Relationship between the anomalous diffusion and the fractal dimension of the environment

A dynamically equivalent atomistic electrochemical paradigm for the larger-scale experiments

Electrochemical systems possess a considerable part of modern technologies, such as the operation of rechargeable batteries and the fabrication of electronic components, which are explored both experimentally and computationally. The largest gap between the experimental observations and atomic-level simulations is their orders-of-magnitude scale difference. While the largest computationally affordable scale of the atomic-level computations is ∼ns and ∼nm, the smallest reachable scale in the typical experiments, using very high-precision devices, is ∼s and ∼μm. In order to close this gap and correlate the studies in the two scales, we establish an equivalent simulation setup for the given general experiment, which excludes the microstructure effects (i.e., solid-electrolyte interface), using the coarse-grained framework. The developed equivalent paradigm constitutes the adjusted values for the equivalent length scale (i.e., lEQ), diffusivity (i.e., DEQ), and voltage (i.e., VEQ). The time scale for the formation and relaxation of the concentration gradients in the vicinity of the electrode matches for both smaller scale (i.e., atomistic) equivalent simulations and the larger scale (i.e., continuum) experiments and could be utilized for exploring the cluster-level inter-ionic events that occur during the extended time periods. The developed model could offer insights for forecasting experiment dynamics and estimating the transition period to the steady-state regime of operation.

Random Walks on Comb-like Structures under Stochastic Resetting

We study the long-time dynamics of the mean squared displacement of a random walker moving on a comb structure under the effect of stochastic resetting. We consider that the walker's motion along the backbone is diffusive and it performs short jumps separated by random resting periods along fingers. We take into account two different types of resetting acting separately: global resetting from any point in the comb to the initial position and resetting from a finger to the corresponding backbone. We analyze the interplay between the waiting process and Markovian and non-Markovian resetting processes on the overall mean squared displacement. The Markovian resetting from the fingers is found to induce normal diffusion, thereby minimizing the trapping effect of fingers. In contrast, for non-Markovian local resetting, an interesting crossover with three different regimes emerges, with two of them subdiffusive and one of them diffusive. Thus, an interesting interplay between the exponents characterizing the waiting time distributions of the subdiffusive random walk and resetting takes place. As for global resetting, its effect is even more drastic as it precludes normal diffusion. Specifically, such a resetting can induce a constant asymptotic mean squared displacement in the Markovian case or two distinct regimes of subdiffusive motion in the non-Markovian case.

Fractal Stochastic Processes on Thin Cantor-Like Sets

We review the basics of fractal calculus, define fractal Fourier transformation on thin Cantor-like sets and introduce fractal versions of Brownian motion and fractional Brownian motion. Fractional Brownian motion on thin Cantor-like sets is defined with the use of non-local fractal derivatives. The fractal Hurst exponent is suggested, and its relation with the order of non-local fractal derivatives is established. We relate the Gangal fractal derivative defined on a one-dimensional stochastic fractal to the fractional derivative after an averaging procedure over the ensemble of random realizations. That means the fractal derivative is the progenitor of the fractional derivative, which arises if we deal with a certain stochastic fractal.

Electrical circuits involving fractal time

In this paper, we develop fractal calculus by defining improper fractal integrals and their convergence and divergence conditions with related tests and by providing examples. Using fractal calculus that provides a new mathematical model, we investigate the effect of fractal time on the evolution of the physical system, for example, electrical circuits. In these physical models, we change the dimension of the fractal time; as a result, the order of the fractal derivative changes; therefore, the corresponding solutions also change. We obtain several analytical solutions that are non-differentiable in the sense of ordinary calculus by means of the local fractal Laplace transformation. In addition, we perform a comparative analysis by solving the governing fractal equations in the electrical circuits and using their smooth solutions, and we also show that when α=1, we get the same results as in the standard version.

Diffusion–Advection Equations on a Comb: Resetting and Random Search

This review addresses issues of various drift–diffusion and inhomogeneous advection problems with and without resetting on comblike structures. Both a Brownian diffusion search with drift and an inhomogeneous advection search on the comb structures are analyzed. The analytical results are verified by numerical simulations in terms of coupled Langevin equations for the comb structure. The subordination approach is one of the main technical methods used here, and we demonstrated how it can be effective in the study of various random search problems with and without resetting.

Written Documents Analyzed as Nature-Inspired Processes: Persistence, Anti-Persistence, and Random Walks—We Remember, as Along Came Writing—T. Holopainen

Written communication is pivotal for societies to develop. However, lexicon and depth of information vary greatly among texts according to their purpose. Scientific texts, diffusion of science reports, general and area-specific news are all written differently. Thus, we explore the characterization of different text categories through a nature-inspired feature known as the Hurst parameter. We contend that the Hurst exponent is useful to unveil the rhetorical structure within written documents. We collected and processed texts in five categories: scientific articles, diffusion of science reports, business news, entertainment news, and random texts. Each category contains 350 documents. We found that the median for scientific texts has the highest value of the Hurst parameter (0.575), followed by business news (0.54); the median for randomly-generated texts is 0.48, which lies in the region associated with random walks. The median value for diffusion texts is 0.49, and for entertainment texts is 0.53. However, these two categories present high dispersion. We conclude that the Hurst parameter is a measure that quantifies the structure of communication in the selected categories of texts. Application of our finding in the field of e-research is discussed.

Quenched and annealed disorder mechanisms in comb models with fractional operators

Recent experimental findings on anomalous diffusion have demanded novel models that combine annealed (temporal) and quenched (spatial or static) disorder mechanisms. The comb model is a simplified description of diffusion on percolation clusters, where the comblike structure mimics quenched disorder mechanisms and yields a subdiffusive regime. Here we extend the comb model to simultaneously account for quenched and annealed disorder mechanisms. To do so, we replace usual derivatives in the comb diffusion equation by different fractional time-derivative operators and the conventional comblike structure by a generalized fractal structure. Our hybrid comb models thus represent a diffusion where different comblike structures describe different quenched disorder mechanisms, and the fractional operators account for various annealed disorder mechanisms. We find exact solutions for the diffusion propagator and mean square displacement in terms of different memory kernels used for defining the fractional operators. Among other findings, we show that these models describe crossovers from subdiffusion to Brownian or confined diffusions, situations emerging in empirical results. These results reveal the critical role of interactions between geometrical restrictions and memory effects on modeling anomalous diffusion.

Diffusion on Middle-ξ Cantor Sets

In this paper, we study Cζ-calculus on generalized Cantor sets, which have self-similar properties and fractional dimensions that exceed their topological dimensions. Functions with fractal support are not differentiable or integrable in terms of standard calculus, so we must involve local fractional derivatives. We have generalized the Cζ-calculus on the generalized Cantor sets known as middle-ξ Cantor sets. We have suggested a calculus on the middle-ξ Cantor sets for different values of ξ with 0<ξ<1. Differential equations on the middle-ξ Cantor sets have been solved, and we have presented the results using illustrative examples. The conditions for super-, normal, and sub-diffusion on fractal sets are given.