Variable-order porous media equations: Application on modeling the S&P500 and Bitcoin price return.

This article reveals a specific category of solutions for the 1+1 variable order (VO) nonlinear fractional Fokker-Planck equations. These solutions are formulated using VO q-Gaussian functions, granting them significant versatility in their application to various real-world systems, such as financial economy areas spanning from conventional stock markets to cryptocurrencies. The VO q-Gaussian functions provide a more robust expression for the distribution function of price returns in real-world systems. Additionally, we analyzed the temporal evolution of the anomalous characteristic exponents derived from our study, which are associated with the long-term (power-law) memory in time series data and autocorrelation patterns.

Space-time fractional porous media equation: Application on modeling of S&P500 price return.

We present the fractional extensions of the porous media equation (PME) with an emphasis on the applications in stock markets. Three kinds of "fractionalization" are considered: local, where the fractional derivatives for both space and time are local; nonlocal, where both space and time fractional derivatives are nonlocal; and mixed, where one derivative is local, and another is nonlocal. Our study shows that these fractional equations admit solutions in terms of generalized q-Gaussian functions. Each solution of these fractional formulations contains a certain number of free parameters that can be fitted with experimental data. Our focus is to analyze stock market data and determine the model that better describes the time evolution of the probability distribution of the price return. We proposed a generalized PME motivated by recent observations showing that q-Gaussian distributions can model the evolution of the probability distribution. Various phases (weak, strong super diffusion, and normal diffusion) were observed on the time evolution of the probability distribution of the price return separated by different fitting parameters [Phys. Rev. E 99, 062313 (2019)1063-651X10.1103/PhysRevE.99.062313]. After testing the obtained solutions for the S&P500 price return, we found that the local and nonlocal schemes fit the data better than the classic porous media equation.

Damage separation model: A replaceable particle method based on strain energy field.

We present a realistic model for simulating particle fragmentation in granular assemblies, the damage separation model (DSM), that addresses the limitations of previous methods by replacing the particle with smaller ones after fragmentation. The method is based on the calculation of the strain energy field inside the particle, and it solves the two major issues of the existing replaceable particle methods: the oversimplification of particle stress, and the unrealistic geometrical constraints needed in postbreakage replacements. Our model is formulated with three modules: (i) a boundary element calculation of stress and strain fields inside the spheropolygons that represent individual particles; (ii) a strain-energy-based theoretical framework to determine the onset of fragmentation; and (iii) an advanced geometrical algorithm, the subset separation method (SSM), to handle the postbreakage replacements in the discrete element simulations. Especially, the SSM effectively calculates the fragments required by the replacement with no geometrical limitation on the number, location, and orientation of the fracture planes. A uniaxial compression test based on laboratory setups is used to validate the method. A comparison is further conducted to study the performance of four different replaceable irregular particle methods. Results indicate that our method overcomes most of the existing issues, including stability, accuracy, and artificial constraints on the number and shape of fragments. The DSM has great potential for capturing the morphological changes of particle breakage and comminution with an unprecedented numerical resolution.

Photocatalytic hydrogen production by ternary heterojunction composites of silver nanoparticles doped FCNT-TiO2.

Silver nanoparticles doped with FCNT-TiO2 heterogeneous catalyst was prepared via one-step chemical reduction process and their efficacy was tested for hydrogen production under solar simulator. Crystallinity, purity, optical properties, and morphologies of the catalysts were examined by X-Ray diffraction, Raman spectroscopy, UV-Visible diffuse reflectance spectra, and Transmission Electron Microscopy. The chemical states and interface interactions were studied by X-ray photoelectron spectroscopy and Fourier transform infrared spectroscopy. The optimized catalyst showed 19.2 mmol g-1 h-1 of hydrogen production, which is 28.5 and 7 times higher than the pristine TiO2 nanoparticles and FCNT-TiO2 nanocomposite, respectively. The optimized catalyst showed stability up to 50 h under the solar simulator irradiation. The natural solar light irradiated catalyst showed ~2.2 times higher hydrogen production rate than the solar simulator irradiation. A plausible reaction mechanism of Ag NPs/FCNT-TiO2 photocatalyst was elucidated by investigating the beneficial co-catalytic role of Ag NPs and FCNTs for enhanced hydrogen production.

Q-Gaussian diffusion in stock markets.

We analyze the Standard & Poor's 500 stock market index from the past 22 years. The probability density function of price returns exhibits two well-distinguished regimes with self-similar structure: the first one displays strong superdiffusion together with short-time correlations and the second one corresponds to weak superdiffusion with weak time correlations. Both regimes are well described by q-Gaussian distributions. The porous media equation-a special case of the Tsallis-Bukman equation-is used to derive the governing equation for these regimes and the Black-Scholes diffusion coefficient is explicitly obtained from the governing equation.

Closed-form solutions for the Lévy-stable distribution.

The Lévy-stable distribution is the attractor of distributions which hold power laws with infinite variance. This distribution has been used in a variety of research areas; for example, in economics it is used to model financial market fluctuations and in statistical mechanics it is used as a numerical solution of fractional kinetic equations of anomalous transport. This function does not have an explicit expression and no uniform solution has been proposed yet. This paper presents a uniform analytical approximation for the Lévy-stable distribution based on matching power series expansions. For this solution, the trans-stable function is defined as an auxiliary function which removes the numerical issues of the calculations of the Lévy-stable distribution. Then, the uniform solution is proposed as a result of an asymptotic matching between two types of approximations called "the inner solution" and "the outer solution." Finally, the results of analytical approximation are compared to the numerical results of the Lévy-stable distribution function, making this uniform solution valid to be applied as an analytical approximation.

Static friction between rigid fractal surfaces.

Using spheropolygon-based simulations and contact slope analysis, we investigate the effects of surface topography and atomic scale friction on the macroscopically observed friction between rigid blocks with fractal surface structures. From our mathematical derivation, the angle of macroscopic friction is the result of the sum of the angle of atomic friction and the slope angle between the contact surfaces. The latter is obtained from the determination of all possible contact slopes between the two surface profiles through an alternative signature function. Our theory is validated through numerical simulations of spheropolygons with fractal Koch surfaces and is applied to the description of frictional properties of Weierstrass-Mandelbrot surfaces. The agreement between simulations and theory suggests that for interpreting macroscopic frictional behavior, the descriptors of surface morphology should be defined from the signature function rather than from the slopes of the contacting surfaces.

Stochastic collision and aggregation analysis of kaolinite in water through experiments and the spheropolygon theory.

An approach based on spheropolygons (i.e., the Minkowski sum of a polygon with N vertices and a disk with spheroradius r) is presented to describe the shape of kaolinite aggregates in water and to investigate interparticle collision dynamics. Spheropolygons generated against images of kaolinite aggregates achieved an error between 0.5% and 20% as compared to at least 32% of equivalent spheres. These spheropolygons were used to investigate the probability of collision (Pr[C]) and aggregation (Pr[A]) under the action of gravitational, viscous, contact (visco-elastic), electrostatic and van der Waals forces. In ortho-axial (i.e., frontal) collision, Pr[A] of equivalent spheres was always 1, however, stochastic analysis of collision among spheropolygons showed that Pr[A] decreased asymptotically with N increasing, and decreased further in peri-axial (i.e., tangential) collision. Trajectory analysis showed that not all collisions occurring within the attraction zone of the double layer resulted in aggregation, neither all those occurring outside it led to relative departure. Rather, the relative motion on surface asperities affected the intensity of contact and attractive forces to an extent to substantially control a collision outcome in either instances. Spheropolygons revealed therefore how external shape can influence particle aggregation, and suggested that this is equally important to contact and double layer forces in determining the probability of particle aggregation.

Simulation of counterflow pedestrian dynamics using spheropolygons.

Pedestrian dynamic models are typically designed for comfortable walking or slightly congested conditions and typically use a single disk or combination of three disks for the shape of a pedestrian. Under crowd conditions, a more accurate pedestrian shape has advantages over the traditional single or three-disks model. We developed a method for simulating pedestrian dynamics in a large dense crowd of spheropolygons adapted to the cross section of the chest and arms of a pedestrian. Our numerical model calculates pedestrian motion from Newton's second law, taking into account viscoelastic contact forces, contact friction, and ground-reaction forces. Ground-reaction torque was taken to arise solely from the pedestrians' orientation toward their preferred destination. Simulations of counterflow pedestrians dynamics in corridors were used to gain insight into a tragic incident at the Madrid Arena pavilion in Spain, where five girls were crushed to death. The incident took place at a Halloween Celebration in 2012, in a long, densely crowded hallway used as entrance and exit at the same time. Our simulations reconstruct the mechanism of clogging in the hallway. The hypothetical case of a total evacuation order was also investigated. The results highlights the importance of the pedestrians' density and the effect of counterflow in the onset of avalanches and clogging and provides an estimation of the number of injuries based on a calculation of the contact-force network between the pedestrians.

Temperature dependence of capillary dynamics: a multiphase and multicomponent adiabatic approach.

We present an analysis of the effect of the temperature on the flow of multiphase systems made of multiple miscible components in uniform cylindrical capillaries in adiabatic conditions. The temperature was explicitly included in the dynamic contact angle, tension at the three-phase contact line, and densities and viscosities of the fluids. The mathematical framework accounted for conservative forces (gravity, inertial, and interfacial tensions), nonconservative forces (viscous dissipation), and fluid retardation effects in the reservoirs at the two capillary ends. Temperature-dependent flow regimes ranged from nonoscillatory to oscillatory in a two-phase binary liquid (water-ethanol) system and in a two-phase pure liquid (ether) system. The Ca-Bo orbits highlighted dynamic attractors that depended on specific system characteristics as well as temperature. We conclude that temperature alone expresses and important role in the dynamical characteristics of capillary rise flow around its equilibrium.