Numerical investigation of the spreading-receding cycle in a concentration-dependent lattice gas automaton diffusion model

This work forms the second part of a numerical study about the dynamics of diffusive fronts in concentration-dependent diffusion processes. We previously demonstrated that one-dimensional spreading of a density front in a concentration-dependent microscopically heterogeneous, macroscopically homogeneous isotropic lattice gas automaton (LGA) substantially deviates from the t^1∕2 relation expected from Fick’s law over large periods of time. The time exponent was found to be larger than 1∕2, i.e., spreading of the density front is enhanced with respect to standard Fickian diffusion. In this note, we specifically investigate the dynamics of receding by using the same LGA model. We show here that the receding process essentially scales as t^1∕2. The LGA simulations of diffusive fronts thus lead to the paradoxical result of Fick’s-compatible receding and anomalous superdiffusive spreading for the same microscopic random structure and the same boundary conditions. The results also suggest that hysteresis of the spreading-receding cycle could arise from the contrasted dynamics between spreading and receding. A conceptual model of “offer and demand” which includes both the diffusivity gradient dD(ρ)∕dρ and the conditions applied at boundaries as main parameters is proposed to tentatively account for the dynamics of diffusive fronts in concentration-dependent diffusion processes.