离散时间下的驱动洛伦兹模型
Driven Lorentz model in discrete time
AI总结：
本文研究二维晶格中障碍物环境下示踪粒子在恒定驱动力下的离散时间随机行走，精确到障碍密度一阶，揭示小力时趋近稳态速度的非解析标度、线性响应失效以及离散步数中首次证明的超扩散行为。
摘要：
我们考虑一个示踪粒子在二维晶格上、存在不可移动硬障碍物的情况下进行随机行走。系统从平衡态出发，施加一个作用在粒子上的恒定拉力，驱动系统进入新的稳态。本研究在离散时间（步数 $N$）中计算任意强度恒定驱动力下的位移矩，结果精确到障碍物密度的一阶。我们发现，对于固定的驱动力 $F$，在小 $F$ 时，趋近终态离散速度的行为按 $\sim N^{-1} \exp(- N F^2 / 16)$ 标度，与线性响应预言的 $\sim N^{-1}$ 显著不同。除了对力的非解析依赖以及爱因斯坦线性响应的失效之外，我们的结果表明，在存在障碍物时，力的方向上的涨落被增强。值得注意的是，当 $F \to \infty$ 时，在中间步数阶段方差按 $\sim N^3$ 增长（超扩散），而在更大步数时回归正常扩散（$\sim N$），这一行为此前曾在连续时间中被观察到，但本文首次在离散步数中予以证明。与指数等待时间情形不同，超扩散区域从 $N=1$ 起立即开始。本文提出的框架允许考虑步间任意类型的等待时间分布，并可通过从属方法过渡到连续时间。我们的发现也通过计算机模拟得到了验证。
英文摘要：
We consider a tracer particle performing a random walk on a two-dimensional lattice in the presence of immobile hard obstacles. Starting from equilibrium, a constant force pulling on the particle is switched on, driving the system to a new stationary state. Our study calculates displacement moments in discrete time (number of steps $N$) for an arbitrarily strong constant driving force, exact to first order in obstacle density. We find that for fixed driving force $F$, the approach to the terminal discrete velocity scales as $\sim N^{-1} \exp(- N F^2 / 16)$ for small $F$, differing significantly from the $\sim N^{-1}$ prediction of linear response. Besides a non-analytic dependence on the force and breakdown of Einstein's linear response, our results show that fluctuations in the directions of the force are enhanced in the presence of obstacles. Notably, the variance grows as $\sim N^3$ (superdiffusion) for $F \to \infty$ at intermediate steps, reverting to normal diffusion ($\sim N$) at larger steps, a behavior previously observed in continuous time but demonstrated here in discrete steps for the first time. Unlike the exponential waiting time case, the superdiffusion regime starts immediately at $N=1$. The framework presented allows considering any type of waiting-time distribution between steps and transition to continuous time using subordination methods. Our findings are also validated through computer simulations.
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