Theory Of Brownian Motion Research Articles

Research on Cutting Tool Wear Based on Fractional Brownian Motion

Cutting tool wear is a very complex process. Various factors have a direct or indirect effect on cutting tool wear, resulting in uncertainty, so it is difficult for experimental data and result to have good stability. However, Vibration analysis is a very important means for condition monitoring and fault diagnosis. This paper aims to study the methods of tool vibration signal processing, pattern recognition and trend prediction. Collected on tool vibration signal at different times, wavelet noise reduction is used to pretreat the vibration signals. Then, for the self-similar vibration signals, we propose the fractional Brownian motion (FBM) theory with long-range dependence (LRD). Combined with Wigner-Ville spectrum, characteristic parameter can be extracted, so the cutting tool wear state can be determined according to fractal dimension and average slope of the fitting curve of the logarithm power spectrum. Finally, we use FBM model to predict the trend of tool vibration signals. Experiments show that the methods have a good effect on tool wear state recognition and trend prediction.

The paradigm of complex probability and the Brownian motion

Andrey N. Kolmogorov's system of axioms can be extended to encompass the imaginary set of numbers and this by adding to his original five axioms an additional three axioms. Hence, any experiment can thus be executed in what is now the complex probability set which is the sum of the real set with its corresponding real probability, and the imaginary set with its corresponding imaginary probability. The objective here is to evaluate the complex probabilities by considering supplementary new imaginary dimensions to the event occurring in the ‘real' laboratory. Whatever the probability distribution of the input random variable in is, the corresponding probability in the whole set is always one, so the outcome of the random experiment in can be predicted totally. The result indicates that chance and luck in is replaced now by total determinism in . This is the consequence of the fact that the probability in is got by subtracting the chaotic factor from the degree of our knowledge of the stochastic system. This novel complex probability paradigm will be applied to the classical theory of Brownian motion and to prove as well the law of large numbers in an original way.

Brownian Motion in the Fluids with Complex Rheology

Theory of Brownian motion of a fine particle in a viscoelastic fluid continuum is developed. The rheology of the embedding medium is described in terms of classical structure (spring-and-damper) schemes. It is shown that a great variety of conceivable ramifications of such structures could be reduced to an effective Jeffreys model: a Maxwell chain shunted by a damper. This scheme comprises just three material parameters: two for viscosities (fast and slow) and one for elasticity. For the thermal motion of a particle in such a fluid, the set of Langevin equations is derived. In the cases of 1D translational and 2D orientational (rotation about fixed axis) motions, the time dependencies of mean-square displacements are found analytically. It is shown that in a Jeffreys fluid, the Brownian motion has three regimes: fast diffusion (short times), slow diffusion (long times) and a crossover of those resulting in virtual localization (dynamic confinement) of the particle. Experimental evidence for that taken from the literature is presented. The developed formalism is applied to the particles with “frozen-in” dipole moments, whose orientational motion is a combination of Brownian diffusion and regular excitation by an AC field. The dynamic susceptibilities are calculated for the ensembles of particles liable for either 2D or 3D rotations. Comparison shows that the out-of-phase part of the susceptibility χ′′(ω) in 3D case differs considerably from that of 2D case. Function χ(ω) for the forced 3D rotary diffusion in a Jeffreys fluid is found for the first time. In our view, it would be useful for realistic theoretical interpretation of microrheology data and the more so for estimating the particle-mediated AC field-induced heating (magnetic hyperthermia) in viscoelastic fluid media.

Synaptic Perturbation and Consciousness

By allowing one artificial neural network to govern the synaptic noise injected into another based upon its appraisal of patterns nucleating from such disturbances, a contemplative form of artificial intelligence is formed whose creativity and pattern delivery closely parallels that of human cognition. Drawing upon the theory of fractional Brownian motion, we may derive an equation, verifiable through statistical mechanics, which governs both the novelty and rhythm of pattern turnover within such neural systems. Through this equation, we gain valuable insight into the process of idea formation within the brain, whether that organ is making sense of its environment or itself. In doing so, a relationship between creativity and consciousness is revealed, along with a potential path toward building conscious machine intelligence.

Black Scholes Option Pricing Model Brownian Motion Approach

Brownian motion has become one of the fundamental building blocks of modern quantitative finance. The mathematical theory of Brownian motion has been applied in contexts ranging far beyond the movement of particles in fluids. Until recently, stock market researchers have confronted the same problem. While they can chart the path of the market on a minute by minute basis it is very hard for them to observe who buys, who sells and how demand and supply affects price fluctuations. There exist many researchers about how the behavior of different investors makes the option price movement in a stock market. The purpose of this paper is to construct the black Scholes option pricing model in the stock markets by using of Brownian motion approach. The main ambition of this study is fourfold: 1) First we begin our approach to construction of Brownian motion from the simple symmetric random walk. 2) Next we introduce the Black – Scholes option pricing model with stock price movement by using of Geometric Brownian motion. 3) Then we extent this Brownian motion approach in the stock market and 4) Finally we construct the model for the generalization based on the deformation of the standard Brownian motion and Black Scholes pricing formula. And this paper will end with conclusion.

Fluctuation and dissipation in de Sitter space

In this paper we study some thermal properties of quantum field theories in de Sitter space by means of holographic techniques. We focus on the static patch of de Sitter and assume that the quantum fields are in the standard Bunch-Davies vacuum. More specifically, we follow the stochastic motion of a massive charged particle due to its interaction with Hawking radiation. The process is described in terms of the theory of Brownian motion in inhomogeneous media and its associated Langevin dynamics. At late times, we find that the particle undergoes a regime of slow diffusion and never reaches the horizon, in stark contrast to the usual random walk behavior at finite temperature. Nevertheless, the fluctuation-dissipation theorem is found to hold at all times.

Influence of primary-particle density in the morphology of agglomerates.

Agglomeration processes occur in many different realms of science, such as colloid and aerosol formation or formation of bacterial colonies. We study the influence of primary-particle density in agglomerate structures using diffusion-controlled Monte Carlo simulations with realistic space scales through different regimes (diffusion-limited aggregation and diffusion-limited colloid aggregation). The equivalence of Monte Carlo time steps to real time scales is given by Hirsch's hydrodynamical theory of Brownian motion. Agglomerate behavior at different time stages of the simulations suggests that three indices (the fractal exponent, the coordination number, and the eccentricity index) characterize agglomerate geometry. Using these indices, we have found that the initial density of primary particles greatly influences the final structure of the agglomerate, as observed in recent experimental works.

Universal statistics of longest lasting records of random walks and Lévy flights

We study the record statistics of random walks after n steps, x0, x1, …, xn, with arbitrary symmetric and continuous distribution p(η) of the jumps ηi = xi − xi − 1. We consider the age of the records, i.e. the time up to which a record survives. Depending on how the age of the current last record is defined, we propose three distinct sequences of ages (indexed by α = I, II, III) associated to a given sequence of records. We then focus on the longest lasting record, which is the longest element among this sequence of ages. To characterize the statistics of these longest lasting records, we compute: (i) the probability that the record of the longest age is broken at step n, denoted by Qα(n), which we call the probability of record breaking and: (ii) the duration of the longest lasting record, . We show that both Qα(n) and the full statistics of are universal, i.e. independent of the jump distribution p(η). We compute exactly the large n asymptotic behaviors of Qα(n) as well as (when it exists) and show that each case gives rise to a different universal constant associated to random walks (including Lévy flights). While two of them appeared before in the excursion theory of Brownian motion, for which we provide here a simpler derivation, the third case gives rise to a non-trivial new constant CIII = 0.241 749… associated to the records of random walks. Other observables characterizing the ages of the records, exhibiting an interesting universal behavior, are also discussed.

An excursion approach to maxima of the Brownian bridge

Distributions of functionals of Brownian bridge arise as limiting distributions in non-parametric statistics. In this paper we will give a derivation of distributions of extrema of the Brownian bridge based on excursion theory for Brownian motion. The idea of rescaling and conditioning on the local time has been used widely in the literature. In this paper it is used to give a unified derivation of a number of known distributions, and a few new ones. Particular cases of calculations include the distribution of the Kolmogorov–Smirnov statistic and the Kuiper statistic.

Stochastically Curtailed Tests Under Fractional Brownian Motion

Stochastic curtailment has been considered for the interim monitoring of group sequential trials (Davis and Hardy, 1994). Statistical boundaries in Davis and Hardy (1994) were derived using theory of Brownian motion. In some clinical trials, the conditions of forming a Brownian motion may not be satisfied. In this paper, we extend the computations of Brownian motion based boundaries, expected stopping times, and type I and type II error rates to fractional Brownian motion (FBM). FBM includes Brownian motion as a special case. Designs under FBM are compared to those under Brownian motion and to those of O’Brien–Fleming type tests. One- and two-sided boundaries for efficacy and futility monitoring are also discussed. Results show that boundary values decrease and error rates deviate from design levels when the Hurst parameter increases from 0.1 to 0.9, these changes should be considered when designing a study under FBM.

How superdiffusion gets arrested: ecological encounters explain shift from Lévy to Brownian movement

Ecological theory uses Brownian motion as a default template for describing ecological movement, despite limited mechanistic underpinning. The generality of Brownian motion has recently been challenged by empirical studies that highlight alternative movement patterns of animals, especially when foraging in resource-poor environments. Yet, empirical studies reveal animals moving in a Brownian fashion when resources are abundant. We demonstrate that Einstein's original theory of collision-induced Brownian motion in physics provides a parsimonious, mechanistic explanation for these observations. Here, Brownian motion results from frequent encounters between organisms in dense environments. In density-controlled experiments, movement patterns of mussels shifted from Lévy towards Brownian motion with increasing density. When the analysis was restricted to moves not truncated by encounters, this shift did not occur. Using a theoretical argument, we explain that any movement pattern approximates Brownian motion at high-resource densities, provided that movement is interrupted upon encounters. Hence, the observed shift to Brownian motion does not indicate a density-dependent change in movement strategy but rather results from frequent collisions. Our results emphasize the need for a more mechanistic use of Brownian motion in ecology, highlighting that especially in rich environments, Brownian motion emerges from ecological interactions, rather than being a default movement pattern.

Potential theory of hyperbolic Brownian motion in tube domains

Potential theory of hyperbolic Brownian motion in tube domains

Step-Stress Accelerated Degradation Testing of NBR Sealing Ring

As an important performance parameter of nitrile-butadiene rubber（NBR）sealing ring, the compression set is a main factor influencing rubber's life. Using the compression set amount as its degradation measurement and the temperature as accelerating stress, we perform the design and actual implementation of a set of accelerated degradation test of NBR considering step-stress. Then we evaluate the results using accelerated degradation model based on the drift theory of Brownian motion. Finally we get the estimation of reliable life of sealing ring under normal temperature. The research in this paper can also offer a new method for the application reliability and life evaluation of rubber products.

Electron dynamics in semiconductors, Markovian and beyond

In earlier calculations of the dynamics of a single electron interacting with lattice vibrations an attempt was made to find a first principles' quantum mechanical description matching the original Haynes–Shockley model for an electron in a semiconductor, a model taken from Einstein's theory of Brownian motion. The equation of motion for the electron's propagator was derived from the exact double path-integral representation of a polaron by Feynman et al. (FHIP). The equation was then converted to the Wigner form of quantum mechanics, and solved in Wigner variables. The purpose of the present calculation is to remove a singularity that arose earlier from scattering by phonons with wave lengths greater than the Einstein length. We find here that the singularity can only be removed by including non-Markovian terms, terms left out in the earlier calculations. The complementary addition of the new terms enhances the plausibility of the modified solution, now no longer Markovian. In this calculation, as a first step, we omit the effect of the electron's recoil energy originating with Markovian scattering by the troublesome long phonon waves. The effect of the neglected recoil energy would be helpful but insufficient in removing the singularities. Of particular interest is the non-Markovian correction to the Einstein diffusion relation, a relationship at the heart of the Haynes–Shockley model. The present calculation, then, may be viewed as yielding an upper limit to the correction, at about 10%.

Water self-diffusion at the surface of silica glasses: effect of hydrophilic to hydrophobic transition

We study water dynamics at a silica aqueous interface. Both hydrophilic (hydroxylated) surfaces and hydrophobic surfaces (dehydroxylated upon irradiation) have been generated from atomistic simulations. A new method for the calculation of the normal self-diffusion coefficients based on the calculation of mean first passage times is proposed. It uses the Smoluchowski theory of Brownian motion and it takes proper account of the layering of the molecules. In the case of parallel self-diffusion coefficients, a decrease is found compared to the bulk values. It can be described in terms of hydrodynamic boundary conditions induced by the surface confinement. This hydrodynamic explanation is not enough to interpret the case of normal self-diffusion coefficients for which an important diminution is found. Normal self-diffusion coefficients appear to depend strongly on the hydrophilicity of the surface. They tend towards their bulk value only at long distances from the surfaces. The first layer of water molecules is found to be partially adsorbed.

Temperature Fluctuations for a System in Contact with a Heat Bath

The problem of how to define and compute temperature fluctuations for a small system in contact with a heat bath is an old one and originates from Einstein’s theory of Brownian motion. Only for a small enough system does their relative size allow a straightforward experimental verification. Here we focus on a mesoscopic system in contact with a heat bath at temperature T ∘ and provide a self-consistent argument showing as to why, and in what sense, the observable standard deviation of temperature from T ∘ equals $\sqrt{k_{\rm B}/C}T_{\circ}$ where C is the mesoscopic system’s heat capacity. Our argument is based on ergodic decomposition, a simple fact that holds for a system in thermodynamic equilibrium and away from phase-transition points. In this way we close a line of argument opened by de Haas-Lorentz a century ago.

A comparison of finite difference and finite volume methods for solving the space-fractional advection-dispersion equation with variable coefficients

Transport processes within heterogeneous media may exhibit non-classical diffusion or dispersion; that is, not adequately described by the classical theory of Brownian motion and Fick's law. We consider a space fractional advection-dispersion equation based on a fractional Fick's law. The equation involves the Riemann-Liouville fractional derivative which arises from assuming that particles may make large jumps. Finite difference methods for solving this equation have been proposed by Meerschaert and Tadjeran. In the variable coefficient case, the product rule is first applied, and then the Riemann-Liouville fractional derivatives are discretised using standard and shifted Grunwald formulas, depending on the fractional order. In this work, we consider a finite volume method that deals directly with the equation in conservative form. Fractionally-shifted Grunwald formulas are used to discretise the fractional derivatives at control volume faces. We compare the two methods for several case studies from the literature, highlighting the convenience of the finite volume approach.

Nucleation kinetics of vapor bubbles in a liquid with arbitrary viscosity

The theory of vapor bubbles nucleation in single-component liquids developed in [J. Phys. Chem. B 116, 9445 (2012)] for the case of high viscosity (the (V,/rho,T)- theory) is extended to the case of arbitrary viscosity. For this purpose, Langevin's approach in the theory of Brownian motion, or Kramers' model of chemical reactions, is employed. The obtained expression for the bubbles nucleation rate is valid for arbitrary relations between the kinetic parameters controlling the nucleation process: viscosity, inertia of a liquid, the rate of evaporation into a bubble and the rate of heat exchange between the bubble and ambient liquid. So, the presented theory together with the (V,/rho,T)- theory gives a complete description of the vapor-bubbles nucleation kinetics in one-component liquids. Limiting cases with respect to the mentioned parameters are considered, in particular, the low viscosity limit. It is shown that the low- and high-viscosity nucleation rates differ from each other qualitatively and quantitatively. The possibility of application of the theory to cavitation in superfluid helium-4 is discussed.

Irreversibility and entropy production in transport phenomena, IV: Symmetry, integrated intermediate processes and separated variational principles for multi-currents

The mechanism of entropy production in transport phenomena is discussed again by emphasizing the role of symmetry of non-equilibrium states and also by reformulating Einstein’s theory of Brownian motion to derive entropy production from it. This yields conceptual reviews of the previous papers [M. Suzuki, Physica A 390 (2011) 1904; 391 (2012) 1074; 392 (2013) 314]. Separated variational principles of steady states for multi external fields {Xi} and induced currents {Ji} are proposed by extending the principle of minimum integrated entropy production found by the present author for a single external field. The basic strategy of our theory on steady states is to take in all the intermediate processes from the equilibrium state to the final possible steady states in order to study the irreversible physics even in the steady states. As an application of this principle, Gransdorff–Prigogine’s evolution criterion inequality (or stability condition) dXP≡∫dr∑iJidXi≤0 is derived in the stronger form dQi≡∫drJidXi≤0 for individual force Xi and current Ji even in nonlinear responses which depend on all the external forces {Xk} nonlinearly. This is called “separated evolution criterion”.Some explicit demonstrations of the present general theory to simple electric circuits with multi external fields are given in order to clarify the physical essence of our new theory and to realize the condition of its validity concerning the existence of the solutions of the simultaneous equations obtained by the separated variational principles. It is also instructive to compare the two results obtained by the new variational theory and by the old scheme based on the instantaneous entropy production. This seems to be suggestive even to the energy problem in the world.

Characterization of anomalous diffusion in porous biological tissues using fractional order derivatives and entropy

In this high-resolution magnetic resonance imaging (MRI) study at 17.6 Tesla of a fixed rat brain, we used the continuous time random walk theory (CTRW) for Brownian motion to characterize anomalous diffusion. The complex mesoporus structure of biological tissues (membranes, organelles, and cells) perturbs the motion of the random walker (water molecules in proton MRI) introducing halts between steps (waiting times) and restrictions on step sizes (jump lengths). When such waiting times and jump lengths are scaled with probability distributions that follow simple inverse power laws (t-(1+α),|x|-(1+β)) non-Gaussian motion gives rise to sub- and super- diffusion. In the CTRW approach, the Fourier transform yields a solution to the generalized diffusion equation that can be expressed by the Mittag–Leffler function (MLF), Eα(-Dα,β|q|βΔ¯α). We interrogated both white and gray matter regions in a 1mm slice of a fixed rat brain (190μm in plane resolution) with diffusion weighted MRI experiments using b-values up to 25,000s/mm2, by independently varying q and Δ. When fitting these data to our model, the fractional order parameters, α and β, and the entropy measure, H(q,Δ¯), were found to provide excellent contrast between white and gray matter and to give results that were sensitive to the type of diffusion experiment performed.