论文标题
与远程互动的一维随机步行
One-dimensional annihilating random walk with long-range interaction
论文作者
论文摘要
我们研究以一个维度的远程相互作用的随机行走。每个粒子在一维环上进行随机行走的方式，以使朝着最近粒子跳跃的概率为$ w = [1-ε（x+μ）^{ - σ}]/2 $（从其最近的粒子移开的可能性为$ 1-W $），从$ x $ $ x $的距离是从$ x $到$ x $的距离粒子到其最大的粒子到其最大的粒子和$ $ $ $ $，$ $，$，$，以及$^$，以及$，$，$，以及$ε$，以及$ $，以及$ $，以及$。对于正（负）$ε$，一个粒子被其最近的粒子有效排斥（吸引），并且每个跳跃通常都有偏见。相遇时，立即将两个粒子从系统中删除。我们首先研究了如果一开始只有两个粒子，则长期限制的生存概率和平均扩散行为。然后，如果所有站点在一开始都占据，我们研究密度衰减如何零。我们发现，渐近行为由七个类别进行分类：（i）$σ> 1 $或$ε= 0 $，（ii）$σ= 1 $和$2ε> 1 $，（iii）$σ= 1 $ and $ 2 $ and $2ε= 1 $，（iv） $ε<0 $，（vii）$ 0 <σ<1 $和$ε<0 $。从某种意义上说，$μ$（有时$ε$）不会影响渐近行为，每个类别中的渐近行为都是普遍的。
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.
