Beyond temporary trapping, particles may become permanently fixed due to irreversible trapping in biomedical systems and chemical catalysis systems, which is named permanent immobilization. To describe anomalous diffusion processes in which particles become permanently immobilized with a certain probability, this paper proposes an immobilization model based on a non-normalized jump-length distribution under the continuous-time random walk framework. By classifying particles into active and immobilized states, we derive a generalized Montroll-Weiss equation incorporating the immobilization mechanism and investigate the diffusion behaviors for Gaussian and power-law jump-length distributions. In the long-time limit, for Gaussian jump lengths, the system evolves toward a stationary state characterized by a Laplace distribution, whereas for power-law jump lengths, the stationary distribution retains a Lévy-type power-law tail. Furthermore, we derive the Fokker-Planck equations governing the particle's displacement distributions. These equations show that the immobilization mechanism introduces only an additional term, $P(x,t)-\delta \left(x\right)$, weighted by the immobilization probability, without changing the fractional derivative operators. We also conduct a detailed comparison on the common quantities between the immobilization mechanism and stochastic resetting due to the similarities between them. This paper provides a theoretical perspective for studying transport processes under irreversible immobilization in complex media.

Close

PrevNext

View figure in article

• Anomalous diffusion
• Continuous-time random walk
• Fokker–Planck equation