Minimal model for anomalous diffusion

Generalization of the Kubo relation for confined motion and ergodicity breakdown

A time-dependent generalized Kubo relation is derived by introducing the notion of a diffusion function for a particle confined in a harmonic potential. The relation reduces to the standard Kubo relation as a special case but holds for anomalous diffusion, nonergodic processes, and bounded motion. We analyze in detail the behaviors of the diffusion and memory functions and report a generalized Stokes-Einstein relation concerning anomalous diffusion. Furthermore, we demonstrate that when a high finite-frequency cutoff is imposed on the noise spectral density, a breakdown in ergodicity accompanied by the appearance of nonstationarity in the velocity autocorrelation function occurs in forced systems. This breakdown is taken as explicit evidence for either decay-spring-memory or recovering-force effects leading to nonexponential relaxation kinematics.

Models of infiltration into homogeneous and fractal porous media with localized sources

We study a random-walk infiltration (RWI) model, in homogeneous and in fractal media, with small localized sources at their boundaries. In this model, particles released at a source, maintained at a constant density value, execute unbiased random walks over a lattice; a model that represents solute infiltration by diffusion into a medium in contact with a reservoir of fixed concentration. A scaling approach shows that the infiltrated length, area, or volume evolves in time as the number of distinct sites visited by a single random walker in the same medium. This is consistent with numerical simulations of the lattice model and exact and numerical solutions of the corresponding diffusion equation. In a Sierpinski carpet, the infiltrated area is expected to evolve as t^{D_{F}/D_{W}} (Alexander-Orbach relation), where D_{F} is the fractal dimension of the medium and D_{W} is the random-walk dimension; the numerical integration of the diffusion equation supports this result with very good accuracy and improves results of lattice random-walk simulations. In a Menger sponge in which D_{F}>D_{W} (i.e., a fractal with a dimension close to 3), a linear time increase of the infiltrated volume is theoretically predicted and confirmed numerically. Thus, no evidence of fractality can be observed in measurements of infiltrated volumes or masses in media where random walks are not recurrent, although the tracer diffusion is anomalous. We compare our findings with results for a fluid infiltration model in which the pressure head is constant at the source and the front displacement is driven by the local gradient of that head. Exact solutions in two and three dimensions and numerical results in a carpet show that this type of fluid infiltration is in the same universality class of RWI, with an equivalence between the head and the particle concentration. These results set a relation between different infiltration processes with localized sources and the recurrence properties of random walks in the same media.

Transient anomalous diffusion regimes in reversible adsorbing systems

A solvable, minimal model of diffusion in the presence of a reversible adsorption site is investigated. We show that the diffusive particles are influenced by the adsorbing site on transient times when they have anomalous subdiffusive behavior. However, the particle dispersion law crosses over to the normal diffusive regime on asymptotically long times. The subdiffusive regime is characterized by a t^{1/4} transient scaling with the same exponent as for the irreversible adsorption. On this transient time scale dominated by particle adsorption, there is a depletion of particles near the adsorbing site, and the typical width of the depletion zone grows in time as t^{1/4} with the same exponent as the subdiffusive dispersion. We show that having a nonzero desorption probability for the adsorbed particles produces a crossover towards normal diffusion on time scales larger than a characteristic reactive time, which we show scales with diffusivity and the adsorption site reactivity.