基于随机速率的Lévy游走模型及其反常扩散动力学研究

Study on the Lévy Walk Model with Random Speeds and Its Anomalous Diffusion Dynamics

摘要: 本研究建立了一个随机速度的Lévy游走模型，通过引入Gamma和Beta分布来描述随机速度的统计特性，当游走时间服从指数分布时，系统会经历从短时超扩散到长时正常扩散的转变；而当游走时间服从幂律分布时，系统则始终保持超扩散状态。研究发现，Gamma分布参数$\left(k,\theta \right)$ 和Beta分布参数$\left({\alpha }_{+},{\alpha }_{-}\right)$ 决定了扩散过程的强度和各向异性程度。该研究不仅为理解复杂系统中的反常扩散现象提供了新的理论框架，也为相关领域的定量分析和预测建模奠定了重要基础。

Abstract: This study establishes a Lévy walk model with random speeds by introducing Gamma and Beta distributions to characterize the statistical properties of stochastic velocities. When the walking time follows an exponential distribution, the system undergoes a transition from short-term superdiffusion to long-term normal diffusion. In contrast, when the walking time follows a power-law distribution, the system persistently exhibits superdiffusive behavior. The research reveals that the parameters of the Gamma and Beta distributions determine the intensity and anisotropy of the diffusion process. This work not only provides a new theoretical framework for understanding anomalous diffusion in complex systems but also lays an important foundation for quantitative analysis and predictive modeling in related fields.