Fractal Diffusion Research Articles

FRACTAL DIFFUSION IN AN ISOTROPOUS MEDIUM

This paper provides a comprehensive examination of fractal diffusion in an isotropic medium. This process has a fascinating temporal memory and an extremely high initial response, which are unquestionably characterized by the Caputo fractional derivative. Its spatio-symmetric property is described by the Riesz fractional derivative. To illustrate the diffusion mechanism, we present an example of a toxic gas leak in a subway station. The numerical results are extremely promising and challenging. They clearly indicate that a MEMS-based gas sensor can track a toxic gas leak immediately due to its extremely fast diffusion at the initial time. This is of great importance for public safety. Furthermore, the diffusion process is primarily contingent upon the density of the air, which we can easily control. The diffusion pattern in an isotropic medium exhibits fractal behavior, which is extremely helpful for designing an emergency system for evacuating people from the center of toxic gas leaks by changing air density or temperature. This paper presents a rigorous mathematical concept for monitoring systems and early warning systems for toxic gas leaks.

Single-Molecule Tracking in Live Cell without Immobilization or without Hydrodynamic Flow by Simulations: Thermodynamic Jitter

Experiments to measure a single molecule/particle, i.e., an individual molecule/particle, at room temperature or under physiological conditions without immobilization—for example, on a surface or without significant hydrodynamic flow—have so far failed. This failure has given impetus to the underlying theory of Brownian molecular motion towards its stochastics due to diffusion. Quantifying the thermodynamic jitter of molecules/particles inspires many and forms the theoretical basis of single-molecule/single-particle biophysics and biochemistry. For the first time, our simulation results for a live cell (cytoplasm) show that the tracks of individual single molecules are localized in Brownian motion, while there is fanning out in fractal diffusion (anomalous diffusion).

Butterfly diffusion over sparse point sets

The graph-based random walk model of fractal diffusion is limited in its use to the connected sets and does not allow for direct fractal dimension estimation based on observed data. We discuss a task of directly obtaining accurate fractal dimension estimates and propose butterfly diffusion as an alternative approach using an explicit relation between walk and fractal dimensions. The validity of the presented approach is evaluated and statistical properties of the dimension estimates are presented. Through experiments on self-similar sets, we demonstrate the effectiveness of this approach in producing unbiased dimension estimates, offering a promising tool for fractal analysis and Monte Carlo simulations. The estimate of fractal dimension can be also created from spectral dimension, but this approach is less general and less accurate.

The visual preference and physiological response to Sierpinski triangle acoustic diffuser: An experimental study using VR technique

Diffusive surfaces should be optimally designed for acoustic and aesthetic purposes. Fractals are commonly used to adapt to the parametric demands of interface design combining mathematical calculation and artistic creation. The author’s previous study proposed an improved fractal acoustic structure based on the Sierpinski-triangle building rule. However, there is insufficient research on quantifying the physiological and visual benefits of such fractal features. Focusing on fractal iteration and randomness of module arrangement as fractal parameters, this study investigates in 3D-virtual reality meeting rooms and identifies the influence of fractal diffusion structures on visual preferences and physiological responses. Variance analysis is used to construct models of fractal parameters, subjective preference factors, and physiological indicators. Results suggest that the Sierpinski-triangle pattern iterated twice with mid-high complexity (quantified by the fractal dimension D = 1.58) is significantly more visually preferred than other fractals. When the modules are at low randomness (quantified by P = 0.25), more random module arrangements elicited better beneficial outcomes. However, as the degree of randomness increased, effects became increasingly irregular. Subjective evaluation factors and physiological indicators show a strong correlation. To sum up, the use of the medium to mid-high complexity fractals in interior spaces would help enhance the visual preferences and stress relief of occupants.

New Model for Coal Gas Diffusion Based on the Fractal Tree-Like Bifurcation Network Structure.

To investigate the nonstationary diffusion and transport law of gas in coal, an innovative fractal diffusion model based on fractal theory and a treelike bifurcation network under different diffusion modes were built. In addition, the quantitative relationships among the diffusion coefficient, temperature, pressure, and structural parameters were determined. The model considers the effect of pore tortuosity and connectivity on gas diffusion, which renders it more realistic than the previously presented single-pore diffusion models. Moreover, each parameter in the model has a definite physical meaning and does not contain any empirical constants. The applicability of the new model was verified by experimental data and other model. Subsequently, a sensitivity analysis of the gas diffusion coefficient was performed to study the effect of the microstructural parameters on gas diffusion. Finally, the dispersion degree of the diffusion coefficients at different temperatures and pressures was analyzed to determine the main influencing factors of gas diffusion at different diffusion stages in coal and study their evolution. The results show that the gas diffusion state is more sensitive to pressure variations. The diffusion behavior of gases in coal-based porous media is more influenced by the temperature and pressure at the beginning of the diffusion process.

A multi-scale fractal model of gas flow considering the evolution of kerogen microstructure and the multi-physical coupling

Shale is a complex porous medium. The migration mechanism within its different microstructural systems is crucial for the efficient recovery of shale gas. The kerogen microstructure influences the adsorption–desorption effect of the reservoir. This study accounts for the heterogeneity of shale and uses fractal dimension to characterize the fracture and inorganic matrix systems. It also establishes a kerogen fractal diffusion model by integrating fractal diffusion with fractal theory. Furthermore, this study analyzes the effect of evolution of micro structural parameters of kerogen on macro permeability, such as: (1) kerogen fractal dimension; (2) effective diffusion coefficient of kerogen; (3) maximum kerogen aperture. Numerical simulation shows that: The maximum pore size and porosity of kerogen in direct proportion to fractal dimension and effective diffusion coefficient. The fractal dimension and effective diffusion coefficient decrease gradually during shale gas extraction. The fractal effective diffusion has a significant impact on cumulative gas extraction.

Analytical solutions of fractal and anomalous diffusion models for pressure and rate transient analysis

In this study, we examine the modeling of the naturally fractured reservoirs based on the fractal, Metzler, and Raghavan anomalous diffusion models which are different from Darcy flux law. The governing equations of these models are presented for a vertical well producing at either a constant rate (CR) or constant bottomhole pressure (CBHP) production in an infinite-acting reservoir. New analytical and early- and late-time asymptotic approximate solutions were derived for the finite-wellbore problem for the Raghavan anomalous model. The approximate solutions identify the fractal and anomalous diffusion parameters that influence behaviors of the early- and late-time pressure and rate transient data. The method of Laplace transformation is used to find the solutions in dimensionless forms which are then inverted to the real-time domain using a numerical inversion algorithm. In addition, we compare the solutions for the fractal, Metzler anomalous, and Raghavan anomalous models and delineate the similarities and differences among these solutions. We show the effect of fractal parameters on diffusion models, the relationship between the fractal model, and Raghavan and Metzler anomalous models, in terms of dimensionless pressure and dimensionless pressure derivative responses in the reservoir. We show that the “dimensionless” pressure defined by Raghavan is not a dimensionless quantity unless the Euclidean dimension d=2 (or for a fractal structure with the mass fractal dimension df=2). To the best of our knowledge, in the literature, there exists no study providing a detailed review and comparing the applicability of each diffusion model on real field well-test data as well as providing new analytical solutions and early- and late-time asymptotic approximate solutions for the anomalous diffusion for both the CR and CBHP cases for the finite-wellbore problem. Hence, this study fills this gap. In addition, we apply the three diffusion models to an interference well test data from the Kizildere geothermal reservoir in Turkey by using the ensemble smoother with multiple data assimilation (ES-MDA) method. The results show that the fractal model in terms of the goodness of fit (root-mean-square error – RMSE) best matches the field test data.

Local Polya fluctuations of Riesz gravitational fields and the Cauchy problem

We consider a pseudodifferential equation of parabolic type with a fractional power of the Laplace operator of order $\alpha\in(0;1)$ acting with respect to the spatial variable. This equation naturally generalizes the well-known fractal diffusion equation. It describes the local interaction of moving objects in the Riesz gravitational field. A simple example of such system of objects is stellar galaxies, in which interaction occurs according to Newton's gravitational law. The Cauchy problem for this equation is solved in the class of continuous bounded initial functions. The fundamental solution of this problem is the Polya distribution of probabilities $\mathcal{P}_\alpha(F)$ of the force $F$ of local interaction between these objects. With the help of obtained solution estimates the correct solvability of the Cauchy problem on the local field fluctuation coefficient under certain conditions is determined. In this case, the form of its classical solution is found and the properties of its smoothness and behavior at the infinity are studied. Also, it is studied the possibility of local strengthening of convergence in the initial condition. The obtained results are illustrated on the $\alpha$-wandering model of the Lévy particle in the Euclidean space $\mathbb{R}^3$ in the case when the particle starts its motion from the origin. The probability of this particle returning to its starting position is investigated. In particular, it established that this probability is a descending to zero function, and the particle "leaves" the space $\mathbb{R}^3$.

CHARACTERIZATION OF CHLORIDE IONS DIFFUSION IN CONCRETE USING FRACTIONAL BROWNIAN MOTION RUN WITH POWER LAW CLOCK

In this paper, we propose a revised fractional Brownian motion run with a nonlinear clock (fBm-nlc) model and utilize it to illustrate the microscopic mechanism analysis of the fractal derivative diffusion model with variable coefficient (VC-FDM). The power-law mean squared displacement (MSD) links the fBm-nlc model and the VC-FDM via the two-parameter power law clock and the Hurst exponent is 0.5. The MSD is verified by using the experimental points of the chloride ions diffusion in concrete. When compared to the linear Brownian motion, the results show that the power law MSD of the fBm-nlc is much better in fitting the experimental points of chloride ions in concrete. The fBm-nlc clearly interprets the VC-FDM and provides a microscopic strategy in characterizing different types of non-Fickian diffusion processes with more different nonlinear functions.

Dynamic studies of copper adsorption on mesoporous alginate beads using an integrated approach of fractal‐like kinetic reaction and diffusion modeling

AbstractThe kinetics of Cu(II) adsorption on calcium alginate (AL) beads was elucidated using reaction‐based and diffusion models. In the present work, the fractal psuedo first‐order kinetic model has been used in combination with the Vermeulen diffusion model. The Cu(II) adsorption parameters were optimized by batch adsorption studies. The AL beads were characterized using field emission scanning electron microscope (FESEM), energy‐dispersive X‐ray (EDAX), and Brunauer‐Emmett‐Teller (BET) surface area analysis. BET studies indicated beads to be mesoporous. The combined studies using fractal reaction and diffusion models helped to trace the path of mass transport of the Cu(II) ions from the bulk of the solution onto the surface of the bead and into the porous mass of the beads. The use of the fractal approach enabled us to determine the heterogeneity of the bead surface and hence gave a better insight into the understanding of the sorption phenomenon. The adsorption rate constants calculated using the equations for the “Fractal” and “Classical” adsorption kinetic models were drastically different, thus pointing toward the explicit need to model the data using diffusion kinetic models attributing to surface heterogeneity. Such combined studies may result in a better understanding of the adsorption of Cu(II) on the AL beads.

Diffusion migration behavior of gas in unsaturated fractured soils: Fractal analytical study

Gas diffusivity is a crucial parameter for modeling the mass flow and transport in the fields of engineering, agriculture and environment. However, existing models were only developed for single-porosity media and underestimated the diffusivity of gas in fractured or cracked media. This study develops a fractal diffusion model of gas in unsaturated dual-porosity media. Based on statistical characteristics, the fracture length is assumed to obey the fractal law and the aperture is defined as a power function of its length. The transition regime of molecular diffusion is adopted to derive the upscaling diffusion model. The developed model considers the heterogeneity of porosity and water distribution by calibrating the relationship between local and bulk properties. 19 sets of data are used to evaluate the robustness of the proposed fractal model. The results show that in dual-porosity media, fractures will serve as preferential channels and significantly enhance gas migration. Besides, the heterogeneity distribution of water content inhibits molecular diffusion by potentially decreasing effective diffusion paths. The influence of fractures on gas diffusion is not only related to fracture volume but also to its scaling fractal dimension and other structures parameters. Finally, the superiority of the proposed fractal model can also be found from the comparison with other models.

Fractal diffusion from a geometric Ricci flow

Fractal diffusion from a geometric Ricci flow

A fluid–solid-chemical coupled fractal model for simulating concrete damage and reinforcement corrosion

Chloride ion invasion into reinforced concrete is a common disease. In this study, we combine the diffusion of free chloride ions, the corrosive expansion of iron, and the destruction of concrete. Considering the fractal characteristics of concrete pore structure, a fractal diffusivity model is established. A fluid–solid-chemical coupling model was established considering the non-uniform corrosion of iron. The correctness of the model is verified by comparison with experimental data. The corrosion thickness and pore structure evolution of concrete under different chloride ion concentration invasion and different porosity were simulated. The results show that: (1) the fractal dimension of tortuosity decreases gradually and then sharply at the initial stage of corrosion; (2) compared with concrete with high porosity, concrete with low porosity invaded by chloride ions has a more significant improvement in its own transport; (3) concrete porosity has nonlinear effect on steel corrosion and concrete pore structure evolution.

First-passage statistics of colloids on fractals: Theory and experimental realization.

In nature and technology, particle dynamics frequently occur in complex environments, for example in restricted geometries or crowded media. These dynamics have often been modeled invoking a fractal structure of the medium although the fractal structure was only indirectly inferred through the dynamics. Moreover, systematic studies have not yet been performed. Here, colloidal particles moving in a laser speckle pattern are used as a model system. In this case, the experimental observations can be reliably traced to the fractal structure of the underlying medium with an adjustable fractal dimension. First-passage time statistics reveal that the particles explore the speckle in a self-similar, fractal manner at least over four decades in time and on length scales up to 20 times the particle radius. The requirements for fractal diffusion to be applicable are laid out, and methods to extract the fractal dimension are established.

Lattice simulation-based diffusion modelling of 3D chromatin structure

Eukaryotic nuclear genome is extensively folded in the nuclei, and the chromatin structure experiences dramatic changes, i.e., condensation and decondensation, during the cell cycle. However, a model to persuasively explain the preserved chromatin interactions during cell cycle remains lacking. In this paper, we developed two simple, lattice-based models that mimic polymer fiber decondensation from initial fractal or anisotropic condensed status, using Markov Chain Monte Carlo (MCMC) methods. By simulating the dynamic decondensation process, we observed about 8.17% and 2.03% of the interactions preserved in the condensation to decondensation transition, in the fractal diffusion and anisotropic diffusion models, respectively. Intriguingly, although interaction hubs, as a physical locus where a certain number of monomers inter-connected, were observed in diffused polymer models in both simulations, they were not associated with the preserved interactions. Our simulation demonstrated that there might exist a small portion of chromatin interactions that preserved during the diffusion process of polymers, while the interacted hubs were more dynamically formed and additional regulatory factors were needed for their preservation.

Simulating stochastic adsorption of diluted solute molecules at interfaces.

This report uses Monte Carlo simulations to connect stochastic single-molecule and ensemble surface adsorption of molecules from dilute solutions. Monte Carlo simulations often use a fundamental time resolution to simulate each discrete step for each molecule. The adsorption rate obtained from such a simulation surprisingly contains an error compared to the results obtained from the traditional method. Simulating adsorption kinetics is interesting in many processes, such as mass transportation within cells, the kinetics of drug–receptor interactions, membrane filtration, and other general reaction kinetics in diluted solutions. Thus, it is important to understand the origin of the disagreement and find a way to correct the results. This report reviews the traditional model, explains the single-molecule simulations, and introduces a method to correct the results of adsorption rate. For example, one can bin finer time steps into time steps of interest to simulate the fractal diffusion or simply introduce a correction factor for the simulations. Then two model systems, self-assembled monolayer (SAM) and biosensing on the patterned surface, are simulated to check the accuracy of the equations. It is found that the adsorption rate of SAM is highly dependent on the conditions and the uncertainty is large. However, the biosensing system is relatively accurate. This is because the concentration gradient near the interface varies significantly with reaction conditions for SAMs while relatively stable for the biosensing system.

Fractal diffusion patterns of periodic points in the Mandelbrot set

The Mandelbrot set is one of the most profound study objects in the fractal field. However, the characteristics of internal structures of the Mandelbrot set are not fully studied, which is an obstacle to further understanding of its convergence mechanism. Here, we demonstrate that if the threshold is not taken when calculating the periodic number in the iteration process, the internal structures formed by the collection of periodic points will show fractal diffusion patterns from the theoretical area to the target area. Even when the threshold is taken, these diffusion characteristics remain until the threshold is larger than a certain value. This critical value of the threshold can be regarded as a transition gate of the internal structure from an unstable state to a stable state. The findings of this study indicate that the fractal patterns of the Mandelbrot set can not only exist in the outer area but also the interior area of the Mandelbrot set.

Fractal neutrons diffusion equation: Uniformization of heat and fuel burn-up in nuclear reactor

In this study, the fractal neutrons diffusion equation is constructed based on the concept of fractal anisotropy and product-like fractal measure introduced by Li and Ostoja-Starzewski using the method of dimensional regularization. In this approach, the global forms of dynamical equations are cast in forms involving integer-order integrals while the local forms are expressed by partial differential equations with derivatives of integer order. This theory is characterized by an effective buckling which varies with distance and a diffusion damping characterized by a spatially variable coefficient. These terms are major in nuclear reactor physics since the magnitude of energy dissipation in fission depends on the position and nature of interaction. We have discussed the cases of parallelepiped and finite cylinder reactors. In both geometrical types of reactors, it was observed that the ratio of the maximum flux to the average flux is too small since the uniformization of the heat fabrication and the fuel burn-up in the nuclear reactor is potential. This has important consequences on safety of the fuel operation and on the types of nuclear materials used in nuclear engineering reactors. Further details are discussed as well.

Trap-controlled fractal diffusion model of an atypical dielectric response

• New model of non-Debye dielectric relaxation, which leads to a new expression for an atypical two-power-law dielectric response, is developed. • A new expression for the complex dielectric permittivity in the frequency domain, which describes well the experimental dielectric spectra demonstrating atypical relaxation behavior, is obtained. • Relationships that relate microscopic parameters of the medium and parameters of atypical dielectric response, such as relaxation time and fractional-power exponents characterizing the shape of the dielectric loss peak are obtained. Within the framework of the multiple trapping and the fractal diffusion equation models a new expression for the complex dielectric permittivity in the frequency domain is found. The new expression allows us to describe an atypical two-power-law dielectric response. An analytical expression for the relaxation time, which is expressed through microscopic parameters of the structural and energy disorder of the system is obtained. Good agreement of the new expression for the complex dielectric permittivity with observed dielectric responses of porous glass composite ADP and nanoscale Si-layered system is demonstrated.

Fractal Characteristics and Its Controlling Factors of Nanopore of Coal from Shanxi Province, North China.

Much attention has been recently paid to the Carboniferous-Permian coal-bearing strata in Shanxi Province, now the largest producing coalbed methane field in China. In this study, a comprehensive approach of mercury injection, low-temperature liquid nitrogen adsorption, and permeability experiments was adopted to investigate the structure and fractal characteristics of nanopores in the Carboniferous-Permian coal (with 0.77%˜3.04% Ro,ran). Based on the fractal model, two fractal dimensions D1 and D₂ corresponding to diffusion pore (<65 nm) and seepage pore (pore size ≥65 nm), respectively, were calculated, and the relationships between the fractal dimensions with the pore structure parameters and permeability are discussed here. The results indicate that the studied coal samples have good fractal characteristics and that the calculated linear correlation coefficients are higher than 0.80. The fractal dimension D1 of the diffusion pores ranges from 2.3777 to 2.4624, with an average of 2.4173, while the fractal dimension D₂ of the seepage pores is between 2.5844 and 2.6256, with an average of 2.5990. The fractal dimensions D1 of the diffusion pores increases with an increase in the BET specific surface area, vitrinite content, and Ro,ran while it decreases with an increase in the permeability, and has a weak correlation with the total pore volume. The correlation coefficients R² for the fractal dimension D₂ of the seepage pores, pore parameters, permeability, and maceral composition ranges from 0.0357 to 0.2551. These results indicate that uncertain relationships exist among these parameters.