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We investigate the diffusion phenomenon of particles in the vicinity of a spherical colloidal particle where particles may be adsorbed/desorbed and react on the surface of the colloidal particle. The mathematical model comprises a generalized diffusion equation to govern bulk dynamics and kinetic equations which can describe non-Debye relaxations and is used for the colloid's surface. For the reaction processes, we also consider the presence of convolution kernels, which offer the flexibility of describing a single process or process with intermediate reactions before forming the final species. Our analysis focuses on analytical and numerical calculations to obtain the particles' behavior on the colloidal particle's surface and to determine how it affects the diffusion of particles around it. The solutions obtained show various behaviors that can be connected to anomalous diffusion phenomena and may be used to describe the ever-richer science of colloidal particles better. [ABSTRACT FROM AUTHOR]

We investigate the H-theorem for a class of generalized kinetic equations with fractional time-derivative, hyperbolic term, and nonlinear diffusion. When the H-theorem is satisfied, we demonstrate that different entropic forms may emerge due to the equation's nonlinearity. We obtain the entropy production related to these entropies and show that its form remains invariant. Furthermore, we investigate some behaviors for these equations from both numerical and analytical perspectives, showing a large class of behaviors connected with anomalous diffusion and their effects on entropy. [ABSTRACT FROM AUTHOR]

Many fundamental physical problems are modeled using differential equations, describing time- and space-dependent variables from conservation laws. Practical problems, such as surface morphology, particle interactions, and memory effects, reveal the limitations of traditional tools. Fractional calculus is a valuable tool for these issues, with applications ranging from membrane diffusion to electrical response of complex fluids, particularly electrolytic cells like liquid crystal cells. This paper presents the main fractional tools to formulate a diffusive model regarding time-fractional derivatives and modify the continuity equations stating the conservation laws. We explore two possible ways to introduce time-fractional derivatives to extend the continuity equations to the field of arbitrary-order derivatives. This investigation is essential, because while the mathematical description of neutral particle diffusion has been extensively covered by various authors, a comprehensive treatment of the problem for electrically charged particles remains in its early stages. For this reason, after presenting the appropriate mathematical tools based on fractional calculus, we demonstrate that generalizing the diffusion equation leads to a generalized definition of the displacement current. This modification has strong implications in defining the electrical impedance of electrolytic cells but, more importantly, in the formulation of the Maxwell equations in material systems. [ABSTRACT FROM AUTHOR]

In this paper, we expand the theory of semi-vector spaces and semi-algebras, both over the semi-field of nonnegative real numbers R 0 + . More precisely, we prove several new results concerning these theories. We introduce to the literature the concept of eigenvalues and eigenvectors of a semi-linear operator, describing how to compute them. The topological properties of semi-vector spaces, such as completeness and separability, are also investigated here. New families of semi-vector spaces derived from the semi-metric, semi-norm and semi-inner product, among others, are exhibited. Furthermore, we show several new results concerning semi-algebras. After this theoretical approach, we apply such a theory in fuzzy automata. More precisely, we describe the semi-algebra of A-fuzzy regular languages and we apply the theory of fuzzy automata for counting patterns in DNA sequences. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We investigate the transient dynamics of the Fisher equation under nonlinear diffusion and fractional operators. Firstly, we investigate the effects of the nonlinear diffusivity parameter in the integer-order Fisher equation, by considering a Gaussian distribution as the initial condition. Measuring the spread of the Gaussian distribution by u (0 , t) − 2 , our results show that the solution reaches a steady state governed by the parameters present in the logistic function in Fisher's equation. The initial transient is an anomalous diffusion process, but a power law cannot describe the whole transient. In this sense, the main novelty of this work is to show that a q-exponential function gives a better description of the transient dynamics. In addition to this result, we extend the Fisher equation via non-integer operators. As a fractional definition, we employ the Caputo fractional derivative and use a discretized system for the numerical approach according to finite difference schemes. We consider the numerical solutions in three scenarios: fractional differential operators acting in time, space, and in both variables. Our results show that the time to reach the steady solution strongly depends on the fractional order of the differential operator, with more influence by the time operator. Our main finding shows that a generalized q-exponential, present in the Tsallis formalism, describes the transient dynamics. The adjustment parameters of the q-exponential depend on the fractional order, connecting the generalized thermostatistics with the anomalous relaxation promoted by the fractional operators in time and space. [ABSTRACT FROM AUTHOR]

The description of neuronal activity has been of great importance in neuroscience. In this field, mathematical models are useful to describe the electrophysical behavior of neurons. One successful model used for this purpose is the Adaptive Exponential Integrate-and-Fire (Adex), which is composed of two ordinary differential equations. Usually, this model is considered in the standard formulation, i.e., with integer order derivatives. In this work, we propose and study the fractal extension of Adex model, which in simple terms corresponds to replacing the integer derivative by non-integer. As non-integer operators, we choose the fractal derivatives. We explore the effects of equal and different orders of fractal derivatives in the firing patterns and mean frequency of the neuron described by the Adex model. Previous results suggest that fractal derivatives can provide a more realistic representation due to the fact that the standard operators are generalized. Our findings show that the fractal order influences the inter-spike intervals and changes the mean firing frequency. In addition, the firing patterns depend not only on the neuronal parameters but also on the order of respective fractal operators. As our main conclusion, the fractal order below the unit value increases the influence of the adaptation mechanism in the spike firing patterns. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The fractional reaction–diffusion equation has been used in many real-world applications in fields such as physics, biology, and chemistry. Motivated by the huge application of fractional reaction–diffusion, we propose a numerical scheme to solve the fractional reaction–diffusion equation under different kernels. Our method can be particularly employed for singular and non-singular kernels, such as the Riemann–Liouville, Caputo, Fabrizio–Caputo, and Atangana–Baleanu operators. Moreover, we obtained general inequalities that guarantee that the stability condition depends explicitly on the kernel. As an implementation of the method, we numerically solved the diffusion equation under the power-law and exponential kernels. For the power-law kernel, we solved by considering fractional time, space, and both operators. In another example, we considered the exponential kernel acting on the time derivative and compared the numerical results with the analytical ones. Our results showed that the numerical procedure developed in this work can be employed to solve fractional differential equations considering different kernels. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This work studies the SIS model extended by fractional and fractal derivatives. We obtain explicit solutions for the standard and fractal formulations; for the fractional case, we study numerical solutions. As a real data example, we consider the Brazilian syphilis data from 2011 to 2021. We fit the data by considering the three variations of the model. Our fit suggests a recovery period of 11.6 days and a reproduction ratio (R 0) equal to 6.5. By calculating the correlation coefficient (r) between the real data and the theoretical points, our results suggest that the fractal model presents a higher r compared to the standard or fractional case. The fractal formulation is improved when two different fractal orders with distinguishing weights are considered. This modification in the model provides a better description of the data and improves the correlation coefficient. [ABSTRACT FROM AUTHOR]

We investigate a three-level system in the context of the fractional Schrödinger equation by considering fractional differential operators in time and space, which promote anomalous relaxations and spreading of the wave packet. We first consider the three-level system omitting the kinetic term, i.e., taking into account only the transition among the levels, to analyze the effect of the fractional time derivative. Afterward, we incorporate a kinetic term and the fractional derivative in space to analyze simultaneous wave packet transition and spreading among the levels. For these cases, we obtain analytical and numerical solutions. Our results show a wide variety of behaviors connected to the fractional operators, such as the non-conservation of probability and the anomalous spread of the wave packet. [ABSTRACT FROM AUTHOR]

We apply an extension of dual-phase-lag in thermal systems to predict the photoacoustic signal for transmission configuration and characteristics of the open photoacoustic cell technique. For this, we consider a particular case from Jeffrey's equation as an extension of the generalized Cattaneo equations. In this context, we obtain the temperature distribution under the effects of fractional differential operators, allowing the calculation of the Photoacoustic signal for the transmission set-up. The results show a rich class of behaviors related to the anomalous diffusion connected to these fractional operators. [ABSTRACT FROM AUTHOR]

In this work, we investigate the effect of the number of available adsorption sites for diffusing particles in a liquid confined between walls where the adsorption (desorption) phenomena occur. We formulate and numerically solve a model for particles governed by Fickian's law of diffusion, where the dynamics at the surfaces obey the Langmuir kinetic equation. The ratio between the available number of adsorption sites and the number of total particles are used as a control parameter. The investigation is carried out in terms of characteristic times of the system for different initial configurations, as well as the cases of identical or non-identical surfaces. We calculate the bulk and surface densities dynamics, as well as the variance of the system, and demonstrate that the number of sites affects the bulk, surface distributions, and diffusive regimes. [ABSTRACT FROM AUTHOR]

We investigate the solutions of a generalized diffusion-like equation by considering a spatial and time fractional derivative and the presence of non-local terms, which can be related to reaction or adsorption–desorption processes. We use the Green function approach to obtain solutions and evaluate the spreading of the system to show a rich class of behaviors. We also connect the results obtained with the anomalous diffusion processes. [ABSTRACT FROM AUTHOR]

The application of fractional calculus in the field of kinetic theory begins with questions raised by Bernoulli, Clausius, and Maxwell about the motion of molecules in gases and liquids. Causality, locality, and determinism underly the early work, which led to the development of statistical mechanics by Boltzmann, Gibbs, Enskog, and Chapman. However, memory and nonlocality influence the future course of molecular interactions (e.g., persistence of velocity and inelastic collisions); hence, modifications to the thermodynamic equations of state, the non-equilibrium transport equations, and the dynamics of phase transitions are needed to explain experimental measurements. In these situations, the inclusion of space- and time-fractional derivatives within the context of the continuous time random walk (CTRW) model of diffusion encodes particle jumps and trapping. Thus, we anticipate using fractional calculus to extend the classical equations of diffusion. The solutions obtained illuminate the structure and dynamics of the materials (gases and liquids) at the molecular, mesoscopic, and macroscopic time/length scales. The development of these models requires building connections between kinetic theory, physical chemistry, and applied mathematics. In this paper, we focus on the kinetic theory of gases and liquids, with particular emphasis on descriptions of phase transitions, inter-phase mixing, and the transport of mass, momentum, and energy. As an example, we combine the pressure–temperature phase diagrams of simple molecules with the corresponding anomalous diffusion phase diagram of fractional calculus. The overlap suggests links between sub- and super-diffusion and molecular motion in the liquid and the vapor phases. [ABSTRACT FROM AUTHOR]

We investigate the solutions of a two-dimensional Schrödinger equation in the presence of geometric constraints, represented by a backbone structure with branches, by taking a position-dependent effective mass for each direction into account. We use Green's function approach to obtain the solutions, which are given in terms of stretched exponential functions. The results can be linked to the properties of the system and show anomalous spreading for the wave packet. We also analyze the interplay between the backbone structure with branches constraining the different directions and the effective mass. In particular, we show how a fractional Schrödinger equation emerges from this scenario. [ABSTRACT FROM AUTHOR]

We propose an anomalous diffusion approach to analyze the electrical impedance response of electrolytic cells using time-fractional derivatives. We establish, in general terms, the conservation laws connected to a modified displacement current entering the fractional approach formulation of the Poisson–Nernst–Planck (PNP) model. In this new formalism, we obtain analytical expressions for the electrical impedance for the case of blocking electrodes and in the presence of general integrodifferential boundary conditions including time-fractional derivatives of distributed order. A conceptual scenario thus emerges aimed at exploring anomalous diffusion and surface effects on the impedance response of the cell to an external stimulus. [ABSTRACT FROM AUTHOR]

We review and extend some results for the fractional Schrödinger equation by considering nonlocal terms or potential given in terms of delta functions. For each case, we have obtained the solution in terms of the Green function approach. [ABSTRACT FROM AUTHOR]

We propose that a kinetic equation of fractional order may be used to account for the frequency dispersion of the effective parameters in the framework of the Langmuir approximation for the adsorption- desorption phenomena. A frequency dependence of these parameters naturally arises in this formalism, indicating that it may play a similar role of a generalization of the Langmuir isotherm obtained from a homogeneous distribution of relaxation times. The fractional approach formalism opens the possibility to consider different relaxation regimes characterizing the interfacial behaviour of electrolytic cells, and may be a powerful tool to interpret complex impedance spectroscopy data. [ABSTRACT FROM AUTHOR]

The use of mathematical models has been longly recognized as a fundamental step for in-depth process studies. Aimed at process control applications, the use of transfer functions identified from plant data represents an interesting modeling approach when compared to the derivation of first-principles-based models. Thermal systems are widely present in scientific and industrial scenarios, with applications in different fields, representing an important component of a cost operation. Towards this, the development of low-cost thermal systems has been reported aimed at understanding basic phenomena and validating modeling and control strategies. In the first part of the work, open loop identification was performed, leading to a fractional order transfer function, with order α=0.838±0.009. In the second part, a classical closed loop identification, using an integer order PI (proportional-integral) controlled, was used to identify an integer order transfer function. In the third part, the closed loop identified transfer function was employed to successfully preview the experimental closed loop behavior of the controlled and manipulated variables using different PI controllers. [ABSTRACT FROM AUTHOR]

We consider a quantum motion governed by the time-dependent Schrödinger equation on a three dimensional comb structure. We derive the corresponding fractional Schrödinger equations for the reduced probability density functions by projection of the three dimensional comb dynamics in the two- and one-dimensional configuration space. This represents another physical example of a system where fractional calculus emerges. We give closed-form solutions of the corresponding equations for the reduced probability density functions in terms of the Fox H-function, by using the Green's function approach. [ABSTRACT FROM AUTHOR]

In this study presents a theoretical and experimental approach to the diffusion of ions and conductometry of ionic solutions. The bases used the related theory of mass transfer, diffusion, of ionic solutions and ionic conductometry. Potassium acetate and tribasic sodium citrate salts at different temperatures (20°C;25°C;35°C) were used to study the mobility of ions. The results indicated a different electrical response that means a distinct behavior of ions when in aqueous solutions of pure salts or mixtures. The study verifies that the mixture of solutions and the temperature directly influenced the conductivity. For the temperature of 20°C the result of the sequence for the conductivity as a function of time was: 10% ethyl acetate - 90% citrate > 90% ethyl acetate - 10% citrate > 50% citrate - 50% acetate > citrate > acetate; to a temperature of 25°C to approximately sequence is: citrate > acetate > 50% ethyl acetate - 50% citrate > 90% ethyl acetate - 10% citrate > 10% ethyl acetate - 90% citrate (up to 10 min) and 35°C conductivity versus time was the following: 10% ethyl acetate - 90% citrate > 50% citrate - 50% acetate > citrate > 90% acetate - 10% citrate > acetate. Moreover, the experimental data obtained were adjusted to the model proposed by Drude. [ABSTRACT FROM AUTHOR]

Corruptive behaviour in politics limits economic growth, embezzles public funds and promotes socioeconomic inequality in modern democracies. We analyse well-documented political corruption scandals in Brazil over the past 27 years, focusing on the dynamical structure of networks where two individuals are connected if they were involved in the same scandal. Our research reveals that corruption runs in small groups that rarely comprise more than eight people, in networks that have hubs and a modular structure that encompasses more than one corruption scandal. We observe abrupt changes in the size of the largest connected component and in the degree distribution, which are due to the coalescence of different modules when new scandals come to light or when governments change. We show further that the dynamical structure of political corruption networks can be used for successfully predicting partners in future scandals. We discuss the important role of network science in detecting and mitigating political corruption. [ABSTRACT FROM AUTHOR]

Fractional order PI control and multivariable systems applied to the binary distillation model of Wood and Berry (1973) were studied. Genetic algorithms were implemented for controller tuning based on the minimization of ITAE of all loops, considering a set-point change in the top and in the bottom compositions. Simulations considered decentralized controllers, random measurement errors in controlled variables, and the presence of decoupling systems, the latter with the purpose of minimizing interaction between control loops A unique set of parameters for the fractional controller, able of handling different real operating situations, was obtained. The fractional PI controller significantly improved the performance of the multivariable control system when compared to the integer order PI controller. The use of fractional control led to a reduction of the ITAE value by 53.6% for a change of set-point of the bottom composition, when compared to the ITAE value obtained with the integer PI controller. Also in regard to the ITAE value, 58.9% of improvement was achieved when comparing systems with and without the loop decouplers. The fractional controllers were stable with errors of up to 10% of the value of the measured variable. [ABSTRACT FROM AUTHOR]

We investigate the diffusion of two different species in a semi-infinite medium considering the presence of linear reaction terms. The dynamics for these species is governed by fractional diffusion equations. We also consider the presence of an adsorption-desorption boundary condition. The solutions for this system are found in terms of the H function of Fox and by analyzing the behavior of the mean square displacement a rich class of diffusion processes is verified. In this sense, we show how the surface effects modify the bulk dynamics and promote an anomalous diffusion of system. [ABSTRACT FROM AUTHOR]

Particulate and granular materials are found in different chemical processes. Measuring and characterizing the mixing degree remains a challenge, s it is an important variable for process performance. The availability of reliable sensors for real time control is still incipient or expensive, an alternative remains on the development of mathematical models for mixing prediction. The most commonly used approach for solid mixing concern on either diffusive or advective/convective processes. Fractional calculus represents a novel approach and a growing research field for process modeling, being based on derivates of arbitrary order. Therefore, it represents an important and alternative tool for mixing process modeling. This work study the use of a fractional diffusion model to describe granular mixing in a rotary cylinder, considering finite type of boundary conditions. Experimental data previously reported were used for validation purposes. The proposed approach could successfully describe the experimental data, thus, can be used as an alternative tool for mixing evaluation. [ABSTRACT FROM AUTHOR]

Lead is a heavy metal effluent pollutant, which can be generated by different chemical plants. Literature reports different approaches for lead removal, however, great attention has been given to water hyacinths. Mathematical modeling of heavy metal sorption represents an important tool for in-depth process studies. This work proposes a new approach for modeling heavy metal isotherm sorption by using fractional calculus formalism. A novel isotherm based on Mittag-Leffler function is developed for lead sorption experimental data equilibrium modeling, in order to provide more accurate equilibrium information for sorption dynamic modeling. Experimental data were obtained using Eicchorniacrassipes as the hyacinth. Simulation results were compared to classical equilibrium sorption models. It was shown that the proposed isotherm, Epslon, provides better results when compared to classical models. [ABSTRACT FROM AUTHOR]

Tilapia (Oreochromis sp.) is the most important and abundant fish species in Brazil due to its adaptability to different environments. The development of tilapia-based products could be an alternative in order to aggregate value and increase fish meat consumption. However, there is little information available on fish burger freezing and cooking in the literature. In this work, the mathematical modeling of the fish burger baking was studied. Previously to the baking process, the fish burgers were assembled in cylindrical shape of height equal to 8 mm and diameter 100 mm and then baked in an electrical oven with forced heat convection at 150 °C. A T-type thermocouple was inserted in the burger to obtain its temperature profile at the central position. In order to describe the temperature of the burger during the baking process, lumped-parameter models of integer and fractional order and also a non-linear model due to heat capacity temperature dependence were considered. The burger physical properties were obtained from the literature. After proper parameter estimation tasks and statistical validation, the fractional order model could better describe the experimental temperature behavior, a value of 0.91 ± 0.02 was obtained for the fractional order of the system with correlation coefficient of 0.99. Therefore, with the better temperature prediction, process control, and economic optimization studies of the baking process can be conducted. [ABSTRACT FROM AUTHOR]

We investigate an intermittent process obtained from the combination of a nonlinear diffusion equation and pauses. We consider the porous media equation with reaction terms related to the rate of switching the particles from the diffusive mode to the resting mode or switching them from the resting to the movement. The results show that in the asymptotic limit of small and long times, the spreading of the system is essentially governed by the diffusive term. The behavior exhibited for intermediate times depends on the rates present in the reaction terms. In this scenario, we show that, in the asymptotic limits, the distributions for this process are given by in terms of power laws which may be related to the q-exponential present in the Tsallis statistics. Furthermore, we also analyze a situation characterized by different diffusive regimes, which emerges when the diffusive term is a mixing of linear and nonlinear terms. [ABSTRACT FROM AUTHOR]

More than a half of world population is now living in cities and this number is expected to be two-thirds by 2050. Fostered by the relevancy of a scientific characterization of cities and for the availability of an unprecedented amount of data, academics have recently immersed in this topic and one of the most striking and universal finding was the discovery of robust allometric scaling laws between several urban indicators and the population size. Despite that, most governmental reports and several academic works still ignore these nonlinearities by often analyzing the raw or the per capita value of urban indicators, a practice that actually makes the urban metrics biased towards small or large cities depending on whether we have super or sublinear allometries. By following the ideas of Bettencourt et al. [PLoS ONE 5 (2010) e13541], we account for this bias by evaluating the difference between the actual value of an urban indicator and the value expected by the allometry with the population size. We show that this scale-adjusted metric provides a more appropriate/informative summary of the evolution of urban indicators and reveals patterns that do not appear in the evolution of per capita values of indicators obtained from Brazilian cities. We also show that these scale-adjusted metrics are strongly correlated with their past values by a linear correspondence and that they also display crosscorrelations among themselves. Simple linear models account for 31%–97% of the observed variance in data and correctly reproduce the average of the scale-adjusted metric when grouping the cities in above and below the allometric laws. We further employ these models to forecast future values of urban indicators and, by visualizing the predicted changes, we verify the emergence of spatial clusters characterized by regions of the Brazilian territory where we expect an increase or a decrease in the values of urban indicators. [ABSTRACT FROM AUTHOR]