Permanent particle immobilization widely emerges across complex transport systems, ranging from intracellular transport to active matter in disordered media, and constitutes a hallmark feature of such systems. Starting from the standard Lévy walk with power-law distributed flight time $\sim t^{-1-\alpha }$, we construct the immobilized Lévy walk framework by introducing a non-normalized power-law distribution to characterize flight durations for both regimes $0<\alpha <1$ and $1<\alpha <2$. In this model, particles may undergo permanent immobilization at renewal events where their flight direction is updated. By separating the particles into active and immobilized states, we derive a general propagator expression that strictly enforces probability conservation. We demonstrate that the immobilized Lévy walks obey the universal scaling $\langle x^2\left(t\right)\rangle \sim t^{2-\alpha }$, which exhibits superdiffusion for $0<\alpha <1$ and subdiffusion for $1<\alpha <2$. The system relaxes toward a power-law stationary distribution $P_{\mathrm{st}}\left(x\right)\sim {\left|x\right|}^{-1-\alpha }$, whose form matches the single-step flight length distribution. We further investigate the time-dependent propagator near the ballistic front via the infinite density formalism, an approach capable of reproducing both the asymptotic mean-squared displacement and stationary distribution. This infinite density captures key long-time statistical signatures of particle dynamics and offers fundamental insights into the dynamical impacts induced by the permanent immobilization mechanism.

Close

PrevNext

View figure in article

• Anomalous diffusion
• Continuous-time random walk
• Diffusion & random walks