Simulation Of Brownian Motion Research Articles

A note on the exact simulation of spherical Brownian motion

We describe an exact simulation algorithm for the increments of Brownian motion on a sphere of arbitrary dimension, based on the skew-product decomposition of the process with respect to the standard geodesic distance. The radial process is closely related to a Wright–Fisher diffusion, increments of which can be simulated exactly using the recent work of Jenkins & Spanò (2017). The rapid spinning phenomenon of the skew-product decomposition then yields the algorithm for the increments of the process on the sphere.

Modification of the spectra of correlated charm-anticharm quark pairs in the quark-gluon plasma * * This work was supported by Chinese government scholarship (CSC) for foreigner graduate students and by NSFC (11675079)

Heavy quarks play an important role in probing the properties of strongly interacting quark-gluon plasma (QGP) created in ultra-relativistic heavy-ion collisions. We study the interactions of single heavy (charm) quarks and correlated charm and anticharm ( ) quark pairs with the medium constituents of QGP by performing fireball+Langevin simulations of the pertinent Brownian motion with elastic collisions. Besides studying the traditional observables, the nuclear modification factor and the elliptic flow of single heavy quarks in QGP for different thermal relaxation rates, we also study the broadening of the azimuthal correlations of charm and anticharm quark pairs in the QGP medium for different relaxation rates and transverse momenta classes. We quantified the smearing of pair azimuthal correlations with an increasing thermal relaxation rate: while the (nearly) back-to-back correlations among pairs are almost completely washed out at low transverse momentum (pT), these correlations at high pT largely survive the pair diffusion. This provides a novel observable for diagnosing the properties of QGP.

Forecast of long‐term EUA price probability using momentum strategy and GBM simulation

AbstractCO2 storage projects are financially intensive and, without the European Union emission trading scheme that encourages CO2 emission reduction projects, they will not be cost effective, that is, feasible. Similar to the huge uncertainties involved in long‐term energy price prediction, CO2 market price is volatile. In this paper, indicators for key CO2 price changes are detected by moving average analysis. To prove the validity of the methods, geometric Brownian motion (GBM) simulations were performed for different drift and volatility values. Drift, the standard deviation of CO2 price changes and CO2 volatility were used on a daily, weekly and monthly basis for short term simulations and on a weekly basis for long term simulations. GBM simulation can help determine the probability of a European emission allowances (EUA) price, which is featured by great variations, depending on the period. Signals for periods were determined by momentum strategy, with 60 days for short and 365 days for the long moving average. They show correlation with natural gas prices, that is, EUA signals appear 13–17 months after the signal in natural gas price. Among the vast number of proposed hybrid prediction models, this correlation allows the prediction of a long‐term price trend. It seems that back‐loading measures obstructed risk estimates because they resulted in extreme price drift in some periods, as the consequence of rebalancing in supply and demand. © 2020 Society of Chemical Industry and John Wiley & Sons, Ltd.

Simulation hedge investment portfolios through options portfolio

<p>This paper presents two hedging strategies with financial options to mitigate the market risk associated with the future purchase of investment portfolios that exhibit the same behavior as Colombia's COLCAP stock index. The first strategy consists in the purchase of a Call plain vanilla option and the second strategy in the purchase of a Call option and the sale of a Call option. The second strategy corresponds to a portfolio of options called Bull Call Spread. To determine the benefits of hedging and the best strategy, the Geometric Brownian Motion and Monte Carlo simulation is used. The results show that the two hedging strategies manage to mitigate market risk and the best strategy is the first one despite the fact that the Bull Call Spread strategy is lower cost.</p>

Exposing the plural nature of molecular clouds

We present the Multiscale non-Gaussian Segmentation (MnGSeg) analysis technique. This wavelet-based method combines the analysis of the probability distribution function (PDF) of map fluctuations as a function of spatial scales and the power spectrum analysis of a map. This technique allows us to extract the non-Gaussianities identified in the multiscaled PDFs usually associated with turbulence intermittency and to spatially reconstruct the Gaussian and the non-Gaussian components of the map. This new technique can be applied on any data set. In the present paper, it is applied on a Herschel column density map of the Polaris flare cloud. The first component has by construction a self-similar fractal geometry similar to that produced by fractional Brownian motion (fBm) simulations. The second component is called the coherent component, as opposed to fractal, and includes a network of filamentary structures that demonstrates a spatial hierarchical scaling (i.e. filaments inside filaments). The power spectrum analysis of the two components proves that the Fourier power spectrum of the initial map is dominated by the power of the coherent filamentary structures across almost all spatial scales. The coherent structures contribute increasingly from larger to smaller scales, without producing any break in the inertial range. We suggest that this behaviour is induced, at least partly, by inertial-range intermittency, a well-known phenomenon for turbulent flows. We also demonstrate that the MnGSeg technique is itself a very sensitive signal analysis technique that allows the extraction of the cosmic infrared background (CIB) signal present in the Polaris flare submillimetre observations and the detection of a characteristic scale for 0.1 ≲ l ≲ 0.3 pc. The origin of this characteristic scale could partly be the transition of regimes dominated by incompressible turbulence versus compressible modes and other physical processes, such as gravity.

Quantum systems for Monte Carlo methods and applications to fractional stochastic processes

Random numbers are a fundamental and useful resource in science and engineering with important applications in simulation, machine learning and cyber-security. Quantum systems can produce true random numbers because of the inherent randomness at the core of quantum mechanics. As a consequence, quantum random number generators are an efficient method to generate random numbers on a large scale. We study in this paper the applications of a viable source of unbiased quantum random numbers (QRNs) whose statistical properties can be arbitrarily programmed without the need for any post-processing and that pass all standard randomness tests of the NIST and Dieharder test suites without any randomness extraction. Our method is based on measuring the arrival time of single photons in shaped temporal modes that are tailored with an electro-optical modulator. The advantages of our QRNs are shown via two applications: simulation of a fractional Brownian motion, which is a non-Markovian process, and option pricing under the fractional SABR model where the stochastic volatility process is assumed to be driven by a fractional Brownian motion. The results indicate that using the same number of random units, our QRNs achieve greater accuracy than those produced by standard pseudo-random number generators. Moreover, we demonstrate the advantages of our method via an increase in computational speed, efficiency, and convergence.

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.

Direct numerical simulation of particle Brownian motion in a fluid with inhomogeneous temperature field

In this work the fluctuating-lattice Boltzmann method was adopted to numerically investigate the Brownian motion of particles in a fluid with inhomogeneous temperature field. It has been found that the Brownian particles are preferential to randomly move into a cold fluid area. Once the particles go into the cold area, the boundary between the hot fluid and cold fluid acts like a temperature barrier, preventing the particles from going out. Most important of all, the Brownian particles can be captured or collected by the cold fluid area if the temperature of cold fluid is lower than a critical value. In addition, the dependence of this critical value on the fluid viscosity is studied.

Simulation of Stochastic Geometric Brownian Motion of Stock Market Using R Programming

Simulation of Stochastic Geometric Brownian Motion of Stock Market Using R Programming

A Molecular Dynamics Simulation of Brownian Motion of a Nanoparticle in a Nanofluid

ABSTRACTBrownian motion of nanoparticles is one of the mechanisms proposed to explain how nanoparticles improve the thermal conductivity of nanofluids where it has been assumed that a large volume of the surrounding base fluid is dragged by the moving nanoparticle, which leads to the heat transfer enhancement. Therefore, the necessary condition for this mechanism to play a key role is that the velocity vector of the moving nanoparticle points in the same direction as the velocity vectors of the surrounding base fluid. Using equilibrium molecular dynamics simulation and modeling a nanofluid, this mechanism is examined where the angles between the velocity vector of the nanoparticle and the velocity vectors of the surrounding base fluid are considered. The results for these angles indicate that the moving nanoparticle cannot considerably drag the surrounding base fluid even with increasing temperature. Furthermore, it is found that the moving nanoparticle may cause a weak thermal dispersion (fluctuation) around itself.

Validation of a Central Approximation in Theories of Regular and Random Electrochemical Electrode Arrays. Extraction of Statistical Information from Amperometric Response of Random Electrode Arrays

Regular electrode arrays generally possess a symmetry which (under certain conditions) allows to reduce their mathematical description to consideration of a single electrode of this array within the laterally bounded domain (known as ‘unit cell’ or ‘diffusion domain’) [1, 2]. In this case by multiplying the simulated current of the single cell by the number of electrodes in the array one would obtain an electrochemical response of the whole array. The electrodes in arrays are generally ordered in a hexagonal or square patterns, which result in hexagonal or square cross-sections of the unit cells, respectively. A straightforward approach to the mass-transport problem in such cells would require complex and time-consuming 3D simulations. Therefore almost always an approximation is made in simulations by replacing the original unit cell by the one with circular cross-section of the same surface area. At one hand, this greatly facilitates the simulations due to the possibility to formulate the mass-transport problem in two dimensions (2D). On the other hand, although this approach was proven to be effective in practice, the error introduced by this approximation, to the best of our knowledge, was never been studied or even estimated. The error associated with the approximation is based on the fact that it is impossible to pave the plane with non-intersecting circles contrary to the perfect plane paving with hexagons or squares as well as due to the different possible diffusion pathways within the single cells with different cross-sections. In order to assess quantitatively this approximation we performed 3D Brownian motion simulations for the single cells of square, hexagon and circular cross-sections equipped with disk electrode. Moreover the situations with inlaid, recessed and protruding electrodes were also considered for each cross-section shape. Notwithstanding the completely different diffusion patterns observed in these systems, the obtained results showed that the approximation gives rise to an experimentally indistinguishable error (less than 5%) for all considered systems [2]. The only exception (i.e. when the error is larger than 5%) was the case when the electrode radius was extremely close to the characteristic size of the cross-section, i.e. the situation when the neighbouring electrodes in the array almost touch each other. Such systems are of low experimental interest and, in addition, exhibit the same behavior as a planar electrode with the surface area of the whole array (except of the very initial times of the experiment). With the same idea in mind, to check the validity of the circular diffusion domain approximation, the random arrays were considered. The arrays with randomly distributed electrodes of the same size were partitioned by means of Voronoi tessellation, which creates a unit cell with polygonal cross-section around each electrode. These polygonal unit cells were replaced then by the unit cells with circular cross-section having the same surface area. Taking into account distribution of the unit cell sizes one can compute the response of the whole array. By comparison the outcome of this procedure with the full scale 3D Brownian motion simulations of the whole array we showed that employment of the circular unit cells in simulations of the responses of the random electrode arrays is validated [3]. These results allow us to develop an easy to evaluate analytical approximation for the electrochemical current of the random array requiring only a distribution of sizes of the unit cells as an input. The analytical approximation in turn makes possible retrieval of the unit cell size distribution from the amperometric response of the random array.

A Brownian motion simulation for the chloride diffusivity of concrete

The purpose of this paper is to present a Brownian motion simulation for the chloride diffusivity of concrete. According to stereological theory, the probability density function and cumulative distribution function for the circular aggregates in terms of the number of aggregates are derived. By coating each aggregate with an interfacial transition zone (ITZ) layer of equal thickness, concrete is then reduced to a two-phase composite material, composed of a bulk cement paste and equivalent aggregates. The equivalent ITZ thickness and the chloride diffusivity of each equivalent aggregate are formulated in an analytical manner. The Brownian motion simulation is used to compute the chloride diffusivity of concrete. Finally, the validity of the numerical simulation is verified with two sets of experimental results and the effects of the aggregate area fraction, the chloride diffusivity of ITZ, and the ITZ thickness on the chloride diffusivity of concrete are evaluated in a quantitative manner. The paper concludes that the numerical simulation can predict the chloride diffusivity of concrete with an average relative error smaller than 6% for the two selected verification examples.

PRINCIPAL COMPONENT ANALYSIS STUDIES OF TURBULENCE IN OPTICALLY THICK GAS

ABSTRACT In this work we investigate the sensitivity of principal component analysis (PCA) to the velocity power spectrum in high-opacity regimes of the interstellar medium (ISM). For our analysis we use synthetic position–position–velocity (PPV) cubes of fractional Brownian motion and magnetohydrodynamics (MHD) simulations, post-processed to include radiative transfer effects from CO. We find that PCA analysis is very different from the tools based on the traditional power spectrum of PPV data cubes. Our major finding is that PCA is also sensitive to the phase information of PPV cubes and this allows PCA to detect the changes of the underlying velocity and density spectra at high opacities, where the spectral analysis of the maps provides the universal −3 spectrum in accordance with the predictions of the Lazarian & Pogosyan theory. This makes PCA a potentially valuable tool for studies of turbulence at high opacities, provided that proper gauging of the PCA index is made. However, we found the latter to not be easy, as the PCA results change in an irregular way for data with high sonic Mach numbers. This is in contrast to synthetic Brownian noise data used for velocity and density fields that show monotonic PCA behavior. We attribute this difference to the PCA's sensitivity to Fourier phase information.

Acceleration and Parallelization of ZENO/Walk-on-Spheres

This paper describes our on-going work to accelerate ZENO, a software tool based on Monte Carlo methods (MCMs), used for computing material properties at nanoscale. ZENO employs three main algorithms: (1) Walk on Spheres (WoS), (2) interior sampling, and (3) surface sampling. We have accelerated the first two algorithms. For the sake of brevity, the paper will discuss our work on the first one only as it is the most commonly used and the acceleration techniques were similar in both cases.WoS is a Brownian motion MCM for solving a class of partial differential equations (PDEs). It provides a stochastic solution to a PDE by estimating the probability that a random walk, which started at infinity, will hit the surface of the material under consideration. WoS is highly effective when the problem's geometry is additive, as this greatly reduces the number of walk steps needed to achieve accurate results. The walks start on the surface of an enclosing sphere and can make much larger jumps than in a direct simulation of Brownian motion. Our current implementation represents the molecular structure of nanomaterials as a union of possibly overlapping spheres. The core processing bottleneck in WoS is a Computational Geometry one, as the algorithm repeatedly determines the distance from query point to the material surface in each step of the random walk.In this paper, we present results from benchmarking spatial data structures, including several open-source implementations of k-D trees, for accelerating WoS algorithmically. The paper also presents results from our multicore and cluster parallel implementation to show that it exhibits linear strong scaling with the number of cores and compute nodes; this implementation delivers up to 4 orders of magnitude speedup compared to the original FORTRAN code when run on 8 nodes (each with dual 6-core Intel Xeon CPUs) with 24 threads per node.

Simulation of Stochastic differential equation of geometric Brownian motion by quasi-Monte Carlo method and its application in prediction of total index of stock market and value at risk

In the prediction of total stock index, we are faced with some parameters as they are uncertain in future and they can undergo changes, and this uncertainty has a few risks, and for a true analysis, the calculations should be performed under risk conditions. One of the evaluation methods under risk and uncertainty conditions is using geometric Brownian motion random differential equation and simulation by Monte Carlo and quasi-Monte Carlo methods as applied in this study. In Monte Carlo method, pseudo-random sequences are used to generate pseudo-random numbers, but in quasi-Monte Carlo method, quasi-random sequences are used with better uniformity and more rapid convergence compared with pseudo-random sequences. The predictions of total stock index and value at risk by this method are better and more exact than Monte Carlo method. This study at first evaluates random differential equation of geometric Brownian motion and its simulation by quasi-Monte Carlo method, and then its application in the predictions of total stock market index and value at risk can be evaluated.

Operational Efficiency Forecasting Model of an Existing Underground Mine Using Grey System Theory and Stochastic Diffusion Processes

Forecasting the operational efficiency of an existing underground mine plays an important role in strategic planning of production. Degree of Operating Leverage (DOL) is used to express the operational efficiency of production. The forecasting model should be able to involve common time horizon, taking the characteristics of the input variables that directly affect the value of DOL. Changes in the magnitude of any input variable change the value of DOL. To establish the relationship describing the way of changing we applied multivariable grey modeling. Established time sequence multivariable response formula is also used to forecast the future values of operating leverage. Operational efficiency of production is often associated with diverse sources of uncertainties. Incorporation of these uncertainties into multivariable forecasting model enables mining company to survive in today’s competitive environment. Simulation of mean reversion process and geometric Brownian motion is used to describe the stochastic diffusion nature of metal price, as a key element of revenues, and production costs, respectively. By simulating a forecasting model, we imitate its action in order to measure its response to different inputs. The final result of simulation process is the expected value of DOL for every year of defined time horizon.

Simulation of the active Brownian motion of a microswimmer

Unlike passive Brownian particles, active Brownian particles, also known as microswimmers, propel themselves with directed motion and thus drive themselves out of equilibrium. Understanding their motion can provide insight into out-of-equilibrium phenomena associated with biological examples such as bacteria, as well as with artificial microswimmers. We discuss how to mathematically model their motion using a set of stochastic differential equations and how to numerically simulate it using the corresponding set of finite difference equations both in homogenous and complex environments. In particular, we show how active Brownian particles do not follow the Maxwell-Boltzmann distribution—a clear signature of their out-of-equilibrium nature—and how, unlike passive Brownian particles, microswimmers can be funneled, trapped, and sorted.

A generalization of the bond fluctuation model to viscoelastic environments

A lattice-based simulation method for polymer diffusion in a viscoelastic medium is presented. This method combines the eight-site bond fluctuation model with an algorithm for the simulation of fractional Brownian motion on the lattice. The method applies to unentangled self-avoiding chains and is probed for anomalous diffusion exponents α between 0.7 and 1.0. The simulation results are in very good agreement with the predictions of the generalized Rouse model of a self-avoiding chain polymer in a viscoelastic medium.

Radiation Damaging of 3C-SiC Lattice: Computer Simulation of Brownian Motion of Non-Point Defects

The formation of porosity in thin layers of coating under its processing using the flux of high energyXe++is happen under the physical processes of nucleation of gaseous defects. The process is described by kinetic equations of Kolmogorov-Feller and Einstein-Smolukhovskii. The initial stage of the phase transition is considered as the superposition of two random processes: clustering of defects and their brownian motion. In the paper, we showthat a damaging phenomenon in layers can be predicted by means of computer experiments. The kinetic distribution function (DF) of defects depends on sizes and depth their penetration into materials under irradiation. Computer simulation of DF allows analysing the porosity and elastic stresses in bilayer caused byXe++blistering. The structures of defects are accumulated inSiClayer due to phase transition as well as a result the action perturbation of elastic forces in thin layers.

Brownian motion simulation: a quantization approach

In this article we suggest a new simulation method for the Wiener process. The main idea is to use functional quantization theory and the decomposition of the Wiener process by Brownian bridges to construct a suitable approximation of the Brownian motion.