Anomalous diffusion in stochastic systems with nonhomogeneously distributed traps

TL;DR

This paper investigates anomalous diffusion in nonhomogeneous media with traps, analyzing how trap density and memory effects influence subdiffusion and superdiffusion, with exact solutions for Gaussian and Lévy statistics.

Contribution

It introduces a position-dependent subordination framework and provides exact solutions for diffusion in fractal trap structures with Gaussian and Lévy statistics.

Findings

System can exhibit subdiffusion or enhanced diffusion depending on parameters.

Exact solutions for trap density and moments in fractal media.

Diffusion regimes differ near boundaries and in the bulk.

Abstract

The stochastic motion in a nonhomogeneous medium with traps is studied and diffusion properties of that system are discussed. The particle is subjected to a stochastic stimulation obeying a general L\'evy stable statistics and experiences long rests due to traps the density of which depends on the position. The memory is taken into account by subordination of that process to a random time; then the subordination equation is position-dependent. The problem is approximated by means of a decoupling of the trap geometry and memory and exactly solved for a power-law trap density, corresponding to a fractal medium structure, in the case of the Gaussian statistics: the density distribution and moments are derived. Depending on geometry and memory parameters, the system may reveal both the subdiffusion and enhanced diffusion. A similar analysis is performed for the L\'evy flights where the finiteness of the variance follows from a multiplicative noise, as a result of impurities and defects at the boundary. Two diffusion regimes are found: in the bulk and near the surface. The anomalous diffusion exponent as a function of the system parameters is derived.