction-diffusion system in two dimensions

We consider a single species reaction diffusion system on a two-dimensional lattice where the particles A are biased to move towards their nearest neighbors and annihilate as they meet. Allowing the bias to take both negative and positive values parametrically, any nonzero bias is seen to drastically affect the behavior of the system compared to the unbiased (simple diffusive) case. For positive bias, a finite number of dimers, which are isolated pairs of particles occurring as nearest neighbors, exist while for negative bias, a finite density of particles survives. Both the quantities vanish in a power-law manner close to the diffusive limit with different exponents. The appearance of dimers is exclusively due to the parallel updating scheme used in the simulation. The results indicate the presence of a continuous phase transition at the diffusive point. In addition, a discontinuity is observed at the fully positive bias limit. The persistence behavior is also analysed for the system.