有限维群集的带噪声多态投票模型
Noisy multistate voter model for flocking in finite dimensions
摘要：
我们研究了一个自推进粒子在成对复制相互作用和噪声作用下的集体行为模型。粒子在二维空间上以恒定速度$v$移动，并且在动力学的单个步骤中，每个粒子采用随机选择的相邻粒子的运动方向，并添加幅度扰动$η$（噪声）。我们研究了粒子的运动和噪声幅度$η$如何影响粒子的排列（顺序）的全局水平。在静态的情况下，粒子被固定在一个正方形晶格的网站，并与他们的第一个邻居相互作用的情况下，我们发现，对于任何噪音$v=0$的系统达到一个稳定的状态完全无序的热力学极限，而$η=0$全序最终实现了系统与任何数量的粒子$N$。因此，当粒子是静态的时，该模型显示在零噪声处的转变，并且因此对于有限噪声（$η>0$）没有有序的稳态。我们发现，有限大小的过渡噪声消失与$N$作为$η_c^{1D} \sim N^{-1}$和$η_c^{2D} \sim \left(N \ln N \right)^{-1/2}$在一维和二维晶格，分别，这是与已知的结果的一种噪声选民模型的催化反应的行为。当粒子以有限的速度$v>0$在空间中运动时，就会出现一个有序相，其特征是一小部分粒子以相似的方向运动。该系统在与$v$成比例的噪声幅度$η_c>0$处表现出有序-无序相变，并且对于$v \ll 1$，其尺度近似为$η_c \sim v \, (-\ln v)^{-1/2}$。这些结果表明，粒子的运动是能够维持一个系统的整体秩序与选民的相互作用。
英文摘要：
We study a model for the collective behavior of self-propelled particles subject to pairwise copying interactions and noise. Particles move at a constant speed $v$ on a two--dimensional space and, in a single step of the dynamics, each particle adopts the direction of motion of a randomly chosen neighboring particle, with the addition of a perturbation of amplitude $η$ (noise). We investigate how the global level of particles' alignment (order) is affected by their motion and the noise amplitude $η$. In the static case scenario $v=0$ where particles are fixed at the sites of a square lattice and interact with their first neighbors, we find that for any noise $η_c>0$ the system reaches a steady state of complete disorder in the thermodynamic limit, while for $η=0$ full order is eventually achieved for a system with any number of particles $N$. Therefore, the model displays a transition at zero noise when particles are static, and thus there are no ordered steady states for a finite noise ($η>0$). We show that the finite-size transition noise vanishes with $N$ as $η_c^{1D} \sim N^{-1}$ and $η_c^{2D} \sim \left(N \ln N \right)^{-1/2}$ in one and two--dimensional lattices, respectively, which is linked to known results on the behavior of a type of noisy voter model for catalytic reactions. When particles are allowed to move in the space at a finite speed $v>0$, an ordered phase emerges, characterized by a fraction of particles moving in a similar direction. The system exhibits an order-disorder phase transition at a noise amplitude $η_c>0$ that is proportional to $v$, and that scales approximately as $η_c \sim v \, (-\ln v)^{-1/2}$ for $v \ll 1$. These results show that the motion of particles is able to sustain a state of global order in a system with voter-like interactions.
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