Brownian Motion Research Articles

Anomalous Random Flights and Time-Fractional Run-and-Tumble Equations

Random flights (also called run-and-tumble walks or transport processes) represent finite velocity random motions changing direction at any Poissonian time. These models in d-dimension, can be studied giving a general formulation of the problem valid at any spatial dimension. The aim of this paper is to extend this general analysis to time-fractional processes arising from a non-local generalization of the kinetic equations. The probabilistic interpretation of the solution of the time-fractional equations leads to a time-changed version of the original transport processes. The obtained results provide a clear picture of the role played by the time-fractional derivatives in this kind of random motions. They display an anomalous behavior and are useful to describe several complex systems arising in statistical physics and biology. In particular, we focus on the one-dimensional random flight, called telegraph process, studying the time-fractional version of the classical telegraph equation and providing a suitable interpretation of its stochastic solutions.

Existence of Tracking Brownian Nanoparticle Technique with Reducing its First Collision Time with One of the Nanosensors in the Interactive Medium

This study discusses the existence of tracking N-dimensional Brownian nanoparticle in an interactive medium, where this tracking is based on Brownian motion analysis. A group of N nanosensors begin the tracking process at the origin of N dimensional space, where the nanosensors serve an important role in detecting and monitoring this nanoparticle. Each nanosensor oscillates as it passes through the origin of its planer surface (to the right and left) in the presence of a succession of random points on each axis. The space's plans cross on a real line, with the origin (0,0,...,0). Because of this uncertainty, we may be able to calculate the overall distance in each tracking step as a function of a discounted effort-reward parameter. We show analytically how this parameter influences demonstrating the existence of this model and lowering the computationally expected value of the first collision time between a nanosensor and a nanoparticle.

A singularly perturbed ruin problem for a two-dimensional Brownian motion in the positive quadrant

Abstract We consider the following problem: the drift of the wealth process of two companies, modelled by a two-dimensional Brownian motion, is controllable such that the total drift adds up to a constant. The aim is to maximize the probability that both companies survive. We assume that the volatility of one company is small with respect to the other, and use methods from singular perturbation theory to construct a formal approximation of the value function. Moreover, we validate this formal result by explicitly constructing a strategy that provides a target functional, approximating the value function uniformly on the whole state space.

Texture-Image-Oriented Coverless Data Hiding Based on Two-Dimensional Fractional Brownian Motion

In an AI-immersing age, scholars look for new possibilities of employing AI technology to their fields, and how to strengthen security and protect privacy is no exception. In a coverless data hiding domain, the embedding capacity of an image generally depends on the size of a chosen database. Therefore, choosing a suitable database is a critical issue in coverless data hiding. A novel coverless data hiding approach is proposed by applying deep learning models to generate texture-like cover images or code images. These code images are then used to construct steganographic images to transmit covert messages. Effective mapping tables between code images in the database and hash sequences are established during the process. The cover images generated by a two-dimensional fractional Brownian motion (2D FBM) are simply called fractional Brownian images (FBIs). The only parameter, the Hurst exponent, of the 2D FBM determines the patterns of these cover images, and the seeds of a random number generator determine the various appearances of a pattern. Through the 2D FBM, we can easily generate as many FBIs of multifarious sizes, patterns, and appearances as possible whenever and wherever. In the paper, a deep learning model is treated as a secret key selecting qualified FBIs as code images to encode corresponding hash sequences. Both different seeds and different deep learning models can pick out diverse qualified FBIs. The proposed coverless data hiding scheme is effective when the amount of secret data is limited. The experimental results show that our proposed approach is more reliable, efficient, and of higher embedding capacity, compared to other coverless data hiding methods.

Soliton wave profiles and dynamical analysis of fractional Ivancevic option pricing model

This study dynamically investigates the mathematical Ivancevic option pricing governing system in terms of conformable fractional derivative, which illustrates a confined Brownian motion identified with a non-linear Schrödinger type equation. This model describes the controlled Brownian motion that comes with a non-linear Schrödinger type equation. The solution to comprehend the market price fluctuations for the suggested model is developed through the application of a mathematical strategy. The modified Kudryashov analytical method is applied to find the fractional analytical exact soliton solution. The restrictions on the parameters required for these solutions to exist were also the result of this approach. The dynamical insights are examined and significant aspects of the phenomenon under study are discussed through the use of the bifurcation analysis. In the related dynamical system, the phase portraits of market price fluctuations are displayed at equilibrium points and for different parameter values. Additionally, the chaos analysis was carried out to show the quasi-periodic and periodic chaotic patterns. In order to track changes in market price, the sensitivity analysis of the studied model is also looked at and presented at different initial conditions. It was discovered that the model experienced price fluctuations as a result of minute changes in initial conditions.

Discrete-time approximation for backward stochastic differential equations driven by G-Brownian motion

Discrete-time approximation for backward stochastic differential equations driven by <i>G</i>-Brownian motion

Measurement of apparent diffusion coefficient (ADC) in fetal organs and placenta using 3 Tesla magnetic resonance imaging (MRI) across gestational ages

Diffusion-weighted imaging (DWI) is a technique used to probe the random microscopic motion of water protons in living tissue, represented by a parameter measurement of apparent diffusion coefficient (ADC) values. This study aimed to measure the ADC values of various fetal organs and placenta using 3T at various gestational ages. This was a prospective observational study. A total of 103 singleton pregnancies from 20 to 38 weeks of gestational age were included. Diffusion-weighted imaging was performed in the axial plane from the fetal head to the trunk with the following parameters: TR: 2000–2500 ms; TE: 88 ms; FOV: 250 mm; 256 matrix; slice thickness: 4 mm with a 0 mm gap; acquisition time: 1 min, 18 s. Diffusion gradient values were b = 0 and b = 700 s/mm2. ADC was measured in fetal brain regions (frontal white matter, occipital white matter, centrum semiovale, pons, thalamus, cerebellum, and fetal organs (lungs, kidney, and placenta). ANOVA was used to calculate the mean ADC values. Karl Pearson’s coefficient of correlation was used to evaluate the correlation between ADC values and increasing gestational age. The mean ADC values of brain regions were: frontal white matter (1.64 ± 0.08 × 10− 3 mm2 /s, F-39.10,p-<0.001), occipital white matter (1.64 ± 0.06 × 10− 3 mm2/s, F-26.14, p-<0.001), centrum semiovale (1.62 ± 0.03 × 10− 3 mm2/s, F-49.88,p-<0.001, pons (1.23 ± 0.09 × 10− 3 mm2/s F-9.14,p-<0.001) ), Thalamus (1.21 ± 0.07 × 10− 3 mm2/s, F-13.54,p-<0.001) and cerebellum (1.36 ± 0.10 × 10− 3 mm2 /s, F-4.19,p-<0.001). The mean ADC values of fetal organs were lung (1.92 ± 0.15 × 10− 3 mm2 /s, F-28.24, p-<0.001), kidney (1.34 ± 0.11 × 10− 3 mm2 /s, F-1.05, p- 0.37) and placenta (1.94 ± 0.11 × 10− 3 mm2 /s, F-160.33, p-<0.001). White-matter regions showed a significant positive correlation with increasing gestational age. Statistically, a negative correlation was observed between increasing gestational age and ADC measurements obtained in the thalamus, cerebellum, pons, and kidney. This will be one of the first few studies to provide the ADC values of the fetal brain and fetal organs using 3T MRI. The current study shows that diffusion-weighted MRI can offer a promising technique to evaluate the structural development of fetal organs and can potentially lead to a biomarker for predicting the functionality of the fetal organs in abnormalities.

Physical information-guided multidirectional gated recurrent unit network fusing attention to solve the Black-Scholes equation

Reasonable option pricing is crucial in the financial derivatives market. Finding analytical solutions for the Black-Scholes (BS) equation, particularly for American options or with fluctuating volatility and interest rates, is challenging. BS equations exhibit strong time-series characteristics, with asset prices typically adhering to geometric Brownian motion. To address the BS equations, we propose a sequence-to-sequence model guided by physical information (PI), called PiMGA. The PiMGA fuses a multidirectional gated recurrent unit (GRU) network with an attention module, where multidirectional GRU enhances the coding performance of the input sequences and the attention module balances the feature weights of the hidden variables. Prior physical knowledge in BS equations is jointly used as a constraint, forming the penalty function for objective optimization. This allows PiMGA to serve as an efficient approximation function in the learning paradigm of physically informed machine learning to solve BS equations. BS equations with various complexities illustrate the accuracy and feasibility of PiMGA for numerical solutions. Furthermore, the out-of-distribution generalization ability of PiMGA is verified by predicting the Nasdaq 100 index.

High-Speed Fluctuation Analysis of Silver-Nanoparticle SERS in Solutions.

We analyzed the fluctuation of surface-enhanced Raman spectra with a temporal resolution of 25 ms using a conventional electron-multiplying charge-coupled device camera experimental setup. The signal-to-noise ratio of the spectra was improved using density-based spatial cluster analysis with noise. Silver nanoparticles (AgNPs) with different sizes were dispersed as surface-enhanced Raman spectroscopy platforms in violet aqueous solutions. The movement of AgNPs and the fluctuation of the spectra were characterized. The fluctuation (signal ON and OFF) was evaluated on the basis of the time intervals between ON and OFF timing. The behavior of each AgNP solution was explained by a two-dimensional random walk model, which means that the phenomenon was mainly governed by the Brownian motion of the AgNPs in the solution. The fluctuation was also compared among three different Raman modes, one of which showed anomalous behavior.

Paradigm on Levenberg–Marquardt neural algorithm analysis of heat conduction optimization for ternary hybrid nanofluid with entropy generation

AbstractThe significance of the present article is to enhance the thermal management and energy efficiency of complex engineering infrastructures such as energy storage systems, modern electric vehicles, thermal insulations, heavy‐duty machinery, and production units. This research aims to understand the intricate relationship between the thermal conductivity performance of ternary () hybrid nanomaterial and entropy generation to optimize material design and efficacy. A synergetic combination of three distinct nanomaterials silicon dioxide, ferric oxide, and titanium oxide with ethylene glycol and water in the ratio 3:2 as a base solvent is comprised of contributing unique thermophysical properties. To elucidate the impact of this hybrid composition on thermal conductivity, various factors are analyzed. The advanced computational technique of Artificial intelligent feed‐forward neural network (AIFFNN) is utilized. The problem governed the system of PDEs, which is transformed into ODEs by dimensionless similarity. Adams method provided the dataset which is filtered and embedded into Marquardt–Levenberg Algorithm (LMA). The study examines the role of nanomaterial constituents, morphology, and boundary conditions on thermal performance and entropy generation. Graphical analysis of velocity, temperature, and entropy is provided with respect to varying parameters, including surface absorption (λ), magnetic strength (Tesla M), radiation parameter (Rd), Brownian motion (Br), and Eckert number (Ec). The findings have practical significance for optimizing material design in engineering and industrial applications.

Variable density and heat generation impact on chemically reactive carreau nanofluid heat-mass transfer over stretching sheet with convective heat condition

The present study focuses on the physical significance of heat generation and chemical reaction on Carreau nanofluid with convective heat conditions. Heat transfer is characterized using convective boundary conditions. The governing partial differential equations (PDEs) are transformed into ordinary differential equations (ODEs) by using well define stream functions and similarity transformations. Using a shooting methodology, the Keller-box method with Newton Raphson scheme is used to elaborate the numerical solutions of physical phenomena. Utilizing a similar technique to find the impact of physical parameter such as the production of heat δ, the rate of reaction Λ, Biot numbers γ, Brownian motion variable Nb, the thermophoresis parameters Nt, the Weissenberg quantity We, Prandtl number Pr, and Lewis number Le on velocity profile, temperature profile and mass transmission profile are determined graphically. The skin-friction coefficient −f″(0), local Nusselt −θ′(0), and Sherwood numbers −ϕ′(0) are analyzed numerically. Increment in fluid velocity and slip temperature are depicted with high Biot number. Maximum magnitude of fluid temperature and fluid concentration function are depicted at high value of temperature dependent density. The magnitude of heat and mass transportation enhanced with maximum choice of Brownian motion.

Significance of magnetic field and Darcy–Forchheimer law on dynamics of non-Newtonian hybrid nanofluid flow over a spinning disc with Arrhenius activation energy and shape factor

Purpose The purpose of the study is to explore the three-dimensional heat and mass transport dynamics of the magneto-hydrodynamic non-Newtonian (Casson fluid) hybrid nanofluid flow comprised of − as nanoparticles suspended in base liquid water as it passes through a flexible spinning disc. The influence of a magnetic field, rotation parameter, porosity, Darcy−Forchheimer, Arrhenius’s activation energy, chemical reaction, Schmidt number and nanoparticle shape effects are substantial physical features of the investigation. Furthermore, the influence of hybrid nanofluid on Brownian motion and thermophoresis features has been represented using the Buongiorno model. The novelty of the work is intended to contribute to a better understanding of Casson non-Newtonian fluid boundary layer flow. Design/methodology/approach The governing mathematical equations that explain the flow and heat and mass transport phenomena for fluid domains include the Navier−Stokes equation, the thermal energy equation and the solutal concentration equations. The governing equations are expressed as partial differential equations, which are then converted into a suitable set of non-linear ordinary differential equations by using the necessary similarity variables. The ordinary differential equations are computed by combining the shooting operation with the three-stage Lobatto BVP4c technique. Findings Graphs and tables are used in the process of analysing the characteristics of velocity distributions, temperature profiles and solutal curves at varying values of the parameters, along with friction drag, heat transfer rate and Sherwood number. It has been revealed that the radial and axial velocities decrease when the Casson parameter value increases and that the rate of heat transmission is higher in hybrid nanofluids with nanoparticles in the shape of a blade. The increase in Brownian motion and thermophoresis parameters causes a rise in the temperature profile. Also, an increase in the activation energy parameter improves the solutal curve. The use of nanoparticles was shown to improve extrusion properties, the rotary heat process and biofuel generation. Originality/value All results are presented graphically and all physical quantities are computed and tabulated. The current results are compared to previous investigations and found to agree significantly with them.

Entropy generation of Williamson hydromagnetic non-linear radiative nanofluid flow near a stagnation point over a porous stretching sheet

Purpose This study aims to analyze the multi-slip effects of entropy generation in steady non-linear magnetohydrodynamics thermal radiation with Williamson nanofluid flow across a porous stretched sheet near a stagnation point. Also, the qualities of viscous dissipation, Cattaneo–Christove heat flux and Arrhenius activation energy are taken into account. Thermophoresis, Brownian motion and Joule heating are also considered. Design/methodology/approach The Navier–Stokes equation, the thermal energy equation and the Solutal concentration equations are the governing mathematical equations that describe the flow and heat and mass transfer phenomena for fluid domains. By using the proper similarity transformations, a set of ordinary differential equationss are retrieved from boundary flow equations. The classical Runge–Kutta fifth-order algorithm along with the shooting technique is implemented to solve the obtained first order differential equations. Findings The study concludes that the temperature distribution boosting for thermal radiation, magnetic field and Eckert number where as the velocity and entropy generation escalate for the Williamson parameter, diffusion parameter and Brinkman number. The skin-friction and heat and mass transfer rate increases with the fluid injection. In addition, tabulated values of friction drag and rate of heat and mass transfer for various values of constraints are provided. Originality/value The comparison of the present results is carried out with the published results and noted a good agreement.

Radiative MHD Casson Non-Newtonian Nanofluid Slip Flow Induced by Stretching Cylinder: A Numerical Approach

Objectives: This article investigates the continuous magnetohydrodynamics slip flow across an extending cylinder, including heat radiation and free stream velocity. The proposed model incorporated the Brownian motion parameter and thermophoresis. Methods: Applying the similarity transformation to a system of partial differential equations yields an ordinary differential equation of first order. Since, there is the nonlinear nature of the modified equations, these ordinary differential equations are addressed by employing mathematical modeling. The shooting process has been embraced to solve changed equations by implementing the following Runge-Kutta Fehlberg method. Findings: Graphical representations are used to show how several fluid factors, such as Casson fluid parameter, free stream velocity, thermophoresis, magnetic parameter, Brownian motion parameter, heat generation parameter, Lewis number, radiation parameter, chemical reaction and curvature parameter affect concentration, temperature and velocity. Results for physical quantity of interest are also computed and reported in tabular form. The study's main findings showed that nanoparticle temperatures rise with greater Brownian motion parameter values, whereas nanoparticle concentration falls with higher heat radiation parameter values. Additionally, it is found that with rise in Lewis number rises as 0.1, 0.4, 0.7, 1.0, 1.3, heat transfer rate falls down by 21.19%. The outcomes are calculated using the MATLAB software. Novelty: This technique's validity is defended by comparing its most recent findings to previously published results and successfully meeting the convergence criteria. The present work emphasizes the analysis of free stream flow in the cylindrical area, specifically in the existence of partial slip and heat radiation, under convective circumstances. This study has significant implications for electronic devices such as processors as well as many businesses and the field of biological and medical sciences. The present investigation aims to support production businesses in achieving the desired level of quality of their products by effectively managing the transport phenomena. Keywords: Stretching cylinder, Heat radiation, Brownian motion, Thermophoresis, Non-Newtonian nanofluid, Shooting technique

Rotational Taylor dispersion in linear flows

The coupling between advection and diffusion in position space can often lead to enhanced mass transport compared with diffusion without flow. An important framework used to characterize the long-time diffusive transport in position space is the generalized Taylor dispersion theory. In contrast, the dynamics and transport in orientation space remains less developed. In this work we develop a rotational Taylor dispersion theory that characterizes the long-time orientational transport of a spheroidal particle in linear flows that is constrained to rotate in the velocity-gradient plane. Similar to Taylor dispersion in position space, the orientational distribution of axisymmetric particles in linear flows at long times satisfies an effective advection–diffusion equation in orientation space. Using this framework, we then calculate the long-time average angular velocity and dispersion coefficient for both simple shear and extensional flows. Analytic expressions for the transport coefficients are derived in several asymptotic limits including nearly spherical particles, weak flow and strong flow. Our analysis shows that at long times the effective rotational dispersion is enhanced in simple shear and suppressed in extensional flow. The asymptotic solutions agree with full numerical solutions of the derived macrotransport equations and results from Brownian dynamics simulations. Our results show that the interplay between flow-induced rotations and Brownian diffusion can fundamentally change the long-time transport dynamics.

Impact of Brownian motion and thermophoresis in magnetohydrodynamic dissipative: Radiative flow of chemically reactive nanoliquid thin films on an unsteady expandable sheet in a composite media

AbstractThis study comprehensively examines magnetohydrodynamic heat transport characteristics within a thin nanofluid film on a stretchable sheet embedded in a composite medium. By considering factors such as the unsteady nature of sheet velocity, Brownian motion, thermophoresis, thermally radiative heat, irregular heat generation/sink, chemical reactions, and dissipation due to viscous fluid, the research provides valuable insights into the variations in fluid velocity, temperature, and nanoparticles concentration. The computational solution utilizes the efficient numerical method that enables accurate predictions of system behavior under varying conditions. Notable findings include the influence of Schmidt numbers on nanoparticle concentration distribution, the opposing impact of thermophoresis parameter values, and the influence of Brownian motion and heat source/sink on temperature profiles in thin nanofluid film. Also, nanoliquid film thickness is reduced by enhancing the porous parameter values and Hartmann number values. The nanoliquid film becomes thinner when the space‐dependent heat source/sink parameter is considered compared to the temperature‐dependent heat source/sink coefficient. In space‐dependent and temperature‐dependent cases, the increase in these parameters leads to a decrease in the temperature gradient. Furthermore, it is observed that higher thermophoresis values correspond to reduced nanoparticle concentration gradient profiles. Also, enhancement in the chemical reaction values leads to an expansion in the solutal boundary region surrounding nanoparticles, and as a consequence, the concentration gradient of nanoparticles is enhanced. This research has significant potential for optimizing heat performance and advancing innovation in industrial and engineering processes.

A magnetohydrodynamic flow of a water-based hybrid nanofluid past a convectively heated rotating disk surface: A passive control of nanoparticles

Abstract One of the basic fluid mechanics problems of fluid flows over a revolving disk has both theoretical and real-world applications. The flow over a rotating disk has been the subject of numerous theoretical studies because it has many real-world applications in areas like rotating machinery, medical equipment, electronic devices, and computer storage. It is also crucial for engineering processes. Therefore, this article deals with a time-independent water-based hybrid nanofluid flow containing copper oxide and silver nanoparticles past a spinning disk. The Newtonian flow is taken into consideration in this analysis. The influence of magnetic field, thermophoresis, nonlinear thermal radiation, Brownian motion, and activation energy has been considered. The present analysis is modeled in a partial differential equation form and is then converted to ordinary differential equations using appropriate variables. A numerical solution using the bvp4c technique is accomplished using MATLAB software. The current results are matched with the previous literature and established a close relationship with previous studies. The purpose of this investigation is to numerically investigate the time-independent hybrid nanofluid flow comprising copper oxide and silver nanoparticles over a rotating disk surface. The results show that the increased magnetic parameters increase the friction force at the surface, which decreases the radial and azimuthal velocity distribution. At the sheet surface, the radial velocity of the hybrid nanofluid shows dominant performance compared to the nanofluid. On the other hand, the magnetic factor has dominant behavior on the azimuthal velocity component of the nanofluid flow compared to the hybrid nanofluid flow. The higher volume fraction and magnetic factor enhance the skin friction at the disk surface. Furthermore, greater surface drag is found for the hybrid nanofluid flow. The higher solid volume fraction, temperature ratio, and Biot number enhance the rate of heat transmission. Also, a higher rate of heat transmission is observed for the hybrid nanofluid flow.

Stochastic thermodynamics of Brownian motion in a flowing fluid

Stochastic thermodynamics of Brownian motion in a flowing fluid

Boundary crossing problems and functional transformations for Ornstein–Uhlenbeck processes

ABSTRACT We are interested in the law of the first passage time of an Ornstein–Uhlenbeck process to time-varying thresholds. We show that this problem is connected to the laws of the first passage time of the process to members of a two-parameter family of functional transformations of a time-varying boundary. For specific values of the parameters, these transformations appear in a realisation of a standard Ornstein–Uhlenbeck bridge. We provide three different proofs of this connection. The first is based on a similar result for Brownian motion, the second uses a generalisation of the so-called Gauss–Markov processes, and the third relies on the Lie group symmetry method. We investigate the properties of these transformations and study the algebraic and analytical properties of an involution operator which is used in constructing them. We also show that these transformations map the space of solutions of Sturm–Liouville equations into the space of solutions of the associated nonlinear ordinary differential equations. Lastly, we interpret our results through the method of images and give new examples of curves with explicit first passage time densities.

On a Planar Random Motion with Asymptotically Correlated Components

On a Planar Random Motion with Asymptotically Correlated Components