Brownian Motion Research Articles

Application of Chelyshkov polynomials in solving stochastic model with fractional Brownian motion

Abstract This paper introduces a computational spectral collocation approach utilizing orthonormal Chelyshkov polynomials to estimate solutions for both linear and nonlinear stochastic differential equations containing fractional Brownian motion characterized by the Hurst parameter H. The method employs Newton–Cotes nodes to transform these integral equations into manageable linear and nonlinear algebraic forms. The reliability of the proposed method is established through theorems on error estimation and convergence analysis. Moreover, some examples are given to demonstrate the viability, credibility, coherence, and dependability of the approach. Also, a comparative evaluation is conducted to contrast the results obtained from the suggested approach with those produced by the spectral collocation method utilizing orthonormal Bernoulli polynomials. Furthermore, a 95 % confidence interval is constructed for the error mean, revealing that the approximations maintain high accuracy.

Detecting rough volatility: a filtering approach

In this paper, we focus on filtering and parameter estimation in stochastic volatility models when observations arise from high-frequency data. We are particularly interested in rough volatility models where spot volatility is driven by fractional Brownian motion with Hurst index H < 1 2 . Since volatility is not directly observable, we rely on particle filtering techniques for statistical inference regarding the current level of volatility and the parameters governing its dynamics. In order to obtain numerically efficient and recursive algorithms, we use the fact that a fractional Brownian motion can be represented through a superposition of Markovian semimartingales (Ornstein-Uhlenbeck processes). We analyze the performance of our approach on simulated data and we compare it to similar studies in the literature. The paper concludes with an empirical case study, where we apply our methodology to high-frequency data of a liquid stock.

Investigation of the Impact of a Chemical Reaction on the Magnetohydrodynamic Boundary Layer Flow of a Radiative Maxwell Fluid over a Stretching Sheet Containing Nanoparticles Employing the Variational Iteration Method

The heat and mass transmission properties of a 2-D electrically conducting incompressible Maxwell fluid past a stretched sheet were studied under thermal radiation, heat generation/absorption, and chemical reactions. This issue has a variety of real-world applications, most notably polymer extrusion and metal thinning. The transport equations account for both Brownian motion and thermophoresis during chemical reactions. Using similarity variables allows for non-dimensionalization of the stream's PDEs and associated boundary conditions. The resulting modified ODEs are solved with the variational iteration approach. The impact of embedded thermo-physical variables on velocity, temperature, and concentration was studied quantitatively. When compared to the RK-Fehlberg approach, the findings are very similar. Raising the chemical reaction parameter narrows the concentration distribution, whereas increasing the temperature increases thermal radiation's impact. As the amount of N_t increases, the thickness of the boundary layer develops, causing the surface temperature to rise, resulting in a temperature increase.

Cascaded lattice Boltzmann simulation of Newtonian and non-Newtonian mixture nanofluids with variable thermophysical properties in a cavity with vertical heat radiator

The central moments-based cascaded lattice Boltzmann method (CLBM) for then Newtonian and non-Newtonian Buongiorno’s model mixture nanofluids (CuO, ZnO, Al2O3-water) has been implemented and applied in a square chamber with a vertical heat radiator using accelerated graphics processing unit (GPU) computing through compute unified device architecture (CUDA) C/C++ platform. Due to the higher numerical stability, the CLBM is a superior numerical tool to the raw moments-based MRT-LBM (multiple-relaxation-time lattice Boltzmann method). Three different models for the viscosity and thermal conductivity of the nanofluids: (i) the Binkmann model for the constant viscosity and the Maxwell model for the constant thermal conductivity (ii) Binkmann and Maxwell model with temperature dependent Brownian motion and (iii) Corcione model with non-Newtonian fluid where the temperature and strain rate determine the nanofluid effective thermal conductivity and viscosity, have been used. The enclosure’s upper and bottom walls are thermally adiabatic, but the left and right walls are uniformly cold. A vertical heater is immersed in the middle position of the cavity. The benchmark results for non-Newtonian, Newtonian, and nanofluids for the various computational domains are used to validate the current code adequately. The Bingham number (Bn), the Rayleigh number (Ra), and The volume fraction of the nanoparticles (ϕ) are the three key parameters that are varied in this investigation to demonstrate the effects of natural convection on the isotherms, streamlines, isolines of nanoparticle volume fractionation, yielded and unyeilded zone, and average Nusselt number (Nu¯). The Brownian motion effects of the nanoparticles augmented the average rate of heat transfer and the use of the Bingham nanofluids reduced the heat transfer enhancement. For the CuO-water nanofluid, the augmentation of the rate of heat transfer is 15.42% from ϕ=0 to 4% while Ra=106 and the corresponding heat transfer enhancement for the ZnO-water nanofluid is 11.11%. For the Bingham fluid, the rate of heat transfer increases 7% from Ra=105 to Ra=106 while ϕ=2% and Bn=0.3. The findings can be applied to optimize automotive radiator systems, which are crucial for maintaining engine temperature.

A new pricing method for integrated energy systems based on geometric Brownian motions under the risk-neutral measure

An integrated energy system (IES) is a promising energy-saving technology. With the advancement of technology and deregulation of the energy market, numerous distributed energy systems (DESs) will be accessed in the IES with various trading modes in the future. This makes it even more challenging to evaluate the fair energy selling price of IES, which is of great concern to market regulators and participants. To address this issue, this study proposes a risk-neutral pricing method for IES based on geometric Brownian motions. First, geometric Brownian motions are introduced to characterize the uncertainty feature of the energy price of the DESs. Then, we propose a stochastic partial differential equation (SPDE) for the energy price of an IES by utilizing the Ito lemma and the properties of the martingale under the risk-neutral measure. Finally, sensitivity analyses of the proposed SPDE on the volatilities and two types of correlation coefficients are conducted by implementing the proposed model in the given scenario.

Asymptotic distribution of entropies and Fisher information measure of ordinal patterns with applications

We present the asymptotic distribution of the Rényi and Tsallis/Havrda–Charvát entropies and the Fisher information measure of ordinal patterns embedding their serial correlation. We study the convergence behavior of the asymptotic variance for some types of dynamics and the permutation entropy to the limit distribution. These results lead to tests for comparing the underlying dynamics of two time series. We apply these tests to discriminate uniform white noise, logistic maps with Gaussian noise, fractional Brownian motion, and f−k noise, with k∈{0.5,1,1.5,2,2.5}. We also applied these tests to cryptocurrency open prices data, with favorable results. We provide the R code that implements the functions.

Approach to stationarity for the KPZ fixed point with boundaries

Abstract Current fluctuations for the one-dimensional totally asymmetric exclusion process (TASEP) connected to reservoirs of particles, and their large scale limit to the KPZ fixed point in finite volume, are studied using exact methods. Focusing on the maximal current phase for TASEP, corresponding to infinite boundary slopes for the KPZ height field, we obtain for general initial condition an exact expression for the late time correction to stationarity, involving extreme value statistics of Brownian paths. In the special cases of stationary and narrow wedge initial conditions, a combination of Bethe ansatz and numerical conjectures alternatively provide fully explicit exact expressions.

Irreversibility Analysis of Radiative TiO2+Al2O3+Cu/H2O Ternary Nanofluid Flow over a Bidirectional Stretching Sheet with Cattaneo-Christov Heat Flux: Nanotechnology Applications

Abstract Ternary nanofluids demonstrate better heat transfer characteristics in contrast to conventional liquids and typical hybrid nanofluids. These are applied in sophisticated cooling systems for electronics, heat exchangers, and automotive engines, along with renewable energy systems such as solar collectors, where efficient heat transfer plays a crucial role. The aim of this research is to investigate the movement of a Casson ternary nanofluid passing through a bidirectional exponential sheet by employing the innovative Cattaneo-Christov heat flux concept. The utilization of the energy equation considers thermal radiation, viscous dissipation, and heat source/sink effects, with the integration of chemical reaction effects into the concentration equation. An analysis of entropy generation is utilized to evaluate the thermodynamic irreversibility within the system. The conversion of transport equations involves a transformation from partial to ordinary differential equations, followed by a numerical solution utilizing the BVP4C solver embedded in the MATLAB package R2022b. The impacts of developed factors on thermal, concentration, and velocity behavior, as well as engineering quantities, are thoroughly explored graphically. The outcome reveals that the velocity gradients diminish as magnetic fields intensify, while it amplified by the Hall factor. The rise in temperature of ternary nanofluid correlates with elevated levels of radiation, and Brownian motion. Concentration intensifies with the rapid development of thermophoresis influences. Heightened values of the Reynolds and Brinkman numbers give rise to amplified entropy production but a decrease in the Bejan number. The ternary nanofluid displays a remarkable 8.92% increase in skin friction on the x-axis and y-axis, influenced by the potent Darcy-Forchheimer factor. The rates of mass and heat transfer in nanofluids undergo a decrease of 8.58% and 12.52%, respectively, due to the heightened influence of Brownian motion and Eckert number. The results could provide valuable insights into the performance of ternary nanofluids in various industrial environments under specific conditions.&amp;#xD;

Regression modeling of Bödewadt slip flow dynamics involving Reiner-Rivlin nanofluid based on a modified Buongiorno approach

Regression models are useful in analyzing rotational flows as they enable accurate predictions of wall shear and heat transfer coefficient. In addition, Bödewadt flow is of paramount importance in fluid dynamics of rotating systems such as turbomachinery and geophysical flows. Moreover, nanofluid’s enhanced heat transfer properties can improve cooling efficiency in applications involving turbines and electronic systems. This study delves into the Bödewadt boundary layer flow of a Reiner-Rivlin fluid containing nanoparticles over a stationary porous disk under slip conditions. The two-phase Buongiorno model is employed, incorporating temperature-dependent diffusion coefficients for enhanced accuracy. To facilitate numerical simulations, the transport equations are converted into an ordinary differential system comprising four unknowns. In the present work, a highly reliable Keller-Box methodology is adopted which agrees very well with the MATLAB built-in program ‘bvp4c’. The computed 2-D and 3-D streamlines vividly capture the Bödewadt flow scenario with Reiner-Rivlin nanofluid. The principle aim to investigate the impact of non-Newtonian behaviour and slip on the flow pattern, while also examining the behavior of temperature/concentration field for nanoparticle working fluids. As thermophoretic diffusion increases, the thermal boundary layer thickens considerably, leading to a notable decrease in the cooling rate of the disk. In contrast, Brownian diffusion has only a minimal impact on the heat transport. In addition, wall suction effect is observed to significantly boost the disk’s cooling rate, though at the expanse of increasing skin friction coefficients. This study introduces linear and quadratic regression models designed to precisely predict both the surface drag and disk cooling rate, which are crucial factors in engineering processes.

Enhancing thermal conductivity of novel ternary nitrate salt mixtures for thermal energy storage (TES) fluid

Efficient thermal energy storage (TES) is crucial for concentrated solar power (CSP) plants, necessitating the exploration of advanced heat transfer fluids with enhanced thermal conductivity. Conventional binary nitrate salt mixtures have limitations in thermal performance, prompting research into ternary mixtures and nanoparticle additives. This study investigates novel ternary nitrate salt mixtures comprising potassium nitrate (KNO₃), lithium nitrate (LiNO₃), and magnesium nitrate hexahydrate (Mg(NO₃)₂·6 H₂O) as high-performance TES fluids during the during the heat absorption phase. Seven different salt compositions were synthesized and characterized using the laser flash technique to evaluate their thermal conductivity over 100–400°C. Results revealed a significant influence of composition on thermal conductivity, with maximum values during melting ranging from 0.0777 W/m·K to 0.7373 W/m·K. Melting points varied from 335 K to 340.39 K, demonstrating tailorability through compositional adjustments. Furthermore, incorporation of 0.5 wt% aluminum oxide (Al₂O₃) nanoparticles resulted in substantial thermal conductivity enhancements, with the most significant increase observed in TSF10 (from 0.0777 W/m·K to 0.491 W/m·K). These improvements are attributed to enhanced phonon transport, increased surface area, and Brownian motion facilitated by Al₂O₃ nanoparticles. The study provides a comprehensive cost analysis, including raw material costs, and discusses the potential efficiency gains for CSP applications. The findings contribute to the development of high-performance and cost-effective TES fluids, advancing the efficiency and viability of sustainable energy generation.

Drift Control of High-Dimensional Reflected Brownian Motion: A Computational Method Based on Neural Networks

Motivated by applications in queueing theory, we consider a stochastic control problem whose state space is the d-dimensional positive orthant. The controlled process Z evolves as a reflected Brownian motion whose covariance matrix is exogenously specified, as are its directions of reflection from the orthant’s boundary surfaces. A system manager chooses a drift vector [Formula: see text] at each time t based on the history of Z, and the cost rate at time t depends on both [Formula: see text] and [Formula: see text]. In our initial problem formulation, the objective is to minimize expected discounted cost over an infinite planning horizon, after which we treat the corresponding ergodic control problem. Extending the earlier work by Han et al. [Han J, Jentzen A, Weinan E (2018) Solving high-dimensional partial differential equations using deep learning. Proc. Natl. Acad. Sci. USA 115(34):8505–8510], we develop and illustrate a simulation-based computational method that relies heavily on deep neural network technology. For the test problems studied thus far, our method is accurate to within a fraction of 1% and is computationally feasible in dimensions up to at least [Formula: see text].

Mechanical Profiling of Biopolymer Condensates through Acoustic Trapping.

Characterizing the mechanical properties of single colloids is a central problem in soft matter physics. It also plays a key role in cell biology through biopolymer condensates, which function as membraneless compartments. Such systems can also malfunction, leading to the onset of a number of diseases, including many neurodegenerative diseases; the functional and pathological condensates are commonly differentiated by their mechanical signature. Probing the mechanical properties of biopolymer condensates at the single particle level has, however, remained challenging. In this study, we demonstrate that acoustic trapping can be used to profile the mechanical properties of single condensates in a contactless manner. We find that acoustic fields exert the acoustic radiation force on condensates, leading to their migration to a trapping point where acoustic potential energy is minimized. Furthermore, our results show that the Brownian motion fluctuation of condensates in an acoustic potential well is an accurate probe for their bulk modulus. We demonstrate that this framework can detect the change in the bulk modulus of polyadenylic acid condensates in response to changes in environmental conditions. Our results show that acoustic trapping opens up a novel path to profile the mechanical properties of soft colloids at the single particle level in a non-invasive manner with applications in biology, materials science, and beyond.

Laplacian operator and its square lattice discretization: Green function vs. Lattice Green function for the flat 2-torus and other related 2D manifolds

Abstract The paper investigates the truncation error between the Green function and the lattice Green function (LGF) for the Laplacian operator defined on the 2-torus and its discretization on a regular square lattice. Extensions to the cylinder and the rectangular domain with free (or Neumann) boundary conditions are also proposed. In each of these instances, the Green function and its discrete analog are given in exact analytical closed-form allowing to infer accurate estimates as the lattice spacing tends to zero. As expected, it is shown that the continuum limit of the LGF coincides well with the Green function in every case. In particular, the issue of logarithmic singularity regularization of the Green function by the lattice discretization is addressed through two related application examples regarding the rectangular domain, and devoted to the computation of corner-to-corner resistance of an electrical conducting square and the mean first-passage time between the diagonally opposite vertices of a square for a standard Brownian motion, both derived considering the continuum limit.

Investigation of unsteady Buongiorno nanofluid in a slanted thermally radiated revolving channel under upstream microbial movement in the absence of chemical reaction

Analysis of Buongiorno nanofluid in a thermally radiated channel in the occurrence of microbes is an enrich research motive. Thermal radiations, dissipation energy, transient effects and upstream microbial acceleration potentially alter the performance of Buongiorno nanofluid. This is a two phase nanofluid model which accommodates the thermophoretic and Brownian movement of the particles along with other integrated physical quantities. Hence, the current research aims to conduct the study of Buongiorno nanofluid in a slanted thermally radiated micro channel in the presence of aforementioned physical phenomenon. The outcomes of the model achieved numerically and discussed deeply. It is investigated that the unsteady and channel revolving parameters enhances the nanofluid movement while adjusting the channel at different inclination also favors the velocity. The velocity G(η) upsurges for larger rotation of the channel while it drops for more transient fluid. Further, the thermophoretic and particles deposition boosts the thermal performance of the setup while thermal radiations depreciated the efficiency. The Schmidt effects in the range of 1.0–7.0 highly contributes in the improvement of mass transport. Moreover, density motile of microbes improved for Lewis and transient effects while Peclet effects are examined excellent to control the microbial influence.

Sticky Brownian motions on star graphs

AbstractThis paper is concerned with the construction of Brownian motions and related stochastic processes in a star graph, which is a non-Euclidean structure where some features of the classical modeling fail. We propose a probabilistic construction of the Sticky Brownian motion by slowing down the Brownian motion when in the vertex of the star graph. Later, we apply a random change of time to the previous construction, which leads to a trapping phenomenon in the vertex of the star graph, with characterization of the trap in terms of a singular measure $$\varPhi $$ Φ . The process associated to this time change is described here and, moreover, we show that it defines a probabilistic representation of the solution to a heat equation type problem on the star graph with non-local dynamic conditions in the vertex that can be written in terms of a Caputo-Džrbašjan fractional derivative defined by the singular measure $$\varPhi $$ Φ . Extensions to general graph structures can be given by applying to our results a localisation technique.

Impact of thermal conductivity of a Williamson’s nanofluid flow due to a stretching sheet

This paper intends to build a mathematical model of a two-dimensional Williamson nanofluid flow due to the stretching of a sheet linearly. The flow geometry is influenced by the magnetic field that is applied externally to the system. The induced magnetic field is infinitesimal and hence neglected. However, the flow structure is incorporated with the Brownian motion and thermophoresis effects as it alter the physics of the flow. The equations that model the flow comprise the highly nonlinear coupled simultaneous system. Thus, a numerical technique namely finite difference accompanied with the Thomas algorithm is adopted to approach the flow system. The motion, temperature, and concentration of the Williamson nanofluid flow are studied for the different flow controlling parameters. The co-efficient of skin-friction, heat, and mass transfer rates is also computed. The streamlines, isotherms, and iso-concentration are plotted to picturise the flow phenomena in the complete domain. The study reveals that the temperature of the flow raises with the Brownian motion of the nanoparticles and the trend is opposite with the concentration. The strength of the streamlines, isotherms, and the concentration contour is identified to be high for the least magnitude of Weissenberg number. The Brownian motion raises the heat transfer rate and slows down the mass transfer rate.

Wiener densities for the Airy line ensemble

AbstractThe parabolic Airy line ensemble is a central limit object in the KPZ (Kardar–Parisi–Zhang) universality class and related areas. On any compact set , the law of the recentered ensemble has a density with respect to the law of independent Brownian motions. We show where is an explicit, tractable, non‐negative function of . We use this formula to show that is bounded above by a ‐dependent constant, give a sharp estimate on the size of the set where as , and prove a large deviation principle for . We also give density estimates that take into account the relative positions of the Airy lines, and prove sharp two‐point tail bounds that are stronger than those for Brownian motion. These estimates are a key input in the classification of geodesic networks in the directed landscape. The paper is essentially self‐contained, requiring only tail bounds on the Airy point process and the Brownian Gibbs property as inputs.

Quantum Bernoulli noises approach to quantum master equations and applications to Ehrenfest-type theorems

This paper investigates the regularity properties of quantum master equations with unbounded coefficients within the space of square-integrable complex-valued Bernoulli functionals. Initially, we demonstrate the existence of a unique regular solution to the linear stochastic Schrödinger equation driven by cylindrical Brownian motions. Subsequently, we establish the existence of regular solutions for the autonomous linear quantum master equation and provide a probabilistic representation of this solution in terms of the stochastic Schrödinger equation. Finally, by taking the expectation of some observables with respect to the solution of the quantum master equation, we derive a system of differential equations that describe the evolution of the mean values of certain quantum observables.

Integrated neural network based simulation of thermo solutal radiative double-diffusive convection of ternary hybrid nanofluid flow in an inclined annulus with thermophoretic particle deposition

Numerical simulation of magnetohydrodynamic radiative double-diffusive convective heat and mass transfer of a ternary hybrid nanofluid with thermophoretic particle deposition in an inclined annulus is studied. Both sides of cylinders are preserved at uniform temperatures, while the remaining sides are thermally insulated. The coupled nonlinear partial differential equations are solved using a finite difference approach. The detailed computational results of heat and mass transfer rate, temperature, concentration and flow fields are presented and discussed for a distinct range of significant physical parameters. The results demonstrated that increasing the angle of the inclination factor increases fluid flow. In light of buoyancy, when the annular cavity is set downward, the enhanced shear velocity is transported upwards, where it reaches its optimal temperature. The higher temperature difference leads to augmented dynamic fluid motion, particularly in the radial direction, as the system seeks to balance the thermal energy through convective transfer. The Brownian motion parameter has abrupt mobility of nanoparticles in liquid, which promotes particle collisions with liquid molecules, yielding kinetic energy. Increased values of thermophoretic parameters augment the thermophoresis force. Thermophoretic particle deposition increases the concentration of nanoparticles in an annulus due to the migration of particles from hot to cold regions under the influence of temperature gradient. The heat and mass transfer characteristics of ternary hybrid nanofluid are forecasted through an artificial neural network-based Levenberg–Marquardt backpropagated algorithm. The built model shows the heat and mass transfer rate root mean squared values through the Levenberg–Marquardt algorithm as one and mean squared error values as 1e-07 and 1e-08 respectively.

From Brownian motion to virtual biopsy: a historical perspective from 40years of diffusion MRI.

Diffusion MRI was introduced in 1985, showing how the diffusive motion of molecules, especially water, could be spatially encoded with MRI to produce images revealing the underlying structure of biologic tissues at a microscopic scale. Diffusion is one of several Intravoxel Incoherent Motions (IVIM) accessible to MRI together with blood microcirculation. Diffusion imaging first revolutionized the management of acute cerebral ischemia by allowing diagnosis at an acute stage when therapies can still work, saving the outcomes of many patients. Since then, the field of diffusion imaging has expanded to the whole body, with broad applications in both clinical and research settings, providing insights into tissue integrity, structural and functional abnormalities from the hindered diffusive movement of water molecules in tissues. Diffusion imaging is particularly used to manage many neurologic disorders and in oncology for detecting and classifying cancer lesions, as well as monitoring treatment response at an early stage. The second major impact of diffusion imaging concerns the wiring of the brain (Diffusion Tensor Imaging, DTI), allowing to obtain from the anisotropic movement of water molecules in the brain white-matter images in 3 dimensions of the brain connections making up the Connectome. DTI has opened up new avenues of clinical diagnosis and research to investigate brain diseases, neurogenesis and aging, with a rapidly extending field of application in psychiatry, revealing how mental illnesses could be seen as Connectome spacetime disorders. Adding that water diffusion is closely associated to neuronal activity, as shown from diffusion fMRI, one may consider that diffusion MRI is ideally suited to investigate both brain structure and function. This article retraces the early days and milestones of diffusion MRI which spawned over 40years, showing how diffusion MRI emerged and expanded in the research and clinical fields, up to become a pillar of modern clinical imaging.