具有重置的扩散行为

Diffusion Behavior with Resetting

摘要: 连续时间随机游走模型是研究反常扩散的重要工具，本文在此基础上引入了随机重置。传统的CTRW模型广泛用于描述反常扩散过程，但其在非平衡稳态、弛豫过程及遍历性破缺等方面的描述能力有限。为此，我们引入了一个具有状态依赖重置的CTRW框架，其中扩散等待时间与重置等待时间分别服从不同参数的指数分布，跳跃步长服从高斯分布。通过傅里叶–拉普拉斯变换方法，我们推导了粒子位置的概率密度函数及其各阶矩的解析表达式。数值模拟结果表明，系统在不同时间尺度上呈现出三个阶段的行为：初始扩散主导的线性增长、中间由重置与扩散竞争主导的弛豫阶段，以及最终达到由重置速率和噪声强度共同决定的非平衡定态。本文模型为理解随机重置在扩散系统中的调控作用提供了新的理论工具，适用于最优搜索、生物分子运动、网络恢复等多领域建模。

Abstract: The continuous-time random walk (CTRW) model is an important tool for studying anomalous diffusion. This paper introduces stochastic resetting on this basis. Traditional CTRW models are widely used to describe anomalous diffusion processes, but their ability to describe non-equilibrium steady states, relaxation processes, and ergodicity breaking is limited. Therefore, we propose a CTRW framework with state-dependent resetting, where the diffusion waiting time and resetting waiting time follow exponential distributions with different parameters, and the jump length follows a Gaussian distribution. Using the Fourier-Laplace transform method, we derive analytical expressions for the probability density function of particle positions and their moments at various orders. Numerical simulation results show that the system exhibits three behavioral stages at different time scales: initial linear growth dominated by diffusion, an intermediate relaxation stage governed by the competition between resetting and diffusion, and ultimately the attainment of a non-equilibrium steady state determined jointly by the resetting rate and noise intensity. This model provides a new theoretical tool for understanding the regulatory role of stochastic resetting in diffusion systems and is applicable to modeling in various fields such as optimal search, biomolecular motion, and network recovery.