One-dimensional annihilating random walk with long-range interaction

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 $W=[1-\epsilon (x+\mu {)}^{-\sigma }]/2$ (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 $\epsilon$, $\mu$, and $\sigma$ are parameters. For positive (negative) $\epsilon$, 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) $\sigma >1$ or $\epsilon =0$, (ii) $\sigma =1$ and $2\epsilon >1$, (iii) $\sigma =1$ and $2\epsilon =1$, (iv) $\sigma =1$ and $2\epsilon <1$, (v) $\sigma <1$ and $\epsilon >0$, (vi) $\sigma =0$ and $\epsilon <0$, and (vii) $0<\sigma <1$ and $\epsilon <0$. The asymptotic behaviors in each category are universal in the sense that $\mu$ (and sometimes $\epsilon$) cannot affect the asymptotic behaviors.

arXiv comment13 pages, 6 figures, 1 table. Published version

References in corpus (22)

• Renormalization Group Calculation for the Reaction $kA\to \varnothing$
• Reaction-Diffusion Processes from Equivalent Integrable Quantum Chains
• EQUIVALENCES BETWEEN STOCHASTIC SYSTEMS
• A new model of binary opinion dynamics: coarsening and effect of disorder
• Disorder-Induced Anomalous Kinetics in the $A+A\to \varnothing$ Reaction
• Reaction Diffusion Models in One Dimension with Disorder
• Driven Pair Contact Process with Diffusion
• Diffusion-limited annihilation in inhomogeneous environments
• On the Relation between One-Species Diffusion-Limited Coalescence and Annihilation in One Dimension
• Solution of classical stochastic one dimensional many-body systems
• The Reaction Process A+A->O in Sinai Disorder
• Numerical Observation of Disorder-Induced Anomalous Kinetics in the A + A -> 0 Reaction
• Annihilating random walks in one-dimensional disordered media
• Universality classes of absorbing phase transitions in generic branching-annihilating particle systems
• Opinion dynamics model with domain size dependent dynamics: novel features and new universality class
• Two-point correlation functions of the diffusion-limited annihilation in one dimension
• $A+A\to \varnothing$ model with a bias towards nearest neighbor
• Random walk weakly attracted to a wall
• Branching annihilating random walks with long-range attraction in one dimension
• Crossover behaviors in branching annihilating attracting walk
• Block renormalization study on the nonequilibrium chiral Ising model
• Coarsening Dynamics of Nonequilibrium Chiral Ising Models

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