Non-Markovian Diffusion and Adsorption-Desorption Dynamics: Analytical and Numerical Results.

The interplay of diffusion with phenomena like stochastic adsorption-desorption, absorption, and reaction-diffusion is essential for life and manifests in diverse natural contexts. Many factors must be considered, including geometry, dimensionality, and the interplay of diffusion across bulk and surfaces. To address this complexity, we investigate the diffusion process in heterogeneous media, focusing on non-Markovian diffusion. This process is limited by a surface interaction with the bulk, described by a specific boundary condition relevant to systems such as living cells and biomaterials. The surface can adsorb and desorb particles, and the adsorbed particles may undergo lateral diffusion before returning to the bulk. Different behaviors of the system are identified through analytical and numerical approaches.

Nonlinear Fokker-Planck Equations, H-Theorem and Generalized Entropy of a Composed System.

We investigate the dynamics of a system composed of two different subsystems when subjected to different nonlinear Fokker-Planck equations by considering the H-theorem. We use the H-theorem to obtain the conditions required to establish a suitable dependence for the system's interaction that agrees with the thermodynamics law when the nonlinearity in these equations is the same. In this framework, we also consider different dynamical aspects of each subsystem and investigate a possible expression for the entropy of the composite system.

Results for Nonlinear Diffusion Equations with Stochastic Resetting.

In this study, we investigate a nonlinear diffusion process in which particles stochastically reset to their initial positions at a constant rate. The nonlinear diffusion process is modeled using the porous media equation and its extensions, which are nonlinear diffusion equations. We use analytical and numerical calculations to obtain and interpret the probability distribution of the position of the particles and the mean square displacement. These results are further compared and shown to agree with the results of numerical simulations. Our findings show that a system of this kind exhibits non-Gaussian distributions, transient anomalous diffusion (subdiffusion and superdiffusion), and stationary states that simultaneously depend on the nonlinearity and resetting rate.

The Kinetics of Sorption-Desorption Phenomena: Local and Non-Local Kinetic Equations.

The kinetics of adsorption phenomena are investigated in terms of local and non-local kinetic equations of the Langmuir type. The sample is assumed in the shape of a slab, limited by two homogeneous planar-parallel surfaces, in such a manner that the problem can be considered one-dimensional. The local kinetic equations in time are analyzed when both saturation and non-saturation regimes are considered. These effects result from an extra dependence of the adsorption coefficient on the density of adsorbed particles, which implies the consideration of nonlinear balance equations. Non-local kinetic equations, arising from the existence of a time delay characterizing a type of reaction occurring between a bulk particle and the surface, are analyzed and show the existence of adsorption effects accompanied by temporal oscillations.

Unusual diffusing regimes caused by different adsorbing surfaces.

A confined liquid with dispersed neutral particles is theoretically studied when the limiting surfaces present different dynamics for the adsorption-desorption phenomena. The investigation considers different non-singular kernels in the kinetic equations at the walls, where the suitable choice of the kernel can account for the relative importance of physisorption or chemisorption. We find that even a small difference in the adsorption-desorption rate of one surface (relative to the other) can drastically affect the behavior of the whole system. The surface and bulk densities and the dispersion are calculated when several scenarios are considered and anomalous-like behaviors are found. The approach described here is closely related to experimental situations, and can be applied in several contexts such as dielectric relaxation, diffusion-controlled relaxation in liquids, liquid crystals, and amorphous polymers.