带休息的不对称性随机化莱维游走与CTRW结合的反常扩散研究

Research on Anomalous Diffusion Combining Asymmetrically Randomized Lévy Walks with Rests and Continuous-Time Random Walks

摘要: 作为描述自然界中间歇性搜索行为的双态随机游走模型，已在物理学等多个领域得到深入研究。本研究提出一种混合模型，该模型结合了方向不对称性经随机化处理的Lévy游走与连续时间随机游走。本研究的意义在于对经典间歇性搜索模型(LWR)进行关键扩展：将传统模型中“绝对静止的休息阶段”替换为具有自身动力学的CTRW过程，以更真实地刻画复杂系统中的多模式输运现象。在物理和生物系统中，粒子常表现为弹道迁移与局部扩散双模式动态交替，而非完美运动与绝对静止的二元切换。为此，我们构建了“弹道飞行阶段”(Lévy游走)与“局部扩散阶段”(CTRW)交替的混合机制，并设定两阶段的持续时间均服从指数分布(速率参数${\lambda }_1$ 和${\lambda }_2$ )，系统分析其反常扩散行为。传统Lévy游走休息模型中，粒子在弹道迁移与静态休息阶段间交替切换，其休息时间分布常导致均方位移呈现复杂标度行为。本模型通过CTRW驱动的局部扩散机制协同调控长等待时间引起的滞留效应，与Lévy飞行及随机跳跃共同实现的长程转移特性，突破了经典LWR模型在多模式输运刻画中的局限性。

Abstract: The two-state random walk model, as a framework for describing intermittent search behavior in natural systems, has been extensively studied in physics and related fields. This work proposes a hybrid model that integrates directionally randomized Lévy walks with continuous-time random walks (CTRW). The significance of this study lies in its key extension of the classical Lévy walk with rests (LWR) model: replacing the traditional “absolutely static resting phase” with a CTRW process possessing intrinsic dynamics, thereby more realistically capturing multi-mode transport phenomena in complex systems. In physical and biological systems, particles often exhibit dynamic alternation between ballistic migration and local diffusion modes, rather than a binary switch between perfect motion and complete rest. To this end, we construct a hybrid mechanism featuring alternating “ballistic flight phases” (Lévy walks) and “local diffusion phases” (CTRW), with the duration of both phases following exponential distributions (rate parameters ${\lambda }_1$ and ${\lambda }_2$ ). We systematically analyze its anomalous diffusion behavior. In traditional Lévy walk with rest models, particles alternate between ballistic migration and static resting phases, where the resting time distribution typically leads to complex scaling behaviors in mean squared displacement. Our model coordinates the trapping effects induced by long waiting times (via CTRW-driven local diffusion) with the long-range transfer capabilities achieved through Lévy flights and random jumps, overcoming the limitations of classical LWR models in characterizing multi-mode transport phenomena.