Three-dimensional Brownian Motion Research Articles

Using Brownian model to study the effect of temperature on diffusion coefficient

This study employs molecular simulation techniques, particularly the Brownian motion model, to investigate the influence of temperature on diffusion coefficients. Leveraging Einsteins relationship for calculating diffusion coefficients and a one-dimensional random walk model, we systematically explore the impact of simulation direction, particle quantity, and simulation duration on diffusion behavior. The research extends its scope to three-dimensional Brownian motion simulations, offering insights into the stochastic motion of particles in a fluid. The primary objective is to analyze the relationship between temperature and diffusion coefficients. Through rigorous linear regression analysis, a significant and strong linear association is identified, demonstrating that an increase in temperature correlates with an increase in the diffusion coefficient. The research not only contributes to the fundamental understanding of molecular motion but also provides practical recommendations for simulation parameters, especially in resource-constrained computational environments. The study acknowledges certain limitations in the current Brownian motion algorithm and proposes avenues for future research to enhance computational efficiency and precision. This abstract encapsulates the key methodologies, findings, and implications of the research, laying the foundation for a comprehensive exploration of temperature-dependent diffusion coefficients in molecular systems.

Valuing three-asset barrier options and autocallable products via exit probabilities of Brownian bridge

This paper discusses pricing barrier options and autocallable structured products on underlying three assets via exit probabilities of a three-dimensional Brownian bridge. We derive the marginal exit and bivariate co-exit probabilities. Despite the impossibility of finding an analytical trivariate co-exit probability, its calculation is not regarded as a purely numerical problem. After specifying the component to be numerically evaluated, logistic regression with the Monte Carlo method is adopted to predict it. Extensive numerical experiments of calculating the exit probability of the three-dimensional Brownian motion and pricing complex products demonstrate the effectiveness and efficiency of our approach.

A novel technique for solving unsteady three-dimensional brownian motion of a thin film nanofluid flow over a rotating surface.

The motion of the fluid due to the swirling of a disk/sheet has many applications in engineering and industry. Investigating these types of problems is very difficult due to the non-linearity of the governing equations, especially when the governing equations are to be solved analytically. Time is also considered a challenge in problems, and times dependent problems are rare. This study aims to investigate the problem related to a transient rotating angled plate through two analytical techniques for the three-dimensional thin film nanomaterials flow. The geometry of research is a swirling sheet with a three-dimensional unsteady nanomaterial thin-film moment. The problem's governing equations of the conservation of mass, momentum, energy, and concentration are partial differential equations (PDEs). Solving PDEs, especially their analytical solution, is considered a serious challenge, but by using similar variables, they can be converted into ordinary differential equations (ODEs). The derived ODEs are still nonlinear, but it is possible to approximate them analytically with semi-analytical methods. This study transformed the governing PDEs into a set of nonlinear ODEs using appropriate similarity variables. The dimensionless parameters such as Prandtl number, Schmidt number, Brownian motion parameter, thermophoretic parameter, Nusselt, and Sherwood numbers are presented in ODEs, and the impact of these dimensionless parameters was considered in four cases. Every case that is considered in this problem was demonstrated with graphs. This study used modified AGM (Akbari-Ganji Method) and HAN (Hybrid analytical and numerical) methods to solve the ODEs, which are the novelty of the current study. The modified AGM is novel and has made the former AGM more complete. The second semi-analytical technique is the HAN method, and because it has been solved numerically in previous articles, this method has also been used. The new results were obtained using the modified AGM and HAN solutions. The validity of these two analytical solutions was proved when compared with the Runge-Kutta fourth-order (RK4) numerical solutions.

A Novel Method for 3D Nanoscale Tracking of 100 nm Polystyrene Particles in Multi-Wavelength Evanescent Fields Microscopy – Absolute Difference Height Verification –

Total internal reflection is an optical imaging technique for nanoparticle tracking and observation employing the scattered light from an evanescent field near the interface or reference surface. Generally, the nanoparticle behavior is the three-dimensional Brownian motion in an aqueous medium. The motion can be traced by an optical microscopy, but it cannot be traced by an electron microscopy technique. In the three-dimensional nanoparticle moving position, the X and Y positions are parallel to the surface, which can be traced by the general microscopy techniques. However, the height position Z of a nanoparticle perpendicular to the surface could not be traced without the longitudinal scanning method. Here, a novel method is proposed to investigate the 3D position of nanoparticles by applying multi-wavelength evanescent fields microscopy, which has a high spatial resolution in the Z-direction without longitudinal scanning. This paper focuses on the verification of measurement in the Z-direction. A piezoelectric actuator was employed to control the nanoparticle displacement in height Z. Standard polystyrene 100 nm particles were randomly adhered on a spherical tip that connected with the piezoelectric actuator. The spherical tip was essentially made from an optical adhesive (n = 1.348) with a refractive index close to the water for decreasing the unnecessary signal from the tip-self during nanoparticle observation in the water. The proposed method could obtain the multi-wavelength scattering lights from the observed nanoparticles by an 8-bit color camera with higher than 50 frames per second recording to investigate the 3D nanoscale tracking. The X and Y positions of nanoparticles were determined by the centroid of the scattering light intensities. The height Z was determined from the logarithm ratios between the detected scattering light intensities of both wavelengths. The measurement repeatability of the absolute difference in height between nanoparticles could be measured less than ±16 nm by using the proposed method. The penetration height measurability range was approximated at 250 nm from the reference surface.

Identification of the Polaron measure in strong coupling and the Pekar variational formula

The path measure corresponding to the Fröhlich polaron appearing in quantum statistical mechanics is defined as the tilted measure \begin{equation*}\mathrm{d}\widehat{\mathbb{P}}_{\varepsilon ,T}=\frac{1}{Z(\varepsilon ,T)}\exp \biggl\{\frac{1}{2}\int _{-T}^{T}\int _{-T}^{T}\frac{\varepsilon \mathrm{e}^{-\varepsilon \vert t-s\vert }}{\vert \omega (t)-\omega (s)\vert }\,\mathrm{d}s\,\mathrm{d}t\biggr\}\,\mathrm{d}\mathbb{P}.\end{equation*} Here, $\varepsilon >0$ is a constant known as the Kac parameter or the inverse-coupling parameter, and $\mathbb{P}$ is the distribution of the increments of the three-dimensional Brownian motion. In (Comm. Pure Appl. Math. 73 (2020) 350–383) it was shown that, when $\varepsilon >0$ is sufficiently small or sufficiently large, the (thermodynamic) limit $\lim_{T\to \infty }\widehat{\mathbb{P}}_{\varepsilon ,T}={\widehat{\mathbb{P}}}_{\varepsilon }$ exists as a process with stationary increments, and this limit was identified explicitly as a mixture of Gaussian processes. In the present article the strong coupling limit or the vanishing Kac parameter limit $\lim_{\varepsilon \to 0}\widehat{\mathbb{P}}_{\varepsilon }$ is investigated. It is shown that this limit exists and coincides with the increments of the so-called Pekar process, a stationary diffusion with generator $\frac{1}{2}\Delta +(\nabla \psi /\psi )\cdot \nabla $, where $\psi $ is the unique (up to spatial translations) maximizer of the Pekar variational problem \begin{equation*}g_{0}=\mathop{\mathrm{sup}}_{\Vert \psi \Vert _{2}=1}\biggl\{\int _{\mathbb{R}^{3}}\int _{\mathbb{R}^{3}}\psi ^{2}(x)\psi ^{2}(y)\vert x-y\vert ^{-1}\,\mathrm{d}x\,\mathrm{d}y-\frac{1}{2}\Vert \nabla \psi \Vert _{2}^{2}\biggr\}.\end{equation*} As the Pekar process was also earlier shown (Ann. Probab. 44 (2016) 3934–3964; Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017) 2214–2228; Comm. Pure Appl. Math. 70 (2017) 1598–1629) to be the limiting object of the mean-field polaron measures, the present identification of the strong coupling limit is a rigorous justification of the mean-field approximation of the polaron problem (on the level of path measures) conjectured by Spohn in (Ann. Physics 175 (1987) 278–318). Replacing the Coulomb potential by continuous function vanishing at infinity and assuming uniqueness (modulo translations) of the relevant variational problem, our proof also shows that path measures coming from a Kac interaction of the above form with translation invariance in space converge to the increments of the corresponding mean-field model.

Systematic Assessment of Burst Impurity in Confocal-Based Single-Molecule Fluorescence Detection Using Brownian Motion Simulations.

Single-molecule fluorescence detection (SMFD) experiments are useful in distinguishing sub-populations of molecular species when measuring heterogeneous samples. One experimental platform for SMFD is based on a confocal microscope, where molecules randomly traverse an effective detection volume. The non-uniformity of the excitation profile and the random nature of Brownian motion, produce fluctuating fluorescence signals. For these signals to be distinguished from the background, burst analysis is frequently used. Yet, the relation between the results of burst analyses and the underlying information of the diffusing molecules is still obscure and requires systematic assessment. In this work we performed three-dimensional Brownian motion simulations of SMFD, and tested the positions at which molecules emitted photons that passed the burst analysis criteria for different values of burst analysis parameters. The results of this work verify which of the burst analysis parameters and experimental conditions influence both the position of molecules in space when fluorescence is detected and taken into account, and whether these bursts of photons arise purely from single molecules, or not entirely. Finally, we show, as an example, the effect of bursts that are not purely from a single molecule on the accuracy in single-molecule Förster resonance energy transfer measurements.

On the Dirichlet form of three-dimensional Brownian motion conditioned to hit the origin

Our concern in this paper is the energy form induced by an eigenfunction of a self-adjoint extension of the restriction of the Laplace operator to $$C_c^\infty ({\mathbb{R}^3}\backslash \{ 0\} )$$). We will prove that this energy form is a regular Dirichlet form with core $$C_c^\infty ({\mathbb{R}^3})$$. The associated diffusion X behaves like a 3-dimensional Brownian motion with a mild radial drift when far from 0, subject to an ever-stronger push toward 0 near that point. In particular, {0} is not a polar set with respect to X. The diffusion X is rotation invariant, and admits a skew-product representation before hitting {0}: its radial part is a diffusion on (0, ∞) and its angular part is a time-changed Brownian motion on the sphere S2. The radial part of X is a “reflected” extension of the radial part of X0 (the part process of X before hitting {0}). Moreover, X is the unique reflecting extension of X0, but X is not a semi-martingale.

Self-Propulsion and Active Motion of Janus Ellipsoids.

The propulsion of platinum-coated polystyrene prolate ellipsoids, as generated by catalytic decomposition of hydrogen peroxide, is characterized by direct visualization of the trajectories of the active particles. These Janus ellipsoids were fabricated by stretching micron-sized polystyrene spheres into different aspect ratios; half of the particle is then capped lengthwise along the ellipsoid's major axis, with platinum deposition. These particles exhibit complex dynamical trajectories in aqueous solutions of hydrogen peroxide of concentration in the range of 2-8% (w/v). In this range, a transition from three-dimensional passive Brownian motion to two-dimensional active motion is observed as the hydrogen peroxide concentration is increased. This transition from passive to active motion is complete by 4% (w/v) hydrogen peroxide. We quantify the effect of particle aspect ratio on the mean-squared displacement and mean-squared angular displacement at the highest hydrogen peroxide concentration. The two-dimensional trajectories of the individual particles were grouped into three categories for dynamical analysis. In the first category, ballistic ellipsoids translate at least 5 times more than purely diffusive ellipsoids at the characteristic time scale of rotational diffusion. In the second category, spinning ellipsoids move only short distances with a dominant rotation about the minor axis; this rotation persists for many revolutions. A third category captures trajectories that include both significant translation and rotation. We consider the physical origins of the observed categories of motion and extract the forces and torques generated by the catalytically generated propulsion as a function of aspect ratio. The particle velocity, and therefore the active force, increases with the aspect ratio.

Security enhancement for OFDM-PON using Brownian motion and chaos in cell.

We propose a novel security enhancement technique for a physical layer secure orthogonal frequency division multiplexed passive optical network (OFDM-PON) based on three-dimensional Brownian motion and chaos in cell (3DBCC). This method confuses an OFDM symbol via transforming it into a 3D symbol matrix and a 3D cell matrix with different size lengths. Different dividing-confusion rules then generate different complementary cumulative distribution functions (CCDFs) of peak-to-average power ratio (PAPR). And we can pre-estimate bit error rate (BER) performance by calculating the CCDF values. We also find that the processing time decreases with the matrix's side length decreasing simultaneously. A new weighted comprehensive value (Qw) is further used to evaluate the overall performance between the processing time and the BER. Finally, an experiment successfully demonstrates a physical layer secure OFDM signal transmission with 22.06-Gb/s data rate over a 25.4-km standard single mode fiber (SSMF). These results indicate that cell (53) has the weighted optimum overall performance, which verifies that the proposed encryption technique is promising for building a physical layer security enhanced OFDM-PON system with a low processing time delay and a good BER for future access network systems.

Transition Probability of Brownian Motion in the Octant and its Application to Default Modelling

ABSTRACTWe derive a semi-analytical formula for the transition probability of three-dimensional Brownian motion in the positive octant with absorption at the boundaries. Separation of variables in spherical coordinates leads to an eigenvalue problem for the resulting boundary value problem in the two angular components. The main theoretical result is a solution to the original problem expressed as an expansion into special functions and an eigenvalue which has to be chosen to allow a matching of the boundary condition. We discuss and test several computational methods to solve a finite-dimensional approximation to this nonlinear eigenvalue problem. Finally, we apply our results to the computation of default probabilities and credit valuation adjustments in a structural credit model with mutual liabilities.

A double color image encryption scheme based on three-dimensional brownian motion

Image encryption is an efficient technique for image protection. This paper presents a double color image encryption (DCIE) scheme based on three-dimensional (3D) Brownian motion. The architecture of diffusion, confusion and diffusion is adopted. The proposed algorithm firstly decomposes two color plain images into R, G, B components and bit planes, then conducts XOR operation between every bit plane and a key bit plane generated from carrier images; next, arranges all the bit planes into a 3D bit matrix, performs block-in 3D Brownian motion confusion and block confusion on it, which can permute the elements across bit planes and across plain images; finally, a subsequent diffusion process is utilized to improve the encryption effect. Simulation results and performance analyses illustrate that the proposed encryption scheme can simultaneously encrypt two color images into noise-like ones that have strong ability of resisting various known attacks.

Dynamics of a magnetic active Brownian particle under a uniform magnetic field.

The dynamics of a magnetic active Brownian particle undergoing three-dimensional Brownian motion, both translation and rotation, under the influence of a uniform magnetic field is investigated. The particle self-propels at a constant speed along its magnetic dipole moment, which reorients due to the interplay between Brownian and magnetic torques, quantified by the Langevin parameter α. In this work, the time-dependent active diffusivity and the crossover time (τ^{cross})-from ballistic to diffusive regimes-are calculated through the time-dependent correlation function of the fluctuations of the propulsion direction. The results reveal that, for any value of α, the particle undergoes a directional (or ballistic) propulsive motion at very short times (t≪τ^{cross}). In this regime, the correlation function decreases linearly with time, and the active diffusivity increases with it. It the opposite time limit (t≫τ^{cross}), the particle moves in a purely diffusive regime with a correlation function that decays asymptotically to zero and an active diffusivity that reaches a constant value equal to the long-time active diffusivity of the particle. As expected in the absence of a magnetic field (α=0), the crossover time is equal to the characteristic time scale for rotational diffusion, τ_{rot}. In the presence of a magnetic field (α>0), the correlation function, the active diffusivity, and the crossover time decrease with increasing α. The magnetic field regulates the regimes of propulsion of the particle. Here, the field reduces the period of time at which the active particle undergoes a directional motion. Consequently, the active particle rapidly reaches a diffusive regime at τ^{cross}≪τ_{rot}. In the limit of weak fields (α≪1), the crossover time decreases quadratically with α, while in the limit of strong fields (α≫1) it decays asymptotically as α^{-1}. The results are in excellent agreement with those obtained by Brownian dynamics simulations.

Mean-field interaction of Brownian occupation measures, I: Uniform tube property of the Coulomb functional

Nous etudions la mesure de trajectoire transformee engendree par l’auto-interaction d’un mouvement Brownien tridimensionnel en utilisant un biais exponentiel par l’energie de Coulomb des mesures d’occupation de ce mouvement au temps $t$. Les asymptotes logarithmiques de la fonction de partition ont ete identifiees dans les annees 1980 par Donsker et Varadhan [Comm. Pure Appl. Math. 505 (1983) 505–528] au moyen d’une formule variationnelle. Recemment, dans (Brownian occupations measures, compactness and large deviations (2014) Preprint), une nouvelle technique pour etudier la mesure de chemins elle-meme a ete introduite. Elle permet de prouver que la mesure d’occupation normalisee se concentre asymptotiquement autour de l’ensemble des maximums de la formule. Dans le present article, nous prouvons que la fonctionnelle de Coulomb de la mesure d’occupation se concentre elle aussi, dans la topologie uniforme, autour de l’ensemble des fonctionnelles de Coulomb correspondant aux maximums. Ceci represente une etape decisive vers une preuve rigoureuse de la convergence des mesures d’occupation normalisees vers un melange explicite des maximums, derivee dans (Mean-field interaction of Brownian occupation measures, II: A rigorous construction of the Pekar process. Preprint). Nos methodes reposent sur l’obtention de la continuite holderienne de la fonctionnelle de Coulomb de la mesure d’occupation avec des probabilites de deviations exponentiellement petites, en invoquant la theorie des grandes deviations developpee dans (Brownian occupations measures, compactness and large deviations (2014) Preprint) pour une certaine fonctionnelle, invariante par dcalage, des mesures d’occupation.

An image encryption scheme based on three-dimensional Brownian motion and chaotic system**Project supported by the National Natural Science Foundation of China (Grant Nos. 41571417 and 61305042), the National Science Foundation of the United States (Grant Nos. CNS-1253424 and ECCS-1202225), the Science and Technology Foundation of Henan Province, China (Grant No. 152102210048), the Foundation and Frontier Project of Henan Province, China

At present, many chaos-based image encryption algorithms have proved to be unsafe, few encryption schemes permute the plain images as three-dimensional (3D) bit matrices, and thus bits cannot move to any position, the movement range of bits are limited, and based on them, in this paper we present a novel image encryption algorithm based on 3D Brownian motion and chaotic systems. The architecture of confusion and diffusion is adopted. Firstly, the plain image is converted into a 3D bit matrix and split into sub blocks. Secondly, block confusion based on 3D Brownian motion (BCB3DBM) is proposed to permute the position of the bits within the sub blocks, and the direction of particle movement is generated by logistic-tent system (LTS). Furthermore, block confusion based on position sequence group (BCBPSG) is introduced, a four-order memristive chaotic system is utilized to give random chaotic sequences, and the chaotic sequences are sorted and a position sequence group is chosen based on the plain image, then the sub blocks are confused. The proposed confusion strategy can change the positions of the bits and modify their weights, and effectively improve the statistical performance of the algorithm. Finally, a pixel level confusion is employed to enhance the encryption effect. The initial values and parameters of chaotic systems are produced by the SHA 256 hash function of the plain image. Simulation results and security analyses illustrate that our algorithm has excellent encryption performance in terms of security and speed.

Entropic forces in Brownian motion

Interest in the concept of entropic forces has risen considerably since Verlinde proposed in 2011 to interpret the force in Newton's second law and gravity as entropic forces. Brownian motion—the motion of a small particle (pollen) driven by random impulses from the surrounding molecules—may be the first example of a stochastic process in which such forces are expected to emerge. In this article, it is shown that at least two types of entropic force can be identified in three-dimensional Brownian motion. This analysis yields simple derivations of known results of Brownian motion, Hooke's law, and—applying an external (non-radial) force—Curie's law and the Langevin-Debye equation.

Rotational microrheology of Maxwell fluids using micron-sized wires

We demonstrate a simple method for rotational microrheology in complex fluids using micrometric wires. The three-dimensional rotational Brownian motion of the wires suspended in Maxwell fluids is measured from their projection on the focal plane of a microscope. We analyze the mean-squared angular displacement of the wires of length between 1 and 40 μm. The viscoelastic properties of the suspending fluids are examined from this analysis and found to be in good agreement with macrorheology data. Viscosities of simple and complex fluids between 10(-2) and 30 Pa s could be measured. As for the elastic modulus, values up to ∼5 Pa could be determined. This simple technique, allowing for a broad range of probed length scales, opens new perspectives in microrheology of heterogeneous materials such as gels, glasses and cells.

Three-dimensional Brownian motion and the golden ratio rule

Let $X=(X_t)_{t\ge0}$ be a transient diffusion process in $(0,\infty)$ with the diffusion coefficient $\sigma>0$ and the scale function $L$ such that $X_t\rightarrow\infty$ as $t\rightarrow \infty$, let $I_t$ denote its running minimum for $t\ge0$, and let $\theta$ denote the time of its ultimate minimum $I_{\infty}$. Setting $c(i,x)=1-2L(x)/L(i)$ we show that the stopping time \[\tau_*=\inf\{t\ge0\vert X_t\ge f_*(I_t)\}\] minimizes $\mathsf{E}(\vert\theta-\tau\vert-\theta)$ over all stopping times $\tau$ of $X$ (with finite mean) where the optimal boundary $f_*$ can be characterized as the minimal solution to \[f'(i)=-\frac{\sigma^2(f(i))L'(f(i))}{c(i,f(i))[L(f(i))-L(i)]}\int_i^{f(i)}\frac{c_i'(i,y)[L(y) -L(i)]}{\sigma^2(y)L'(y)}\,dy\] staying strictly above the curve $h(i)=L^{-1}(L(i)/2)$ for $i>0$. In particular, when $X$ is the radial part of three-dimensional Brownian motion, we find that \[\tau_ *=\inf\biggl\{t\ge0\Big\vert\frac{X_t-I_t}{I_t}\ge\varphi\biggr\},\] where $\varphi=(1+\sqrt{5})/2=1.61\ldots$ is the golden ratio. The derived results are applied to problems of optimal trading in the presence of bubbles where we show that the golden ratio rule offers a rigorous optimality argument for the choice of the well-known golden retracement in technical analysis of asset prices.

Maximum likelihood estimation of spatially correlated signal-dependent noise in hyperspectral images

A new algorithm is described for estimating the noise model parameters in hyperspectral data when neither noise components variance nor noise spatial/spectral correlation priors are available. A maximum likelihood (ML) technique is introduced for checking the noise spatial correlation hypothesis and estimating the spatial correlation function width alongside with estimating signal-independent and signal-dependent noise components variance. The hyperspectral image is assumed to match a limited set of assumptions. A three-dimensional (3-D) fractional Brownian motion (fBm) model is introduced for describing locally the texture of the 3-D image noisy textural fragment. Nonstationarity of the useful image signal is taken into account by performing the estimation locally on a 3-D block-by-block basis. The accuracy of the proposed algorithm is first illustrated for synthetic images obtained from either pure fBm or almost noise-free AVIRIS hyperspectral images artificially degraded with spatially correlated noise. The results obtained for synthetic images demonstrate appropriate accuracy and robustness of the proposed method. Then results obtained for real life AVIRIS hyperspectral data sets confirm the noise spatial uncorrelation hypothesis for images acquired by the AVIRIS sensor. Conclusions and open problems are outlined.

Influence of confinement on dynamical heterogeneities in dense colloidal samples

We study a dense colloidal suspension confined between two quasiparallel glass plates as a model system for a supercooled liquid in confined geometries. We directly observe the three-dimensional Brownian motion of the colloidal particles using laser scanning confocal microscopy. The particles form dense layers along the walls, but crystallization is avoided as we use a mixture of two particle sizes. A normally liquidlike sample, when confined, exhibits slower diffusive motion. Particle rearrangements are spatially heterogeneous, and the shapes of the rearranging regions are strongly influenced by the layering. These rearranging regions become more planar upon confinement. The wall-induced layers and changing character of the spatially heterogeneous dynamics appear strongly connected to the confinement-induced glassiness.

3-D Brownian Motion Simulator for High-Sensitivity Nanobiotechnological Applications

A wide variety of nanobiotechnologic applications are being developed for nanoparticle based in vitro diagnostic and imaging systems. Some of these systems make possible highly sensitive detection of molecular biomarkers. Frequently, the very low concentration of the biomarkers makes impossible the classical, partial differential equation-based mathematical simulation of the motion of the nanoparticles involved. We present a three-dimensional Brownian motion simulation tool for the prediction of the movement of nanoparticles in various thermal, viscosity, and geometric settings in a rectangular cuvette. For nonprofit users the server is freely available at the site http://brownian.pitgroup.org.