Decomposition of Anomalous Diffusion in generalized two step random walks

Diffusive behaviour is generally described by the Central Limit Theorem (CLT), which predicts that the displacement distribution would be gaussian in large time limit with a variance that grows proportionally to the square root of time. However, numerous experimental systems exhibit anomalous diffusion that deviates from the CLT. The fundamental causes of such behaviour can be decomposed into three constitutive effects, known as the Joseph, Noah, and Moses effects. These effects are associated with the violation of CLT. While several theoretical frameworks exist to describe anomalous diffusion, real-world scenarios often require hybrid approaches that combine distinct mechanisms. In this work, we discuss about such an approach, Generalized Two-Step Random Walk (GTSRW) model that alternates between continuous-time random walk (CTRW) and Drude-like dynamics [Hu, Yuhang, and Jian Liu. Physical Review E 111.1 (2025): 014148]. In the CTRW phase, the walker executes instantaneous Gaussian-distributed steps after waiting periods drawn from a power-law distribution. In the Drude-like phase, the walker’s velocity is explicitly coupled to the elapsed duration, with durations sampled from a different power-law distribution. We show that the anomalous diffusion produced by GTSRWs is a complex combination of Joseph, Noah, and Moses effects.