Brownian Motion Research Articles

Geometrical multiscale tortuosity of desert ant walking trajectories.

Desert ants stand out as some of the most intriguing insect navigators, having captured the attention of scientists for decades. This includes the structure of walking trajectories during goal approach and search behaviour for the nest and familiar feeding sites. We analyse such trajectories with regard to changes in walking direction. The directional change of the ants is quantified, i.e. an angle θ between trajectory increments of a given arclength λ is computed. This was done for different length scales λ, according to our goal of analysing desert ant path characteristics with respect to length scale. (i) Varying λ through more than two orders of magnitude demonstrated Brownian motion characteristics typical of the random walk component of search behaviour. Unexpectedly, this random walk component was also present in - supposedly rather linear - approach trajectories. (ii) There were small but notable deviations from a uniform angle distribution that is characteristic of random walks. This was true for specific search situations, mostly close to the (virtual) goal position. (iii) Experience with a feeder position resulted in straighter approaches and more focussed searches, which was also true for nest searches, albeit to lesser extents. Taken together, these results both verify and extend previous studies on desert ant path characteristics. Of particular interest are the ubiquitous Brownian motion signatures and specific deviations thereof close to the goal position, indicative of unexpectedly structured search behaviour.

​Mean reflected BSDE driven by a marked point process and application in insurance risk management

This paper aims to solve a super-hedging problem along with insurance re-payment under running risk management constraints. The initial endowment for the super-heding problem is characterized by a class of mean reflected backward stochastic differential equation driven by a marked point process (MPP) and a Brownian motion. By Lipschitz assumptions on the generators and proper integrability on the terminal value, we give the well-posedness of this kind of BSDEs by combining a representation theorem with the fixed point argument.

Effects of nanoparticle size on surface dynamics and thermal performance in film boiling of Al[formula omitted]O[formula omitted] water-based nanofluids

This study investigates the impact of nanoparticle diameter on film boiling in Al2O3 water-based nanofluids along a vertical cylinder, focusing on nanoparticle sizes of 5 nm, 10 nm, 30 nm, and 50 nm. Employing a Continuous-Species-Transfer method within a Computational Multi-Fluid Dynamics framework, it simulates the behavior of nanoparticles in liquid and vapor states, emphasizing Brownian motion and thermophoretic effects. A 2D axisymmetric analysis reveals how nanoparticle sizes influence temperature gradients, deposition patterns, and thermophysical properties, with a particular focus on the Nusselt number for assessing heat transfer efficiency. Findings indicate that smaller nanoparticles accumulate on surfaces more over time because of increased Brownian motion. Additionally, nanofluids with smaller particles improve thermal conductivity and concentration near the heat source, which decreases surface tension in the vapor phase. In contrast, larger nanoparticles lead to reduced fluid viscosity. Interestingly, nanofluids with 30 nm and 50 nm nanoparticles perform similarly, indicating a size threshold beyond which further increases do not significantly improve boiling heat transfer. In contrast, nanofluids containing 5 nm and 10 nm nanoparticles exhibit markedly superior thermal removal efficiency, underscoring the critical role of nanoparticle size in optimizing film boiling performance.

Thermal case study of magnetic radiative flow impacts on Newtonian nanofluid over a stretchable plate in absorbent: Box approach

Nanofluid technology represents a significant breakthrough in thermal engineering, with widespread applications spanning heat exchangers, cancer treatment, and heat storage systems. Despite its success in improving heat transfer rates in diverse devices, the challenge of enhancing thermal conductivity using nanoparticles looms large. This study aims to elucidate the 2D flow of nanofluid under suction/injection over a stretching wedge. It considers factors such as Brownian and thermophoretic diffusions, nonlinear heat generation, and thermal radiation. Additionally, it examines the effects of viscous dissipation and Joule heating. Through the conversion of governing nonlinear coupled partial differential equations into a system of coupled ordinary differential equations using similarity transformations, numerical solutions are obtained via the Keller-box method. The analysis delves into the significant impacts of relevant parameters on velocity, temperature, concentration, and microorganism fields, presented graphically. Notably, the findings underscore the role of solid particles in enhancing the thermal efficiency of base fluids. These investigation outcomes are vital for controlling heat transferal rates and fluid velocities throughout a variation of manufacturing processes and industrial sectors, guaranteeing the required quality of the ending product. The perceptions garnered from this research hold capacity for a broad range of industrial purposes and carry consequences through countless engineering disciplines, mostly within the range of nanotechnology.

On the second law of thermodynamics: a global flow spontaneously induced by a locally nonchaotic energy barrier

It is well known that, inherently, certain nonchaotic particle movements cannot reach thermodynamic equilibrium (Section 2). Usually, they are small-scale and their behaviors are “trivial”. In current research, we show that, beyond the boundary of the second law of thermodynamics where Boltzmann’s H-theorem does not apply, there are also large-sized systems of nontrivial energy properties (Section 3). The key concept is local nonchaoticity, demonstrated by using a narrow energy barrier. The barrier width is much less than the nominal particle mean free path, so that inside the barrier, particle-particle collision is sparse and the particle trajectories tend to be locally nonchaotic. Across the barrier, the steady-state particle flux ratio is intrinsically in a non-Boltzmann form. With a step-ramp structure, the nonequilibrium effect spreads to the entire system, and a global flow is generated spontaneously from the random thermal motion. The deviation from thermodynamic equilibrium is steady and significant, and compatible with the basic principle of maximum entropy. Although the direct experimental realization of the model may be difficult, the theoretical and numerical analyses could shed light on the fundamentals of thermodynamics and statistical mechanics.

Numerics with coordinate transforms for efficient Brownian dynamics simulations

Many stochastic processes in the physical and biological sciences can be modelled as Brownian dynamics with multiplicative noise. However, numerical integrators for these processes can lose accuracy or even fail to converge when the diffusion term is configuration-dependent. One remedy is to construct a transform to a constant-diffusion process and sample the transformed process instead. In this work, we explain how coordinate-based and time-rescaling-based transforms can be used either individually or in combination to map a general class of variable-diffusion Brownian motion processes into constant-diffusion ones. The transforms are invertible, thus allowing recovery of the original dynamics. We motivate our methodology using examples in one dimension before then considering multivariate diffusion processes. We illustrate the benefits of the transforms through numerical simulations, demonstrating how the right combination of integrator and transform can improve computational efficiency and the order of convergence to the invariant distribution. Notably, the transforms that we derive are applicable to a class of multibody, anisotropic Stokes-Einstein diffusion that has applications in biophysical modelling.

Self-normalized inference for stationarity of irregular spatial data

A self-normalized approach for testing the stationarity of a d-dimensional random field is considered in this paper. Because the discrete Fourier transforms (DFT) at fundamental frequencies of a second-order stationary random field are asymptotically uncorrelated (see (Bandyopadhyay and Subba Rao, 2017), one can construct a stationarity test based on the sample covariance of the DFTs. Such a test is usually inferior because it involves an overestimated scale parameter that leads to low size and power. To circumvent this shortcoming, this paper proposes two self-normalized statistics based on extreme value and partial sum of the sample covariance of the DFTs. Under certain regularity conditions, it is shown that the proposed tests converge to functionals of Brownian motion. Simulations and a data analysis demonstrate the outstanding performance of the proposed tests.

Evaluating plasma fluctuation by collisional-radiative model using Malliavin derivative

Abstract Non-equilibrium plasma has garnered significant attention due to its involvement in short-term phenomena. One example is plasma fluctuation, encompassing variations in electron temperature or density. This phenomenon is explored not only in nuclear fusion but also in semiconductor etching and thin film deposition. This study introduces an evaluation method employing Malliavin derivatives to predict plasma fluctuations based on a collisional-radiative model. This model delineates the chemical kinetics of excited states within the plasma. Given that electron impact excitation and atomic collisional processes are stochastic, they exhibit characteristics similar to a Wiener process. Furthermore, the properties of fractional Brownian motion are applied to the Malliavin derivative. The revised Wiener process is utilized to analyze the changes in excited-level populations within short durations. This approach assesses electron or atomic density fluctuations by considering the contributions of electron collisions and those of ground-state atoms.

Identification of the continuum field structure at multiple scale levels.

For continuum fields such as turbulence, analyses of the field structure offer insights into their kinematic and dynamic properties. To ensure the analyses are quantitative rather than merely illustrative, two conditions are essential: space-filling and structure quantification. A pertinent example is the dissipation element (DE) structure, which is however susceptible to noisy interference, rendering it inefficient for extracting the large-scale features of the field. In this study, the multi-level DE structure is proposed based on the multi-level extremal point concept. At a given scale level, the entire field can be decomposed into the corresponding space-filling and non-overlapping DEs, each characterized by its length scale l and the scalar difference Δϕ between its two extremal points. We will first elaborate on the fundamental principles of this method. Results from an artificially constructed two-scale field indicate that the decomposed units adequately represent the geometry of the original field. In examining the fractal Brownian motion, a structure function equivalent ⟨Δϕ|l⟩ and an energy spectrum equivalent are introduced. The scaling relation derived from ⟨Δϕ|l⟩ corresponds with the Hurst number. Furthermore, the multi-level DE structure distinctly reveals the two different inertial ranges in two-dimensional turbulence. Overall, this novel structure identification approach holds significant potential for complex analyses concerning the field geometry.

On the separation cut-off phenomenon for Brownian motions on high dimensional spheres

On the separation cut-off phenomenon for Brownian motions on high dimensional spheres

Integral sliding mode control and stability for Markov jump systems with structured perturbations and time-varying delay driven by fractional Brownian motion

The issues of stability and sliding mode control (SMC) for time-varying delay Markov jump systems (MJSs) with structured perturbations constrained by fractional Brownian motion (fBm) are explored. First, constructing a novel Lyapunov-Krasovskii functional (LKF) with exponential terms that contain the double-integral term, the pth moment exponential stability conditions are derived by utilizing the generalized fractional Itoˆ formula and conditional mathematical expectation. Subsequently, by designing the innovative integral sliding mode surface (SMS) associated with time-varying delay and the SMC law, the state trajectories of the dynamic systems can reach the designed SMS within a finite time. Ultimately, the numerical experiment is executed to confirm and ensure the accuracy and reliability of the obtained results.

Asymptotic behaviors of solutions to Sobolev-type stochastic differential equations

This paper is devoted to studying the Sobolev-type stochastic differential equations with Lévy noise and mixed fractional Brownian motion. Applying a method (principle) of comparability of functions by character of Shcherbakov recurrence, it characters at least one (or exactly one) solution with the same properties as the coefficients of the equation. We establish the existence of Poisson stable solutions for the Sobolev-type equation, which includes periodic solutions, quasi-periodic solutions, almost periodic solutions, almost automorphic solutions, etc. We also obtain the global asymptotical stability of bounded Poisson stable solutions and present an example to illustrate our theoretical results.

The extremal point process of branching Brownian motion in Rd

The extremal point process of branching Brownian motion in Rd

Parameter estimation for the stochastic heat equation with multiplicative noise from local measurements

For the stochastic heat equation with multiplicative noise we consider the problem of estimating the diffusivity parameter in front of the Laplace operator. Based on local observations in space, we first study an estimator, derived in Altmeyer and Reiß (2021) for additive noise. A stable central limit theorem shows that this estimator is consistent and asymptotically mixed normal. By taking into account the quadratic variation, we propose two new estimators. Their limiting distributions exhibit a smaller (conditional) variance and the last estimator also works for vanishing noise levels. The proofs are based on local approximation results to overcome the intricate nonlinearities and on a stable central limit theorem for stochastic integrals with respect to cylindrical Brownian motion. Simulation results illustrate the theoretical findings.

Collagenase motors in gelatine-based hydrogels.

Nano/micromotors outperform Brownian motion due to their self-propulsive capabilities and hold promise as carriers for drug delivery across biological barriers such as the extracellular matrix. This study employs poly(2-(diethylamino)ethyl methacrylate) polymer brushes to enhance the collagenase-loading capacity of silica particle-based motors with the aim to systematically investigate the impact of gelatine viscosity, motors' size, and morphology on their propulsion velocity. Notably, 500 nm and 1 μm motors achieve similar speeds as high as ∼15 μm s-1 in stiff gelatine-based hydrogels when triggered with calcium. Taken together, our findings highlight the potential of collagenase-based motors for navigating the extracellular matrix, positioning them as promising candidates for efficient drug delivery.

Fluctuation-dissipation relation in cosmic microwave background

We study the fluctuation-dissipation relation for sound waves in the cosmic microwave background (CMB), employing effective field theory (EFT) for fluctuating hydrodynamics.Treating sound waves as the linear response to thermal radiation, we establish the fluctuation-dissipation relation within a cosmological framework.While dissipation is elucidated in established linear cosmological perturbation theory, the standard Boltzmann theory overlooks the associated noise, possibly contributing to inconsistencies in Lambda Cold Dark Matter (ΛCDM) cosmology. This paper employs EFT for fluctuating hydrodynamics in cosmological perturbation theory, deriving sound wave noise.Notably, the long-time limit of the noise spectrum is independent of viscosity details, resembling a Brownian motion bounded in a harmonic potential.The net energy transfer between the sound wave system and the radiation environment reaches a balance within Hubble time, suggesting the thermal equilibrium of the sound waves themselves.The induced density power spectrum is characterized as white noise dependent on the inverse of the entropy density, which is negligibly small on the CMB scale. The energy density of the entire sound wave system scales as a -4, akin to radiation. While the numerical factor is not determined in the present calculation, the back reaction of the sound wave system to the background radiation may not be negligible, serving as a potential source for various fitting issues in ΛCDM cosmology.

Pricing vulnerable lookback options using Laplace transforms

This study develops a closed-form solution for pricing vulnerable lookback options, which combines the Black–Scholes model for the underlying asset and a correlated jump-diffusion model for the issuer’s asset to account for default risks. Our approach relies on applying Laplace transforms to establish a closed-form solution and compute them numerically using Laplace inversion algorithms. To determine the price, we derive the joint distribution of the first passage time of a drifted Brownian motion and the value of the correlated jump diffusion. Through numerical examples, we demonstrate the accuracy of the numerical solutions obtained through our method and the stability of the algorithm. Subsequently, we conduct a numerical analysis to understand how counterparty risk influences option prices based on our formula.

Study of nanofluid flow in a stationary cone–disk system with temperature-dependent viscosity and thermal conductivity

The substantial temperature gradient experienced by systems operating at relatively high temperatures significantly impacts the transport characteristics of fluids. Hence, considering temperature-dependent fluid properties is critical for obtaining realistic prediction of fluid behavior and optimizing system performance. The current study focuses on the flow of nanofluids in a stationary cone–disk system (SCDS), taking into account temperature-dependent thermal conductivity and viscosity. The influence of Brownian motion, thermophoresis, and Rosseland radiative flux on the heat transport features are also examined. The Reynolds model for viscosity and Chiam's model for thermal conductivity are employed. The Navier–Stokes equation, the energy equation, the incompressibility condition, and the continuity equation for nanoparticles constitute the governing system. The Lie-group transformations lead the self-similar ordinary differential equations, which are then solved numerically. Multi-variate non-linear regression models for the rate of heat and mass transfers on the disk surface were developed. Our study reveals a notable decrease in the rate of heat and mass transfer when pre-swirl exists in the flow. The significant influence of nanofluid slip mechanisms on the effective temperature and nanofluid volume fraction (NVF) within the system is highlighted. Furthermore, the variable viscosity property enhances the temperature and NVF of the SCDS.

Mean-squared exponential stability of distributed almost periodic sequence for nonlocal space-time discrete shunting inhibitory cellular neural networks with Brownian motions

The discrete-time and discrete-space model for stochastic nonlocal shunting inhibitory cellular neural networks with reaction diffusions are modelled in the first time. Owing to the consideration of spatial variables, the discrete model derived in this article is more complex than the traditional ordinary difference model. In accordance with the constant-variation-formula for discrete-time and discrete-space model, Banach contracting mapping principle, the method of proof by contradiction and stochastic calculus, we also obtain the existence of a unique bounded almost periodic sequence for the discrete model, which is exponentially stable in the mean-square sense. Noting that it is the first time to consider the dynamics of almost periodicity in distribution of time-space discrete neural network models. The practicability of the present results is demonstrated by means of an illustration.

EUROPEAN OPTION PRICING IN THE GENERALIZED MIXED WEIGHTED FRACTIONAL BROWNIAN MOTION

In order to describe the self-similarity and long-range dependence of financial asset prices, this paper adopts a new fractional-type process, i.e, the generalized mixed weighted fractional Brownian motion to describe the dynamic change process of risky asset prices. A European option pricing model driven by the generalized mixed weighted fractional Brownian motion is constructed, and explicit solutions to the pricing formulas of European call options and European put options are derived by using the arbitrage-free pricing theory. Finally, through numerical simulation, the influence of the parameter on the option price is analyzed.