The contact process with diffusion (PCPD) defined by the binary reactions $B+B\to B+B+B, B+B\to \varnothing$ and diffusive particle spreading exhibits an unusual active to absorbing phase transition whose universality class has long been disputed. Multiple studies have indicated that an explicit account of particle pair degrees of freedom may be required to properly capture this system's effective long-time, large-scale behavior. We introduce a two-species representation for the PCPD in which single particles $B$ and particle pairs $A$ are dynamically coupled according to the stochastic reaction processes $B+B\to A, A\to A+B, A\to \varnothing$, and $A\to B+B$, with each particle type diffusing independently. Mean-field analysis reveals that the phase transition of this model is driven by competition and balance between the two species. We employ Monte Carlo simulations in one, two, and three dimensions to demonstrate that this model consistently captures the pertinent features of the PCPD. In the inactive phase, $A$ particles rapidly go extinct, effectively leaving the $B$ species to undergo pure diffusion-limited pair annihilation kinetics $B+B\to \varnothing$. At criticality, both $A$ and $B$ densities decay with the same exponents (within numerical errors) as the corresponding order parameters of the original PCPD, and display mean-field scaling above the upper critical dimension $d_c=2$. In one dimension, the critical exponents for the $B$ species obtained from seed simulations also agree well with previously reported exponent value ranges. We demonstrate that the scaling properties of consecutive particle pairs in the PCPD are identical with that of the $A$ species in the coupled model. This two-species picture resolves the conceptual difficulty for seed simulations in the original PCPD and naturally introduces multiple length scales and timescales to the system, which are also the origin of strong corrections to scaling. The extracted moment ratios from our simulations indicate that our model displays the same temporal crossover behavior as the PCPD, which further corroborates its full dynamical equivalence with our coupled model.

中文翻译：

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具扩散对接触过程的两种种群耦合模型

二元反应定义的扩散接触过程（PCPD） $乙+乙\to 乙+乙+乙， 乙+乙\to \varnothing$扩散粒子的扩散表现出不同寻常的活性到吸收相变，其普遍性类别一直存在争议。多项研究表明，可能需要明确说明粒子对的自由度，才能正确捕获该系统的有效长期，大规模行为。我们为PCPD引入了两种表示，其中单个粒子$乙$ 和粒子对 $\mathrm{一种}$ 根据随机反应过程动态耦合 $乙+乙\to \mathrm{一种}， \mathrm{一种}\to \mathrm{一种}+乙， \mathrm{一种}\to \varnothing$和 $\mathrm{一种}\to 乙+乙$，每种粒子类型都独立扩散。平均场分析显示，该模型的相变是由两个物种之间的竞争和平衡驱动的。我们在一维，二维和三维中采用了蒙特卡洛模拟，以证明该模型能够始终如一地捕获PCPD的相关特征。在非活动阶段，$\mathrm{一种}$ 粒子迅速消失，有效地离开了 $乙$ 种经历纯扩散限制对pair灭动力学 $乙+乙\to \varnothing$。都处于临界状态$\mathrm{一种}$ 和 $乙$ 密度以与原始PCPD的相应阶次参数相同的指数（在数值误差范围内）衰减，并在上临界尺寸之上显示均场标度 $d_C=2$。在一个维度上，$乙$通过种子模拟获得的物种也与先前报道的指数值范围非常吻合。我们证明了PCPD中连续粒子对的缩放性质与$\mathrm{一种}$耦合模型中的物种。这两个物种的图片解决了原始PCPD中种子模拟的概念难度，并且自然地向系统引入了多个长度尺度和时间尺度，这也是对尺度进行强校正的源头。从我们的仿真中提取的力矩比率表明，我们的模型显示出与PCPD相同的时间跨接行为，这进一步证明了我们的耦合模型具有完全的动力学等效性。