Characterizing unstructured data with the nearest neighbor permutation entropy.

Permutation entropy and its associated frameworks are remarkable examples of physics-inspired techniques adept at processing complex and extensive datasets. Despite substantial progress in developing and applying these tools, their use has been predominantly limited to structured datasets such as time series or images. Here, we introduce the k-nearest neighbor permutation entropy, an innovative extension of the permutation entropy tailored for unstructured data, irrespective of their spatial or temporal configuration and dimensionality. Our approach builds upon nearest neighbor graphs to establish neighborhood relations and uses random walks to extract ordinal patterns and their distribution, thereby defining the k-nearest neighbor permutation entropy. This tool not only adeptly identifies variations in patterns of unstructured data but also does so with a precision that significantly surpasses conventional measures such as spatial autocorrelation. Additionally, it provides a natural approach for incorporating amplitude information and time gaps when analyzing time series or images, thus significantly enhancing its noise resilience and predictive capabilities compared to the usual permutation entropy. Our research substantially expands the applicability of ordinal methods to more general data types, opening promising research avenues for extending the permutation entropy toolkit for unstructured data.

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.

Entropy Production in a Fractal System with Diffusive Dynamics.

We study the entropy production in a fractal system composed of two subsystems, each of which is subjected to an external force. This is achieved by using the H-theorem on the nonlinear Fokker-Planck equations (NFEs) characterizing the diffusing dynamics of each subsystem. In particular, we write a general NFE in terms of Hausdorff derivatives to take into account the metric of each system. We have also investigated some solutions from the analytical and numerical point of view. We demonstrate that each subsystem affects the total entropy and how the diffusive process is anomalous when the fractal nature of the system is considered.

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.