Brownian semistationary processes and volatility/intermittency
DEFF Research Database (Denmark)
Barndorff-Nielsen, Ole Eiler; Schmiegel, Jürgen
A new class of stochastic processes, termed Brownian semistationary processes (BSS), is introduced and discussed. This class has similarities to that of Brownian semimartingales (BSM), but is mainly directed towards the study of stationary processes, and BSS processes are not in general of the...... turbulent velocity fields and is the purely temporal version of the general tempo-spatial framework of ambit processes. The latter, which may have applications also to the finance of energy markets, is briefly considered at the end of the paper, again with reference to the question of inference on the...
Hybrid scheme for Brownian semistationary processes
DEFF Research Database (Denmark)
Bennedsen, Mikkel; Lunde, Asger; Pakkanen, Mikko S.
We introduce a simulation scheme for Brownian semistationary processes, which is based on discretizing the stochastic integral representation of the process in the time domain. We assume that the kernel function of the process is regularly varying at zero. The novel feature of the scheme is to...... approximate the kernel function by a power function near zero and by a step function elsewhere. The resulting approximation of the process is a combination of Wiener integrals of the power function and a Riemann sum, which is why we call this method a hybrid scheme. Our main theoretical result describes the...... asymptotics of the mean square error of the hybrid scheme and we observe that the scheme leads to a substantial improvement of accuracy compared to the ordinary forward Riemann-sum scheme, while having the same computational complexity. We exemplify the use of the hybrid scheme by two numerical experiments...
Brownian semi-stationary processes, turbulence and smooth processes
DEFF Research Database (Denmark)
Urbina, José Ulises Márquez
This thesis analysis the use of Brownian semi-stationary (BSS) processes to model the main statistical features present in turbulent time series, and some asymptotic properties of certain classes of smooth processes. Turbulence is a complex phenomena governed by the Navier-Stokes equations. These...... equations do not represent a fully functional model and, consequently, it has been necessary to develop phenomenological models capturing main aspects of turbulent dynamics. The BSS processes were proposed as an option to model turbulent time series. In this thesis we proved, through a simulation....... We also studied the distributional properties of the increments of BSS processes with the intent to better understand why the BSS processes seem to accurately reproduce the temporal turbulent dynamics. BSS processes in general are not semimartingales. However, there are conditions which make a BSS...
Asymptotic theory for Brownian semi-stationary processes with application to turbulence
DEFF Research Database (Denmark)
Corcuera, José Manuel; Hedevang, Emil; Pakkanen, Mikko S.;
2013-01-01
This paper presents some asymptotic results for statistics of Brownian semi-stationary (BSS) processes. More precisely, we consider power variations of BSS processes, which are based on high frequency (possibly higher order) differences of the BSS model. We review the limit theory discussed in...... [Barndorff-Nielsen, O.E., J.M. Corcuera and M. Podolskij (2011): Multipower variation for Brownian semistationary processes. Bernoulli 17(4), 1159-1194; Barndorff-Nielsen, O.E., J.M. Corcuera and M. Podolskij (2012): Limit theorems for functionals of higher order differences of Brownian semi......-stationary processes. In "Prokhorov and Contemporary Probability Theory", Springer.] and present some new connections to fractional diffusion models. We apply our probabilistic results to construct a family of estimators for the smoothness parameter of the BSS process. In this context we develop estimates with gaps...
A weak limit theorem for numerical approximation of Brownian semi-stationary processes
DEFF Research Database (Denmark)
Podolskij, Mark; Thamrongrat, Nopporn
In this paper we present a weak limit theorem for a numerical approximation of Brownian semi-stationary processes studied in [14]. In the original work of [14] the authors propose to use Fourier transformation to embed a given one dimensional (Levy) Brownian semi-stationary process into a two-par....../drift process needs to be numerically simulated. In particular, weak approximation errors for smooth test functions can be obtained from our asymptotic theory.......In this paper we present a weak limit theorem for a numerical approximation of Brownian semi-stationary processes studied in [14]. In the original work of [14] the authors propose to use Fourier transformation to embed a given one dimensional (Levy) Brownian semi-stationary process into a two......-parameter stochastic field. For the latter they use a simple iteration procedure and study the strong approximation error of the resulting numerical scheme given that the volatility process is fully observed. In this work we present the corresponding weak limit theorem for the setting, where the volatility...
Discretization of Lévy semistationary processes with application to estimation
DEFF Research Database (Denmark)
Bennedsen, Mikkel; Lunde, Asger; Pakkanen, Mikko
Motivated by the construction of the Ito stochastic integral, we consider a step function method to discretize and simulate volatility modulated Lévy semistationary processes. Moreover, we assess the accuracy of the method with a particular focus on integrating kernels with a singularity at the...... origin. Using the simulation method, we study the finite sample properties of some recently developed estimators of realized volatility and associated parametric estimators for Brownian semistationary processes. Although the theoretical properties of these estimators have been established under high...
Assessing Gamma kernels and BSS/LSS processes
DEFF Research Database (Denmark)
Barndorff-Nielsen, Ole E.
This paper reviews the roles of gamma type kernels in the theory and modelling for Brownian and Lévy semistationary processes. Applications to financial econometrics and the physics of turbulence are pointed out.......This paper reviews the roles of gamma type kernels in the theory and modelling for Brownian and Lévy semistationary processes. Applications to financial econometrics and the physics of turbulence are pointed out....
Random Brownian scaling identities and splicing of Bessel processes
Pitman, Jim; Yor, Marc
1998-01-01
An identity in distribution due to Knight for Brownian motion is extended in two different ways: first by replacing the supremum of a reflecting Brownian motion by the range of an unreflected Brownian motion and second by replacing the reflecting Brownian motion by a recurrent Bessel process. Both extensions are explained in terms of random Brownian scaling transformations and Brownian excursions. The first extension is related to two different constructions of Itô’s law of ...
Brownian motion an introduction to stochastic processes
Schilling, René L; Böttcher, Björn
2014-01-01
Stochastic processes occur everywhere in sciences and engineering, and need to be understood by applied mathematicians, engineers and scientists alike. This is a first course introducing the reader gently to the subject. Brownian motions are a stochastic process, central to many applications and easy to treat.
Noncolliding Brownian Motion and Determinantal Processes
Katori, Makoto; Tanemura, Hideki
2007-12-01
A system of one-dimensional Brownian motions (BMs) conditioned never to collide with each other is realized as (i) Dyson's BM model, which is a process of eigenvalues of hermitian matrix-valued diffusion process in the Gaussian unitary ensemble (GUE), and as (ii) the h-transform of absorbing BM in a Weyl chamber, where the harmonic function h is the product of differences of variables (the Vandermonde determinant). The Karlin-McGregor formula gives determinantal expression to the transition probability density of absorbing BM. We show from the Karlin-McGregor formula, if the initial state is in the eigenvalue distribution of GUE, the noncolliding BM is a determinantal process, in the sense that any multitime correlation function is given by a determinant specified by a matrix-kernel. By taking appropriate scaling limits, spatially homogeneous and inhomogeneous infinite determinantal processes are derived. We note that the determinantal processes related with noncolliding particle systems have a feature in common such that the matrix-kernels are expressed using spectral projections of appropriate effective Hamiltonians. On the common structure of matrix-kernels, continuity of processes in time is proved and general property of the determinantal processes is discussed.
Monitoring autocorrelated process: A geometric Brownian motion process approach
Li, Lee Siaw; Djauhari, Maman A.
2013-09-01
Autocorrelated process control is common in today's modern industrial process control practice. The current practice of autocorrelated process control is to eliminate the autocorrelation by using an appropriate model such as Box-Jenkins models or other models and then to conduct process control operation based on the residuals. In this paper we show that many time series are governed by a geometric Brownian motion (GBM) process. Therefore, in this case, by using the properties of a GBM process, we only need an appropriate transformation and model the transformed data to come up with the condition needs in traditional process control. An industrial example of cocoa powder production process in a Malaysian company will be presented and discussed to illustrate the advantages of the GBM approach.
Stochastic calculus for fractional Brownian motion and related processes
Mishura, Yuliya S
2008-01-01
The theory of fractional Brownian motion and other long-memory processes are addressed in this volume. Interesting topics for PhD students and specialists in probability theory, stochastic analysis and financial mathematics demonstrate the modern level of this field. Among these are results about Levy characterization of fractional Brownian motion, maximal moment inequalities for Wiener integrals including the values 0
Brownian motion vs. pure-jump processes for individual stocks
Benoît Sévi; César Baena
2011-01-01
Using recent activity signature function methodology developed in Todorov and Tauchen (2010), we provide empirical evidence that individual stocks from the New York Stock Exchange are adequately represented by a Brownian motion plus medium to large (rare) jumps thus invalidating the pure-jump process hypothesis proposed in numerous contributions. This result improves our understanding of the fine structure of asset prices and has implications for derivatives pricing.
Feller Processes: The Next Generation in Modeling. Brownian Motion, L\\'evy Processes and Beyond
Böttcher, Björn
2010-01-01
We present a simple construction method for Feller processes and a framework for the generation of sample paths of Feller processes. The construction is based on state space dependent mixing of L\\'evy processes. Brownian Motion is one of the most frequently used continuous time Markov processes in applications. In recent years also L\\'evy processes, of which Brownian Motion is a special case, have become increasingly popular. L\\'evy processes are spatially homogeneous, but empirical data ofte...
Brownian Motion as a Limit to Physical Measuring Processes
DEFF Research Database (Denmark)
Niss, Martin
2016-01-01
In this paper, we examine the history of the idea that noise presents a fundamental limit to physical measuring processes. This idea had its origins in research aimed at improving the accuracy of instruments for electrical measurements. Out of these endeavors, the Swedish physicist Gustaf A. Ising...... formulated a general conclusion concerning the nature of physical measurements, namely that there is a definite limit to the ultimate sensitivity of measuring instruments beyond which we cannot advance, and that this limit is determined by Brownian motion. Ising’s conclusion agreed with experiments and...... received widespread recognition, but his way of modeling the system was contested by his contemporaries. With the more embracing notion of noise that developed during and after World War II, Ising’s conclusion was reinterpreted as showing that noise puts a limit on physical measurement processes. Hence...
International Nuclear Information System (INIS)
The purpose of this work was to construct a Brownian motion with values in simplicial complexes with piecewise differential structure. After a martingale theory attempt, we constructed a family of continuous Markov processes with values in an admissible complex; we named every process of this family, isotropic transport process. We showed that the family of the isotropic processes contains a subsequence, which converged weakly to a measure; we named it the Wiener measure. Then, we constructed, thanks to the finite dimensional distributions of the Wiener measure a new continuous Markov process with values in an admissible complex: the Brownian motion. We finished with a geometric analysis of this Brownian motion, to determinate, under hypothesis on the complex, the recurrent or transient behavior of such process. (author)
QUANTUM STOCHASTIC PROCESSES: BOSON AND FERMION BROWNIAN MOTION
Directory of Open Access Journals (Sweden)
A.E.Kobryn
2003-01-01
Full Text Available Dynamics of quantum systems which are stochastically perturbed by linear coupling to the reservoir can be studied in terms of quantum stochastic differential equations (for example, quantum stochastic Liouville equation and quantum Langevin equation. In order to work it out one needs to define the quantum Brownian motion. As far as only its boson version has been known until recently, in the present paper we present the definition which makes it possible to consider the fermion Brownian motion as well.
Arithmetic Brownian motion subordinated by tempered stable and inverse tempered stable processes
Wyłomańska, Agnieszka
2012-01-01
In the last decade the subordinated processes have become popular and found many practical applications. Therefore in this paper we examine two processes related to time-changed (subordinated) classical Brownian motion with drift (called arithmetic Brownian motion). The first one, so called normal tempered stable, is related to the tempered stable subordinator, while the second one - to the inverse tempered stable process. We compare the main properties (such as probability density functions, Laplace transforms, ensemble averaged mean squared displacements) of such two subordinated processes and propose the parameters' estimation procedures. Moreover we calibrate the analyzed systems to real data related to indoor air quality.
(Quantum) Fractional Brownian Motion and Multifractal Processes under the Loop of a Tensor Networks
Descamps, Benoît
2016-01-01
We derive fractional Brownian motion and stochastic processes with multifractal properties using a framework of network of Gaussian conditional probabilities. This leads to the derivation of new representations of fractional Brownian motion. These constructions are inspired from renormalization. The main result of this paper consists of constructing each increment of the process from two-dimensional gaussian noise inside the light-cone of each seperate increment. Not only does this allows us to derive fractional Brownian motion, we can introduce extensions with multifractal flavour. In another part of this paper, we discuss the use of the multi-scale entanglement renormalization ansatz (MERA), introduced in the study critical systems in quantum spin lattices, as a method for sampling integrals with respect to such multifractal processes. After proper calibration, a MERA promises the generation of a sample of size $N$ of a multifractal process in the order of $O(N\\log(N))$, an improvement over the known method...
DEFF Research Database (Denmark)
E. Barndorff-Nielsen, Ole; Benth, Fred Espen; Szozda, Benedykt
This paper generalizes the integration theory for volatility modulated Brownian-driven Volterra processes onto the space G* of Potthoff-Timpel distributions. Sufficient conditions for integrability of generalized processes are given, regularity results and properties of the integral are discussed...
The oscillation of the occupation time process of super- Brownian motion on Sierpinski gasket
Institute of Scientific and Technical Information of China (English)
郭军义
2000-01-01
The occupation time process of super-Brownian motion on the Sierpinski gasket is studied. It is shown that this process does not possess stable property in the long run, but oscillates periodically in some sense. Other convergence properties are also studied.
The oscillation of the occupation time process of super-Brownian motion on Sierpinski gasket
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
The occupation time process of super-Brownian motion on the Sierpinski gasket is studied. It is shown that this process does not possess stable property in the long run, but oscillates periodically in some sense. Other convergence properties are also studied.
Fractional Brownian motion, the Matern process, and stochastic modeling of turbulent dispersion
Lilly, J M; Early, J J; Olhede, S C
2016-01-01
Stochastic process exhibiting power-law slopes in the frequency domain are frequently well modeled by fractional Brownian motion (fBm). In particular, the spectral slope at high frequencies is associated with the degree of small-scale roughness or fractal dimension. However, a broad class of real-world signals have a high-frequency slope, like fBm, but a plateau in the vicinity of zero frequency. This low-frequency plateau, it is shown, implies that the temporal integral of the process exhibits diffusive behavior, dispersing from its initial location at a constant rate. Such processes are not well modeled by fBm, which has a singularity at zero frequency corresponding to an unbounded rate of dispersion. A more appropriate stochastic model is a much lesser-known random process called the Matern process, which is shown herein to be a damped version of fractional Brownian motion. This article first provides a thorough introduction to fractional Brownian motion, then examines the details of the Matern process and...
Optimal control of a stochastic processing system driven by a fractional Brownian motion input
Ghosh, Arka P.; Roitershtein, Alexander; Weerasinghe, Ananda
2010-01-01
We consider a stochastic control model driven by a fractional Brownian motion. This model is a formal approximation to a queueing network with an ON-OFF input process. We study stochastic control problems associated with the long-run average cost, the infinite-horizon discounted cost, and the finite-horizon cost. In addition, we find a solution to a constrained minimization problem as an application of our solution to the long-run average cost problem. We also establish Abel...
Harnack Inequality and Regularity for a Product of Symmetric Stable Process and Brownian Motion
Karli, Deniz
2010-01-01
In this paper, we consider a product of a symmetric stable process in $\\mathbb{R}^d$ and a one-dimensional Brownian motion in $\\mathbb{R}^+$. Then we define a class of harmonic functions with respect to this product process. We show that bounded non-negative harmonic functions in the upper-half space satisfy Harnack inequality and prove that they are locally H\\"older continuous. We also argue a result on Littlewood-Paley functions which are obtained by the $\\alpha$-harmonic extension of an $L...
Chernov, N.; Dolgopyat, D.
2008-01-01
A classical model of Brownian motion consists of a heavy molecule submerged into a gas of light atoms in a closed container. In this work we study a 2D version of this model, where the molecule is a heavy disk of mass M and the gas is represented by just one point particle of mass m = 1, which interacts with the disk and the walls of the container via elastic collisions. Chaotic behavior of the particles is ensured by convex (scattering) walls of the container. We prove that the position and ...
Fractional Brownian motion time-changed by gamma and inverse gamma process
Kumar, A; Połoczański, R; Sundar, S
2016-01-01
Many real time-series exhibit behavior adequate to long range dependent data. Additionally very often these time-series have constant time periods and also have characteristics similar to Gaussian processes although they are not Gaussian. Therefore there is need to consider new classes of systems to model these kind of empirical behavior. Motivated by this fact in this paper we analyze two processes which exhibit long range dependence property and have additional interesting characteristics which may be observed in real phenomena. Both of them are constructed as the superposition of fractional Brownian motion (FBM) and other process. In the first case the internal process, which plays role of the time, is the gamma process while in the second case the internal process is its inverse. We present in detail their main properties paying main attention to the long range dependence property. Moreover, we show how to simulate these processes and estimate their parameters. We propose to use a novel method based on re...
Generalization of Brownian Motion with Autoregressive Increments
Fendick, Kerry
2011-01-01
This paper introduces a generalization of Brownian motion with continuous sample paths and stationary, autoregressive increments. This process, which we call a Brownian ray with drift, is characterized by three parameters quantifying distinct effects of drift, volatility, and autoregressiveness. A Brownian ray with drift, conditioned on its state at the beginning of an interval, is another Brownian ray with drift over the interval, and its expected path over the interval is a ray with a slope that depends on the conditioned state. This paper shows how Brownian rays can be applied in finance for the analysis of queues or inventories and the valuation of options. We model a queue's net input process as a superposition of Brownian rays with drift and derive the transient distribution of the queue length conditional on past queue lengths and on past states of the individual Brownian rays comprising the superposition. The transient distributions of Regulated Brownian Motion and of the Regulated Brownian Bridge are...
Harmonic functions on Walsh's Brownian motion
Jehring, Kristin Elizabeth
2009-01-01
In this dissertation we examine a variation of two- dimensional Brownian motion introduced in 1978 by Walsh. Walsh's Brownian motion can be described as a Brownian motion on the spokes of a (rimless) bicycle wheel. We will construct such a process by randomly assigning an angle to the excursions of a reflecting Brownian motion from 0. With this construction we see that Walsh's Brownian motion in R² behaves like one-dimensional Brownian motion away from the origin, but at the origin behaves di...
Van den Broeck, C; Kawai, R
2006-06-01
Onsager symmetry implies that a Brownian motor, driven by a temperature gradient, will also perform a refrigerator function upon loading. We analytically calculate the corresponding heat flow for an exactly solvable microscopic model and compare it with molecular dynamics simulations. PMID:16803223
Archimedes’ principle for Brownian liquid
Burdzy, Krzysztof; Chen, Zhen-Qing; Pal, Soumik
2011-01-01
We consider a family of hard core objects moving as independent Brownian motions confined to a vessel by reflection. These are subject to gravitational forces modeled by drifts. The stationary distribution for the process has many interesting implications, including an illustration of the Archimedes' principle. The analysis rests on constructing reflecting Brownian motion with drift in a general open connected domain and studying its stationary distribution. In dimension two we utilize known ...
Archimedes' principle for Brownian liquid
Burdzy, Krzysztof; Pal, Soumik
2009-01-01
We consider a family of hard core objects moving as independent Brownian motions confined to a vessel by reflection. These are subject to gravitational forces modeled by drifts. The stationary distribution for the process has many interesting implications, including an illustration of the Archimedes' principle. The analysis rests on constructing reflecting Brownian motion with drift in a general open connected domain and studying its stationary distribution. In dimension two we utilize known results about sphere packing.
Trefan, Gyorgy
1993-01-01
The goal of this thesis is to contribute to the ambitious program of the foundation of developing statistical physics using chaos. We build a deterministic model of Brownian motion and provide a microscopic derivation of the Fokker-Planck equation. Since the Brownian motion of a particle is the result of the competing processes of diffusion and dissipation, we create a model where both diffusion and dissipation originate from the same deterministic mechanism--the deterministic interaction of that particle with its environment. We show that standard diffusion which is the basis of the Fokker-Planck equation rests on the Central Limit Theorem, and, consequently, on the possibility of deriving it from a deterministic process with a quickly decaying correlation function. The sensitive dependence on initial conditions, one of the defining properties of chaos insures this rapid decay. We carefully address the problem of deriving dissipation from the interaction of a particle with a fully deterministic nonlinear bath, that we term the booster. We show that the solution of this problem essentially rests on the linear response of a booster to an external perturbation. This raises a long-standing problem concerned with Kubo's Linear Response Theory and the strong criticism against it by van Kampen. Kubo's theory is based on a perturbation treatment of the Liouville equation, which, in turn, is expected to be totally equivalent to a first-order perturbation treatment of single trajectories. Since the boosters are chaotic, and chaos is essential to generate diffusion, the single trajectories are highly unstable and do not respond linearly to weak external perturbation. We adopt chaotic maps as boosters of a Brownian particle, and therefore address the problem of the response of a chaotic booster to an external perturbation. We notice that a fully chaotic map is characterized by an invariant measure which is a continuous function of the control parameters of the map
Directory of Open Access Journals (Sweden)
Davide Mercadante
Full Text Available Pectin methylesterases (PMEs hydrolyze the methylester groups that are found on the homogalacturonan (HG chains of pectic polysaccharides in the plant cell wall. Plant and bacterial PMEs are especially interesting as the resulting de-methylesterified (carboxylated sugar residues are found to be arranged contiguously, indicating a so-called processive nature of these enzymes. Here we report the results of continuum electrostatics calculations performed along the molecular dynamics trajectory of a PME-HG-decasaccharide complex. In particular it was observed that, when the methylester groups of the decasaccharide were arranged in order to mimic the just-formed carboxylate product of de-methylesterification, a net unidirectional sliding of the model decasaccharide was subsequently observed along the enzyme's binding groove. The changes that occurred in the electrostatic binding energy and protein dynamics during this translocation provide insights into the mechanism by which the enzyme rectifies Brownian motions to achieve processivity. The free energy that drives these molecular motors is thus demonstrated to be incorporated endogenously in the methylesterified groups of the HG chains and is not supplied exogenously.
Directory of Open Access Journals (Sweden)
De-Lei Sheng
2014-01-01
Full Text Available This paper investigates the excess-of-loss reinsurance and investment problem for a compound Poisson jump-diffusion risk process, with the risk asset price modeled by a constant elasticity of variance (CEV model. It aims at obtaining the explicit optimal control strategy and the optimal value function. Applying stochastic control technique of jump diffusion, a Hamilton-Jacobi-Bellman (HJB equation is established. Moreover, we show that a closed-form solution for the HJB equation can be found by maximizing the insurer’s exponential utility of terminal wealth with the independence of two Brownian motions W(t and W1(t. A verification theorem is also proved to verify that the solution of HJB equation is indeed a solution of this optimal control problem. Then, we quantitatively analyze the effect of different parameter impacts on optimal control strategy and the optimal value function, which show that optimal control strategy is decreasing with the initial wealth x and decreasing with the volatility rate of risk asset price. However, the optimal value function V(t;x;s is increasing with the appreciation rate μ of risk asset.
Entropic forces in Brownian motion
Roos, Nico
2013-01-01
The interest in the concept of entropic forces has risen considerably since E. Verlinde proposed 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 note it is shown that at least two types of entropic motion can be identified in the case of 3D Brownian motion (or random walk). This yields simple derivations of known results of Brownian motion, Hook s law and, applying an external (nonradial) force, Curie s law and the Langevin-Debye equation.
Entropic forces in Brownian motion
Roos, Nico
2014-12-01
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.
Feynman Rules For Brownian Motion
Hatamian, S T
2003-01-01
We present a perturbation theory extending a prescription due to Feynman for computing the probability density function in Brownian-motion. The method used, can be applied to a wide variety of otherwise difficult circumstances in Brownian-motion. The exact moments and kurtosis, if not the distribution itself, for many important cases can be summed for arbitrary times. As expected, the behavior at early time regime, for the sample processes considered, deviate significantly from the usual diffusion theory; a fact with important consequences in various applications such as financial physics. A new class of functions dubbed "Damped-exponential-integrals" are also identified.
Kaj, Ingemar; Taqqu, M. S.
2008-01-01
It has become common practice to use heavy-tailed distributions in order to describe the variations in time and space of network traffic workloads. The asymptotic behavior of these workloads is complex; different limit processes emerge depending on the specifies of the work arrival structure and the nature of the asymptotic scaling. We focus on two variants of the infinite source Poisson model and provide a coherent and unified presentation of the scaling theory by using integral representati...
A multiscale guide to Brownian motion
Grebenkov, Denis S.; Belyaev, Dmitry; Jones, Peter W.
2016-01-01
We revise the Lévy construction of Brownian motion as a simple though rigorous approach to operate with various Gaussian processes. A Brownian path is explicitly constructed as a linear combination of wavelet-based ‘geometrical features’ at multiple length scales with random weights. Such a wavelet representation gives a closed formula mapping of the unit interval onto the functional space of Brownian paths. This formula elucidates many classical results about Brownian motion (e.g., non-differentiability of its path), providing an intuitive feeling for non-mathematicians. The illustrative character of the wavelet representation, along with the simple structure of the underlying probability space, is different from the usual presentation of most classical textbooks. Similar concepts are discussed for the Brownian bridge, fractional Brownian motion, the Ornstein-Uhlenbeck process, Gaussian free fields, and fractional Gaussian fields. Wavelet representations and dyadic decompositions form the basis of many highly efficient numerical methods to simulate Gaussian processes and fields, including Brownian motion and other diffusive processes in confining domains.
Canonical active Brownian motion
Gluck, Alexander; Huffel, Helmuth; Ilijic, Sasa
2008-01-01
Active Brownian motion is the complex motion of active Brownian particles. They are active in the sense that they can transform their internal energy into energy of motion and thus create complex motion patterns. Theories of active Brownian motion so far imposed couplings between the internal energy and the kinetic energy of the system. We investigate how this idea can be naturally taken further to include also couplings to the potential energy, which finally leads to a general theory of cano...
Cubero, David; Renzoni, Ferruccio
2016-01-01
Part I. Historical Overview and Early Developments: 1. Limitations imposed by the second law of thermodynamics; 2. Fundamental models of ratchet devices; 3. General relevance of the concept of ratchet; Part II. Theoretical Foundations: 4. Classical ratchets; 5. Quantum ratchets; 6. Energetics and characterization; Part III. Experimental Realizations of Ratchet Devices: 7. Ratchets for colloidal particles; 8. Cold atom ratchets; 9. Solid state ratchets; 10. Bio-inspired molecular motors; Appendix A. Stochastic processes techniques; Appendix B. Symmetries in a 1D overdamped system; Appendix C. Floquet theory; References; Index.
Lawler, Gregory F.; Werner, Wendelin
2003-01-01
We define a natural conformally invariant measure on unrooted Brownian loops in the plane and study some of its properties. We relate this measure to a measure on loops rooted at a boundary point of a domain and show how this relation gives a way to ``chronologically add Brownian loops'' to simple curves in the plane.
Brownian motion and stochastic calculus
Karatzas, Ioannis
1998-01-01
This book is designed as a text for graduate courses in stochastic processes. It is written for readers familiar with measure-theoretic probability and discrete-time processes who wish to explore stochastic processes in continuous time. The vehicle chosen for this exposition is Brownian motion, which is presented as the canonical example of both a martingale and a Markov process with continuous paths. In this context, the theory of stochastic integration and stochastic calculus is developed. The power of this calculus is illustrated by results concerning representations of martingales and change of measure on Wiener space, and these in turn permit a presentation of recent advances in financial economics (option pricing and consumption/investment optimization). This book contains a detailed discussion of weak and strong solutions of stochastic differential equations and a study of local time for semimartingales, with special emphasis on the theory of Brownian local time. The text is complemented by a large num...
Continuum limits of random matrices and the Brownian carousel
Valko, Benedek; Virag, Balint
2007-01-01
We show that at any location away from the spectral edge, the eigenvalues of the Gaussian unitary ensemble and its general beta siblings converge to Sine_beta, a translation invariant point process. This process has a geometric description in term of the Brownian carousel, a deterministic function of Brownian motion in the hyperbolic plane. The Brownian carousel, a description of the a continuum limit of random matrices, provides a convenient way to analyze the limiting point processes. We sh...
Dynamical 3-Space: Anisotropic Brownian Motion Experiment
Cahill R. T.
2015-01-01
In 2014 Jiapei Dai reported evidence of anisotropic Brownian motion of a toluidine blue colloid solution in water. In 2015 Felix Scholkmann analysed the Dai data and detected a sidereal time dependence, indicative of a process driving the preferred Brownian mo- tion diffusion direction to a star-based preferred direction. Here we further analyse the Dai data and extract the RA and Dec of that preferred direction, and relate the data to previous determinations from NASA Spacecr...
Brownian Motion in Minkowski Space
Directory of Open Access Journals (Sweden)
Paul O'Hara
2015-06-01
Full Text Available We construct a model of Brownian motion in Minkowski space. There are two aspects of the problem. The first is to define a sequence of stopping times associated with the Brownian “kicks” or impulses. The second is to define the dynamics of the particle along geodesics in between the Brownian kicks. When these two aspects are taken together, the Central Limit Theorem (CLT leads to temperature dependent four dimensional distributions defined on Minkowski space, for distances and 4-velocities. In particular, our processes are characterized by two independent time variables defined with respect to the laboratory frame: a discrete one corresponding to the stopping times when the impulses take place and a continuous one corresponding to the geodesic motion in-between impulses. The subsequent distributions are solutions of a (covariant pseudo-diffusion equation which involves derivatives with respect to both time variables, rather than solutions of the telegraph equation which has a single time variable. This approach simplifies some of the known problems in this context.
Brownian ratchets and Parrondo's games
Harmer, Gregory P.; Abbott, Derek; Taylor, Peter G.; Parrondo, Juan M. R.
2001-09-01
Parrondo's games present an apparently paradoxical situation where individually losing games can be combined to win. In this article we analyze the case of two coin tossing games. Game B is played with two biased coins and has state-dependent rules based on the player's current capital. Game B can exhibit detailed balance or even negative drift (i.e., loss), depending on the chosen parameters. Game A is played with a single biased coin that produces a loss or negative drift in capital. However, a winning expectation is achieved by randomly mixing A and B. One possible interpretation pictures game A as a source of "noise" that is rectified by game B to produce overall positive drift—as in a Brownian ratchet. Game B has a state-dependent rule that favors a losing coin, but when this state dependence is broken up by the noise introduced by game A, a winning coin is favored. In this article we find the parameter space in which the paradoxical effect occurs and carry out a winning rate analysis. The significance of Parrondo's games is that they are physically motivated and were originally derived by considering a Brownian ratchet—the combination of the games can be therefore considered as a discrete-time Brownian ratchet. We postulate the use of games of this type as a toy model for a number of physical and biological processes and raise a number of open questions for future research.
Noncommutative Brownian motion
Santos, Willien O; Souza, Andre M C
2016-01-01
We investigate the Brownian motion of a particle in a two-dimensional noncommutative (NC) space. Using the standard NC algebra embodied by the sympletic Weyl-Moyal formalism we find that noncommutativity induces a non-vanishing correlation between both coordinates at different times. The effect itself stands as a signature of spatial noncommutativity and offers further alternatives to experimentally detect the phenomena.
Meurs, P.; Broeck, C. Van Den
2005-01-01
Recently, a thermal Brownian motor was introduced [Van den Broeck, Kawai and Meurs, Phys. Rev. Lett. (2004)], for which an exact microscopic analysis is possible. The purpose of this paper is to review some further properties of this construction, and to discuss in particular specific issues including the relation with macroscopic response and the efficiency at maximum power.
Brownian coagulation at high particle concentrations
Trzeciak, T. M.
2012-01-01
The process of Brownian coagulation, whereby particles are brought together by thermal motion and grow by collisions, is one of the most fundamental processes influencing the final properties of particulate matter in a variety of technically important systems. It is of importance in colloids, emulsi
Gomez-Marin, A.; Sancho, J. M.
2004-01-01
In this paper we present a model of a symmetric Brownian motor (SBM) which changes the sign of its velocity when the temperature gradient is inverted. The velocity, external work and efficiency are studied as a function of the temperatures of the baths and other relevant parameters. The motor shows a current reversal when another parameter (a phase shift) is varied. Analytical predictions and results from numerical simulations are performed and agree very well. Generic properties of this type...
RESEARCH NOTES On the support of super-Brownian motion with super-Brownian immigration
Institute of Scientific and Technical Information of China (English)
洪文明; 钟惠芳
2001-01-01
The support properties of the super Brownian motion with random immigration Xρ1 are considered,where the immigration rate is governed by the trajectory of another super-Brownian motion ρ. When both the initial state Xρo of the process and the immigration rate process ρo are of finite measure and with compact supports, the probability of the support of the process Xρi dominated by a ball is given by the solutions of a singular elliptic boundary value problem.
Brownian Motion, "Diverse and Undulating"
Duplantier, Bertrand
2016-01-01
We describe in detail the history of Brownian motion, as well as the contributions of Einstein, Sutherland, Smoluchowski, Bachelier, Perrin and Langevin to its theory. The always topical importance in physics of the theory of Brownian motion is illustrated by recent biophysical experiments, where it serves, for instance, for the measurement of the pulling force on a single DNA molecule. In a second part, we stress the mathematical importance of the theory of Brownian motion, illustrated by two chosen examples. The by-now classic representation of the Newtonian potential by Brownian motion is explained in an elementary way. We conclude with the description of recent progress seen in the geometry of the planar Brownian curve. At its heart lie the concepts of conformal invariance and multifractality, associated with the potential theory of the Brownian curve itself.
The Local Fractional Bootstrap
DEFF Research Database (Denmark)
Bennedsen, Mikkel; Hounyo, Ulrich; Lunde, Asger;
new resampling method, the local fractional bootstrap, relies on simulating an auxiliary fractional Brownian motion that mimics the fine properties of high frequency differences of the Brownian semistationary process under the null hypothesis. We prove the first order validity of the bootstrap method...
Combinatorial fractal Brownian motion model
Institute of Scientific and Technical Information of China (English)
朱炬波; 梁甸农
2000-01-01
To solve the problem of how to determine the non-scaled interval when processing radar clutter using fractal Brownian motion (FBM) model, a concept of combinatorial FBM model is presented. Since the earth (or sea) surface varies diversely with space, a radar clutter contains several fractal structures, which coexist on all scales. Taking the combination of two FBMs into account, via theoretical derivation we establish a combinatorial FBM model and present a method to estimate its fractal parameters. The correctness of the model and the method is proved by simulation experiments and computation of practial data. Furthermore, we obtain the relationship between fractal parameters when processing combinatorial model with a single FBM model. Meanwhile, by theoretical analysis it is concluded that when combinatorial model is observed on different scales, one of the fractal structures is more obvious.
Brownian coagulation at high particle concentrations
Trzeciak, T. M.
2012-01-01
The process of Brownian coagulation, whereby particles are brought together by thermal motion and grow by collisions, is one of the most fundamental processes influencing the final properties of particulate matter in a variety of technically important systems. It is of importance in colloids, emulsions, flocculation, air pollution, soot formation, materials manufacture and growth of interstellar dust, to name a few of its applications. With continuous progress in particulate matter processing...
Convergence rates of posterior distributions for Brownian semimartingale models
F.H. van der Meulen; A.W. van der Vaart; J.H. van Zanten
2006-01-01
Key words and Phrases: Bayesian estimation, Continuous semimartingale, Dirichlet process, Hellinger distance, Infinite dimensional model, Rate of convergence, Wavelets. We consider the asymptotic behavior of posterior distributions based on continuous observations from a Brownian semimartingale mode
The open quantum Brownian motions
International Nuclear Information System (INIS)
Using quantum parallelism on random walks as the original seed, we introduce new quantum stochastic processes, the open quantum Brownian motions. They describe the behaviors of quantum walkers—with internal degrees of freedom which serve as random gyroscopes—interacting with a series of probes which serve as quantum coins. These processes may also be viewed as the scaling limit of open quantum random walks and we develop this approach along three different lines: the quantum trajectory, the quantum dynamical map and the quantum stochastic differential equation. We also present a study of the simplest case, with a two level system as an internal gyroscope, illustrating the interplay between the ballistic and diffusive behaviors at work in these processes. Notation Hz: orbital (walker) Hilbert space, CZ in the discrete, L2(R) in the continuum Hc: internal spin (or gyroscope) Hilbert space Hsys=Hz⊗Hc: system Hilbert space Hp: probe (or quantum coin) Hilbert space, Hp=C2 ρttot: density matrix for the total system (walker + internal spin + quantum coins) ρ-bar t: reduced density matrix on Hsys: ρ-bar t=∫dxdy ρ-bar t(x,y)⊗|x〉z〈y| ρ-hat t: system density matrix in a quantum trajectory: ρ-hat t=∫dxdy ρ-hat t(x,y)⊗|x〉z〈y|. If diagonal and localized in position: ρ-hat t=ρt⊗|Xt〉z〈Xt| ρt: internal density matrix in a simple quantum trajectory Xt: walker position in a simple quantum trajectory Bt: normalized Brownian motion ξt, ξt†: quantum noises (paper)
Oscillatory Fractional Brownian Motion and Hierarchical Random Walks
Bojdecki, Tomasz; Gorostiza, Luis G.; Talarczyk, Anna
2012-01-01
We introduce oscillatory analogues of fractional Brownian motion, sub-fractional Brownian motion and other related long range dependent Gaussian processes, we discuss their properties, and we show how they arise from particle systems with or without branching and with different types of initial conditions, where the individual particle motion is the so-called c-random walk on a hierarchical group. The oscillations are caused by the discrete and ultrametric structure of the hierarchical group,...
Conformal invariance, universality, and the dimension of the Brownian frontier
Lawler, Gregory
2003-01-01
This paper describes joint work with Oded Schramm and Wendelin Werner establishing the values of the planar Brownian intersection exponents from which one derives the Hausdorff dimension of certain exceptional sets of planar Brownian motion. In particular, we proof a conjecture of Mandelbrot that the dimension of the frontier is 4/3. The proof uses a universality principle for conformally invariant measures and a new process, the stochastic Loewner evolution ($SLE$), introduced by Schramm. Th...
Speckle Patterns and 2-Dimensional Brownian Motion
International Nuclear Information System (INIS)
We present the results of a Monte Carlo simulation of Brownian Motion on a 2-dimensional lattice with nearest-neighbor interactions described by a linear model. These nearest-neighbor interactions lead to a spatial variance structure on the lattice. The resulting Brownian pattern fluctuates in value from point to point in a manner characteristic of a stationary stochastic process. The value at a lattice point is interpreted as an intensity level. The difference in values in neighboring cells produces a fluctuating intensity pattern on the lattice. Changing the size of the mesh changes the relative size of the speckles. Increasing the mesh size tends to average out the intensity in the direction of the mean of the stationary process. (Author)
A group action on increasing sequences of set-indexed Brownian motions
Yosef, Arthur
2015-01-01
We prove that a square-integrable set-indexed stochastic process is a set-indexed Brownian motion if and only if its projection on all the strictly increasing continuous sequences are one-parameter $G$-time-changed Brownian motions. In addition, we study the "sequence-independent variation" property for group stationary-increment stochastic processes in general and for a set-indexed Brownian motion in particular. We present some applications.
Assessing Relative Volatility/Intermittency/Energy Dissipation
DEFF Research Database (Denmark)
Barndorff-Nielsen, Ole E.; Pakkanen, Mikko; Schmiegel, Jürgen
process in particular. While this estimation method is motivated by the assessment of relative energy dissipation in empirical data of turbulence, we apply it also to energy price data. Moreover, we develop a probabilistic asymptotic theory for relative power variations of Brownian semistationary......We introduce the notion of relative volatility/intermittency and demonstrate how relative volatility statistics can be used to estimate consistently the temporal variation of volatility/intermittency even when the data of interest are generated by a non-semimartingale, or a Brownian semistationary...... processes and Ito semimartingales and discuss how it can be used for inference on relative volatility/intermittency....
Brownian Motion Theory and Experiment
Basu, K; Basu, Kasturi; Baishya, Kopinjol
2003-01-01
Brownian motion is the perpetual irregular motion exhibited by small particles immersed in a fluid. Such random motion of the particles is produced by statistical fluctuations in the collisions they suffer with the molecules of the surrounding fluid. Brownian motion of particles in a fluid (like milk particles in water) can be observed under a microscope. Here we describe a simple experimental set-up to observe Brownian motion and a method of determining the diffusion coefficient of the Brownian particles, based on a theory due to Smoluchowski. While looking through the microscope we focus attention on a fixed small volume, and record the number of particles that are trapped in that volume, at regular intervals of time. This gives us a time-series data, which is enough to determine the diffusion coefficient of the particles to a good degree of accuracy.
Brownian motion, martingales, and stochastic calculus
Le Gall, Jean-François
2016-01-01
This book offers a rigorous and self-contained presentation of stochastic integration and stochastic calculus within the general framework of continuous semimartingales. The main tools of stochastic calculus, including Itô’s formula, the optional stopping theorem and Girsanov’s theorem, are treated in detail alongside many illustrative examples. The book also contains an introduction to Markov processes, with applications to solutions of stochastic differential equations and to connections between Brownian motion and partial differential equations. The theory of local times of semimartingales is discussed in the last chapter. Since its invention by Itô, stochastic calculus has proven to be one of the most important techniques of modern probability theory, and has been used in the most recent theoretical advances as well as in applications to other fields such as mathematical finance. Brownian Motion, Martingales, and Stochastic Calculus provides a strong theoretical background to the reader interested i...
On collisions of Brownian particles
Ichiba, Tomoyuki; Karatzas, Ioannis
2010-01-01
We examine the behavior of $n$ Brownian particles diffusing on the real line with bounded, measurable drift and bounded, piecewise continuous diffusion coefficients that depend on the current configuration of particles. Sufficient conditions are established for the absence and for the presence of triple collisions among the particles. As an application to the Atlas model for equity markets, we study a special construction of such systems of diffusing particles using Brownian motions with refl...
Anomalous Brownian refrigerator
Rana, Shubhashis; Pal, P. S.; Saha, Arnab; Jayannavar, A. M.
2016-02-01
We present a detailed study of a Brownian particle driven by Carnot-type refrigerating protocol operating between two thermal baths. Both the underdamped as well as the overdamped limits are investigated. The particle is in a harmonic potential with time-periodic strength that drives the system cyclically between the baths. Each cycle consists of two isothermal steps at different temperatures and two adiabatic steps connecting them. Besides working as a stochastic refrigerator, it is shown analytically that in the quasistatic regime the system can also act as stochastic heater, depending on the bath temperatures. Interestingly, in non-quasistatic regime, our system can even work as a stochastic heat engine for certain range of cycle time and bath temperatures. We show that the operation of this engine is not reliable. The fluctuations of stochastic efficiency/coefficient of performance (COP) dominate their mean values. Their distributions show power law tails, however the exponents are not universal. Our study reveals that microscopic machines are not the microscopic equivalent of the macroscopic machines that we come across in our daily life. We find that there is no one to one correspondence between the performance of our system under engine protocol and its reverse.
Martínez, I. A.; Roldán, É.; Dinis, L.; Petrov, D.; Parrondo, J. M. R.; Rica, R. A.
2016-01-01
The Carnot cycle imposes a fundamental upper limit to the efficiency of a macroscopic motor operating between two thermal baths. However, this bound needs to be reinterpreted at microscopic scales, where molecular bio-motors and some artificial micro-engines operate. As described by stochastic thermodynamics, energy transfers in microscopic systems are random and thermal fluctuations induce transient decreases of entropy, allowing for possible violations of the Carnot limit. Here we report an experimental realization of a Carnot engine with a single optically trapped Brownian particle as the working substance. We present an exhaustive study of the energetics of the engine and analyse the fluctuations of the finite-time efficiency, showing that the Carnot bound can be surpassed for a small number of non-equilibrium cycles. As its macroscopic counterpart, the energetics of our Carnot device exhibits basic properties that one would expect to observe in any microscopic energy transducer operating with baths at different temperatures. Our results characterize the sources of irreversibility in the engine and the statistical properties of the efficiency--an insight that could inspire new strategies in the design of efficient nano-motors.
Parlar, Mahmut
2004-01-01
Brownian motion is an important stochastic process used in modelling the random evolution of stock prices. In their 1973 seminal paper--which led to the awarding of the 1997 Nobel prize in Economic Sciences--Fischer Black and Myron Scholes assumed that the random stock price process is described (i.e., generated) by Brownian motion. Despite its…
Metastable states in Brownian energy landscape
Cheliotis, Dimitris
2015-01-01
Random walks and diffusions in symmetric random environment are known to exhibit metastable behavior: they tend to stay for long times in wells of the environment. For the case that the environment is a one-dimensional two-sided standard Brownian motion, we study the process of depths of the consecutive wells of increasing depth that the motion visits. When these depths are looked in logarithmic scale, they form a stationary renewal cluster process. We give a description of the structure of t...
Brownian movement and molecular reality
Perrin, Jean
2005-01-01
How do we know that molecules really exist? An important clue came from Brownian movement, a concept developed in 1827 by botanist Robert Brown, who noticed that tiny objects like pollen grains shook and moved erratically when viewed under a microscope. Nearly 80 years later, in 1905, Albert Einstein explained this ""Brownian motion"" as the result of bombardment by molecules. Einstein offered a quantitative explanation by mathematically estimating the average distance covered by the particles over time as a result of molecular bombardment. Four years later, Jean Baptiste Perrin wrote Brownia
Some Brownian functionals and their laws
Donati-Martin, C.; Yor, M.
1997-01-01
We develop some topics about Brownian motion with a particular emphasis on the study of principal values of Brownian local times. We show some links between principal values and Doob’s $h$-transforms of Brownian motion, for nonpositive harmonic functions $h$. We also give a survey and complement some martingale approaches to Ray–Knight theorems for local times.
A Brownian model for recurrent earthquakes
Matthews, M.V.; Ellsworth, W.L.; Reasenberg, P.A.
2002-01-01
We construct a probability model for rupture times on a recurrent earthquake source. Adding Brownian perturbations to steady tectonic loading produces a stochastic load-state process. Rupture is assumed to occur when this process reaches a critical-failure threshold. An earthquake relaxes the load state to a characteristic ground level and begins a new failure cycle. The load-state process is a Brownian relaxation oscillator. Intervals between events have a Brownian passage-time distribution that may serve as a temporal model for time-dependent, long-term seismic forecasting. This distribution has the following noteworthy properties: (1) the probability of immediate rerupture is zero; (2) the hazard rate increases steadily from zero at t = 0 to a finite maximum near the mean recurrence time and then decreases asymptotically to a quasi-stationary level, in which the conditional probability of an event becomes time independent; and (3) the quasi-stationary failure rate is greater than, equal to, or less than the mean failure rate because the coefficient of variation is less than, equal to, or greater than 1/???2 ??? 0.707. In addition, the model provides expressions for the hazard rate and probability of rupture on faults for which only a bound can be placed on the time of the last rupture. The Brownian relaxation oscillator provides a connection between observable event times and a formal state variable that reflects the macromechanics of stress and strain accumulation. Analysis of this process reveals that the quasi-stationary distance to failure has a gamma distribution, and residual life has a related exponential distribution. It also enables calculation of "interaction" effects due to external perturbations to the state, such as stress-transfer effects from earthquakes outside the target source. The influence of interaction effects on recurrence times is transient and strongly dependent on when in the loading cycle step pertubations occur. Transient effects may
The relativistic Brownian motion: Interdisciplinary applications
International Nuclear Information System (INIS)
Relativistic Brownian motion theory will be applied to the study of analogies between physical and economic systems, emphasizing limiting cases in which Gaussian distributions are no longer valid. The characteristic temperatures of the particles will be associated with the concept of variance, and this will allow us to choose whether the pertinent distribution is classical or relativistic, while working specific situations. The properties of particles can be interpreted as economic variables, in order to study the behavior of markets in terms of Levy financial processes, since markets behave as stochastic systems. As far as we know, the application of the Juettner distribution to the study of economic systems is a new idea.
Brownian shape dynamics in fission
Randrup Jørgen; Möller Peter
2013-01-01
It was recently shown that remarkably accurate fission-fragment mass distributions are obtained by treating the nuclear shape evolution as a Brownian walk on previously calculated five-dimensional potentialenergy surfaces; the current status of this novel method is described here.
Brownian shape dynamics in fission
Directory of Open Access Journals (Sweden)
Randrup Jørgen
2013-12-01
Full Text Available It was recently shown that remarkably accurate fission-fragment mass distributions are obtained by treating the nuclear shape evolution as a Brownian walk on previously calculated five-dimensional potentialenergy surfaces; the current status of this novel method is described here.
Perturbative theory for Brownian vortexes.
Moyses, Henrique W; Bauer, Ross O; Grosberg, Alexander Y; Grier, David G
2015-06-01
Brownian vortexes are stochastic machines that use static nonconservative force fields to bias random thermal fluctuations into steadily circulating currents. The archetype for this class of systems is a colloidal sphere in an optical tweezer. Trapped near the focus of a strongly converging beam of light, the particle is displaced by random thermal kicks into the nonconservative part of the optical force field arising from radiation pressure, which then biases its diffusion. Assuming the particle remains localized within the trap, its time-averaged trajectory traces out a toroidal vortex. Unlike trivial Brownian vortexes, such as the biased Brownian pendulum, which circulate preferentially in the direction of the bias, the general Brownian vortex can change direction and even topology in response to temperature changes. Here we introduce a theory based on a perturbative expansion of the Fokker-Planck equation for weak nonconservative driving. The first-order solution takes the form of a modified Boltzmann relation and accounts for the rich phenomenology observed in experiments on micrometer-scale colloidal spheres in optical tweezers. PMID:26172698
On the weak convergence of super-Brownian motion with immigration
Institute of Scientific and Technical Information of China (English)
2009-01-01
We prove fluctuation limit theorems for the occupation times of super-Brownian motion with immigration. The weak convergence of the processes is established, which improves the results in references. The limiting processes are Gaussian processes.
Institute of Scientific and Technical Information of China (English)
王剑君
2011-01-01
In this paper, a new kind of hybrid model is presented. Under the hypothesis of underlying asset price submitting to multidimensional fractional Brownian motions and Poisson processes, the pricing formulas of two kinds of exotic options are obtained by means of the generalized pricing formula of European contingent claim of the model.%文章假设标的资产价格服从受分数布朗运动和泊松过程共同驱动的一类混合模型,通过这一模型的欧式未定权益的一般定价公式,求出了2种奇异期权的定价公式.
Stochastic flows in the Brownian web and net
Czech Academy of Sciences Publication Activity Database
Schertzer, E.; Sun, R.; Swart, Jan M.
2014-01-01
Roč. 227, č. 1065 (2014), s. 1-160. ISSN 0065-9266 R&D Projects: GA ČR GA201/07/0237; GA ČR GA201/09/1931 Institutional support: RVO:67985556 Keywords : Brownian web * Brownian net * stochastic flow of kernels * measure-valued process * Howitt-Warren flow * linear system * random walk in random environment * finite graph representation Subject RIV: BA - General Mathematics Impact factor: 1.727, year: 2014 http://library.utia.cas.cz/separaty/2013/SI/swart-0396636.pdf
Jakubowski, Jacek
2011-01-01
The aim of this paper is to present the new results concerning some functionals of Brownian motion with drift and present their applications in financial mathematics. We find a probabilistic representation of the Laplace transform of special functional of geometric Brownian motion using the squared Bessel and radial Ornstein-Uhlenbeck processes. Knowing the transition density functions of the above we obtain computable formulas for certain expectations of the concerned functional. As an example we find the moments of processes representing an asset price in the lognormal volatility ans Stein models. We also present links among the geometric Brownian motion, the Markov processes studied by Matsumoto and Yor and the hyperbolic Bessel processes.
Efficiency of Brownian heat engines.
Derényi, I; Astumian, R D
1999-06-01
We study the efficiency of one-dimensional thermally driven Brownian ratchets or heat engines. We identify and compare the three basic setups characterized by the type of the connection between the Brownian particle and the two heat reservoirs: (i) simultaneous, (ii) alternating in time, and (iii) position dependent. We make a clear distinction between the heat flow via the kinetic and the potential energy of the particle, and show that the former is always irreversible and it is only the third setup where the latter is reversible when the engine works quasistatically. We also show that in the third setup the heat flow via the kinetic energy can be reduced arbitrarily, proving that even for microscopic heat engines there is no fundamental limit of the efficiency lower than that of a Carnot cycle. PMID:11969723
Radiation Reaction on Brownian Motions
Seto, Keita
2016-01-01
Tracking the real trajectory of a quantum particle is one of the interpretation problem and it is expressed by the Brownian (stochastic) motion suggested by E. Nelson. Especially the dynamics of a radiating electron, namely, radiation reaction which requires us to track its trajectory becomes important in the high-intensity physics by PW-class lasers at present. It has been normally treated by the Furry picture in non-linear QED, but it is difficult to draw the real trajectory of a quantum particle. For the improvement of this, I propose the representation of a stochastic particle interacting with fields and show the way to describe radiation reaction on its Brownian motion.
The maximum of Brownian motion with parabolic drift
Janson, Svante; Louchard, Guy; Martin-Löf, Anders
2010-01-01
We study the maximum of a Brownian motion with a parabolic drift; this is a random variable that often occurs as a limit of the maximum of discrete processes whose expectations have a maximum at an interior point. We give new series expansions and integral formulas for the distribution and the first two moments, together with numerical values to high precision.
Hänggi, Peter; Marchesoni, Fabio
2005-01-01
In the year 1905 Albert Einstein published four papers that raised him to a giant in the history of science of all times. These works encompass the photon hypothesis (for which he obtained the Nobel prize in 1921), his first two papers on (special) relativity theory and, of course, his first paper on Brownian motion, entitled "\\"Uber die von der molekularkinetischen Theorie der W\\"arme geforderte Bewegung von in ruhenden Fl\\"ussigkeiten suspendierten Teilchen'' (submitted on May 11, 1905). Th...
Cooperative Transport of Brownian Particles
Derenyi, Imre; Vicsek, Tamas
1998-01-01
We consider the collective motion of finite-sized, overdamped Brownian particles (e.g., motor proteins) in a periodic potential. Simulations of our model have revealed a number of novel cooperative transport phenomena, including (i) the reversal of direction of the net current as the particle density is increased and (ii) a very strong and complex dependence of the average velocity on both the size and the average distance of the particles.
Kingman's coalescent and Brownian motion
Berestycki, J.; Berestycki, N
2009-01-01
We describe a simple construction of Kingman's coalescent in terms of a Brownian excursion. This construction is closely related to, and sheds some new light on, earlier work by Aldous and Warren. Our approach also yields some new results: for instance, we obtain the full multifractal spectrum of Kingman's coalescent. This complements earlier work on Beta-coalescents by the authors and Schweinsberg. Surprisingly, the thick part of the spectrum is not obtained by taking the limit as $\\alpha \\t...
Operator Fractional Brownian Motion and Martingale Differences
Directory of Open Access Journals (Sweden)
Hongshuai Dai
2014-01-01
Full Text Available It is well known that martingale difference sequences are very useful in applications and theory. On the other hand, the operator fractional Brownian motion as an extension of the well-known fractional Brownian motion also plays an important role in both applications and theory. In this paper, we study the relation between them. We construct an approximation sequence of operator fractional Brownian motion based on a martingale difference sequence.
G- Brownian motion and Its Applications
EBRAHIMBEYGI, Atena; DASTRANJ, Elham
2015-01-01
Abstract. The concept of G-Brownian motion and G-Ito integral has been introduced by Peng. Also Ito isometry lemma is proved for Ito integral and Brownian motion. In this paper we first investigate the Ito isometry lemma for G-Brownian motion and G-Ito Integral. Then after studying of MG2,0-class functions [4], we introduce Stratonovich integral for G-Brownian motion,say G- Stratonovich integral. Then we present a special construction for G- Stratonovich integral.
Maximum of a Fractional Brownian Motion: Analytic Results from Perturbation Theory.
Delorme, Mathieu; Wiese, Kay Jörg
2015-11-20
Fractional Brownian motion is a non-Markovian Gaussian process X_{t}, indexed by the Hurst exponent H. It generalizes standard Brownian motion (corresponding to H=1/2). We study the probability distribution of the maximum m of the process and the time t_{max} at which the maximum is reached. They are encoded in a path integral, which we evaluate perturbatively around a Brownian, setting H=1/2+ϵ. This allows us to derive analytic results beyond the scaling exponents. Extensive numerical simulations for different values of H test these analytical predictions and show excellent agreement, even for large ϵ. PMID:26636835
Pricing European option under the time-changed mixed Brownian-fractional Brownian model
Guo, Zhidong; Yuan, Hongjun
2014-07-01
This paper deals with the problem of discrete time option pricing by a mixed Brownian-fractional subdiffusive Black-Scholes model. Under the assumption that the price of the underlying stock follows a time-changed mixed Brownian-fractional Brownian motion, we derive a pricing formula for the European call option in a discrete time setting.
Brownian motion of helical flagella.
Hoshikawa, H; Saito, N
1979-07-01
We develops a theory of the Brownian motion of a rigid helical object such as bacterial flagella. The statistical properties of the random forces acting on the helical object are discussed and the coefficients of the correlations of the random forces are determined. The averages , and are also calculated where z and theta are the position along and angle around the helix axis respectively. Although the theory is limited to short time interval, direct comparison with experiment is possible by using the recently developed cinematography technique. PMID:16997210
Quantum trajectories for Brownian motion
Strunz, W T; Gisin, Nicolas; Yu, T; Strunz, Walter T.; Diosi, Lajos; Gisin, Nicolas
1999-01-01
We present the stochastic Schroedinger equation for the dynamics of a quantum particle coupled to a high temperature environment and apply it the dynamics of a driven, damped, nonlinear quantum oscillator. Apart from an initial slip on the environmental memory time scale, in the mean, our result recovers the solution of the known non-Lindblad quantum Brownian motion master equation. A remarkable feature of our approach is its localization property: individual quantum trajectories remain localized wave packets for all times, even for the classically chaotic system considered here, the localization being stronger the smaller $\\hbar$.
Intersection Exponents for Planar Brownian Motion
Lawler, Gregory F.; Werner, Wendelin
1999-01-01
We derive properties concerning all intersection exponents for planar Brownian motion and we define generalized exponents that, loosely speaking, correspond to noninteger numbers of Brownian paths. Some of these properties lead to general conjectures concerning the exact value of these exponents.
Nonlinear Brownian motion - mean square displacement
Directory of Open Access Journals (Sweden)
W.Ebeling
2004-01-01
Full Text Available The stochastic dynamics of self-propelled Brownian particles is studied by means of the Langevin and the Fokker-Planck approach. We model the driving by a nonlinear friction function which has a negative part at small velocities, leading to active Brownian motion of the particles. The mean square displacement is estimated analytically and compared with numerical simulations.
Dimensional Properties of Fractional Brownian Motion
Institute of Scientific and Technical Information of China (English)
Dong Sheng WU; Yi Min XIAO
2007-01-01
Let Bα = {Bα(t),t ∈ RN} be an (N,d)-fractional Brownian motion with Hurst index α∈ (0, 1). By applying the strong local nondeterminism of Bα, we prove certain forms of uniform Hausdorff dimension results for the images of Bα when N > αd. Our results extend those of Kaufman for one-dimensional Brownian motion.
Communication: Memory effects and active Brownian diffusion
International Nuclear Information System (INIS)
A self-propelled artificial microswimmer is often modeled as a ballistic Brownian particle moving with constant speed aligned along one of its axis, but changing direction due to random collisions with the environment. Similarly to thermal noise, its angular randomization is described as a memoryless stochastic process. Here, we speculate that finite-time correlations in the orientational dynamics can affect the swimmer’s diffusivity. To this purpose, we propose and solve two alternative models. In the first one, we simply assume that the environmental fluctuations governing the swimmer’s propulsion are exponentially correlated in time, whereas in the second one, we account for possible damped fluctuations of the propulsion velocity around the swimmer’s axis. The corresponding swimmer’s diffusion constants are predicted to get, respectively, enhanced or suppressed upon increasing the model memory time. Possible consequences of this effect on the interpretation of the experimental data are discussed
From Molecular Dynamics to Brownian Dynamics
Erban, Radek
2014-01-01
Three coarse-grained molecular dynamics (MD) models are investigated with the aim of developing and analyzing multiscale methods which use MD simulations in parts of the computational domain and (less detailed) Brownian dynamics (BD) simulations in the remainder of the domain. The first MD model is formulated in one spatial dimension. It is based on elastic collisions of heavy molecules (e.g. proteins) with light point particles (e.g. water molecules). Two three-dimensional MD models are then investigated. The obtained results are applied to a simplified model of protein binding to receptors on the cellular membrane. It is shown that modern BD simulators of intracellular processes can be used in the bulk and accurately coupled with a (more detailed) MD model of protein binding which is used close to the membrane.
Communication: Memory effects and active Brownian diffusion.
Ghosh, Pulak K; Li, Yunyun; Marchegiani, Giampiero; Marchesoni, Fabio
2015-12-01
A self-propelled artificial microswimmer is often modeled as a ballistic Brownian particle moving with constant speed aligned along one of its axis, but changing direction due to random collisions with the environment. Similarly to thermal noise, its angular randomization is described as a memoryless stochastic process. Here, we speculate that finite-time correlations in the orientational dynamics can affect the swimmer's diffusivity. To this purpose, we propose and solve two alternative models. In the first one, we simply assume that the environmental fluctuations governing the swimmer's propulsion are exponentially correlated in time, whereas in the second one, we account for possible damped fluctuations of the propulsion velocity around the swimmer's axis. The corresponding swimmer's diffusion constants are predicted to get, respectively, enhanced or suppressed upon increasing the model memory time. Possible consequences of this effect on the interpretation of the experimental data are discussed. PMID:26646861
Communication: Memory effects and active Brownian diffusion
Energy Technology Data Exchange (ETDEWEB)
Ghosh, Pulak K. [Department of Chemistry, Presidency University, Kolkata 700073 (India); Li, Yunyun, E-mail: yunyunli@tongji.edu.cn [Center for Phononics and Thermal Energy Science, Tongji University, Shanghai 200092 (China); Marchegiani, Giampiero [Dipartimento di Fisica, Università di Camerino, I-62032 Camerino (Italy); Marchesoni, Fabio [Center for Phononics and Thermal Energy Science, Tongji University, Shanghai 200092 (China); Dipartimento di Fisica, Università di Camerino, I-62032 Camerino (Italy)
2015-12-07
A self-propelled artificial microswimmer is often modeled as a ballistic Brownian particle moving with constant speed aligned along one of its axis, but changing direction due to random collisions with the environment. Similarly to thermal noise, its angular randomization is described as a memoryless stochastic process. Here, we speculate that finite-time correlations in the orientational dynamics can affect the swimmer’s diffusivity. To this purpose, we propose and solve two alternative models. In the first one, we simply assume that the environmental fluctuations governing the swimmer’s propulsion are exponentially correlated in time, whereas in the second one, we account for possible damped fluctuations of the propulsion velocity around the swimmer’s axis. The corresponding swimmer’s diffusion constants are predicted to get, respectively, enhanced or suppressed upon increasing the model memory time. Possible consequences of this effect on the interpretation of the experimental data are discussed.
Quantum mechanics and the square root of the Brownian motion
Frasca, Marco
2014-01-01
Using the Euler--Maruyama technique, we show that a class of Wiener processes exists that are obtained by computing an arbitrary positive power of them. This can be accomplished with a proper set of definitions that makes meaningful the realization at discrete times of these processes and make them computable. Then, we are able to show that quantum mechanics is not directly a stochastic process characterizing Brownian motion but rather its square root. Schr\\"odinger equation is immediately derived without further assumptions as the Fokker--Planck equation for this process. This generalizes without difficulty to a Clifford algebra that makes immediate the introduction of spin and a generalization to the Dirac equation. A relevant conclusion is that the introduction of spin is essential to recover the Brownian motion from its square root.
Brownian motion meets Riemann curvature
International Nuclear Information System (INIS)
The general covariance of the diffusion equation is exploited in order to explore the curvature effects appearing in Brownian motion over a d-dimensional curved manifold. We use the local frame defined by the so-called Riemann normal coordinates to derive a general formula for the mean-square geodesic distance (MSD) at the short-time regime. This formula is written in terms of O(d) invariants that depend on the Riemann curvature tensor. We study the n-dimensional sphere case to validate these results. We also show that the diffusion for positive constant curvature is slower than the diffusion in a plane space, while the diffusion for negative constant curvature turns out to be faster. Finally the two-dimensional case is emphasized, as it is relevant for single-particle diffusion on biomembranes
Blending Brownian motion and heat equation
Cristiani, Emiliano
2015-01-01
In this short communication we present an original way to couple the Brownian motion and the heat equation. More in general, we suggest a way for coupling the Langevin equation for a particle, which describes a single realization of its trajectory, with the associated Fokker-Planck equation, which instead describes the evolution of the particle's probability density function. Numerical results show that it is indeed possible to obtain a regularized Brownian motion and a Brownianized heat equation still preserving the global statistical properties of the solutions. The results also suggest that the more macroscale leads the dynamics the more one can reduce the microscopic degrees of freedom.
Renewal Structure of the Brownian Taut String
Schertzer, Emmanuel
2015-01-01
In a recent paper, M. Lifshits and E. Setterqvist introduced the taut string of a Brownian motion $w$, defined as the function of minimal quadratic energy on $[0,T]$ staying in a tube of fixed width $h>0$ around $w$. The authors showed a Law of Large Number (L.L.N.) for the quadratic energy spent by the string for a large time $T$. In this note, we exhibit a natural renewal structure for the Brownian taut string, which is directly related to the time decomposition of the Brownian motion in te...
The maximum of Brownian motion with parabolic drift (Extended abstract)
Janson, Svante; Louchard, Guy; Martin-Löf, Anders
2010-01-01
We study the maximum of a Brownian motion with a parabolic drift; this is a random variable that often occurs as a limit of the maximum of discrete processes whose expectations have a maximum at an interior point. This has some applications in algorithmic and data structures analysis. We give series expansions and integral formulas for the distribution and the first two moments, together with numerical values to high precision.
Onsager coefficients of a Brownian Carnot cycle
Izumida, Yuki; Okuda, Koji
2010-01-01
We study a Brownian Carnot cycle introduced by T. Schmiedl and U. Seifert [Europhys. Lett. \\textbf{81}, 20003 (2008)] from a viewpoint of the linear irreversible thermodynamics. By considering the entropy production rate of this cycle, we can determine thermodynamic forces and fluxes of the cycle and calculate the Onsager coefficients for general protocols, that is, arbitrary schedules to change the potential confining the Brownian particle. We show that these Onsager coefficients contain the...
Stochastic Calculus with respect to multifractional Brownian motion
Lebovits, Joachim
2011-01-01
Stochastic calculus with respect to fractional Brownian motion (fBm) has attracted a lot of interest in recent years, motivated in particular by applications in finance and Internet traffic modeling. Multifractional Brownian motion (mBm) is a Gaussian extension of fBm that allows to control the pointwise regularity of the paths of the process and to decouple it from its long range dependence properties. This generalization is obtained by replacing the constant Hurst parameter H of fBm by a function h(t). Multifractional Brownian motion has proved useful in many applications, including the ones just mentioned. In this work we extend to mBm the construction of a stochastic integral with respect to fBm. This stochastic integral is based on white noise theory, as originally proposed in [15], [6], [4] and in [5]. In that view, a multifractional white noise is defined, which allows to integrate with respect to mBm a large class of stochastic processes using Wick products. It\\^o formulas (both for tempered distribut...
Coulomb Friction Driving Brownian Motors
International Nuclear Information System (INIS)
We review a family of models recently introduced to describe Brownian motors under the influence of Coulomb friction, or more general non-linear friction laws. It is known that, if the heat bath is modeled as the usual Langevin equation (linear viscosity plus white noise), additional non-linear friction forces are not sufficient to break detailed balance, i.e. cannot produce a motor effect. We discuss two possibile mechanisms to elude this problem. A first possibility, exploited in several models inspired to recent experiments, is to replace the heat bath's white noise by a “collisional noise”, that is the effect of random collisions with an external equilibrium gas of particles. A second possibility is enlarging the phase space, e.g. by adding an external potential which couples velocity to position, as in a Klein—Kramers equation. In both cases, non-linear friction becomes sufficient to achieve a non-equilibrium steady state and, in the presence of an even small spatial asymmetry, a motor effect is produced. (general)
Brownian relaxation of an inelastic sphere in air
Bird, G. A.
2016-06-01
The procedures that are used to calculate the forces and moments on an aerodynamic body in the rarefied gas of the upper atmosphere are applied to a small sphere of the size of an aerosol particle at sea level. While the gas-surface interaction model that provides accurate results for macroscopic bodies may not be appropriate for bodies that are comprised of only about a thousand atoms, it provides a limiting case that is more realistic than the elastic model. The paper concentrates on the transfer of energy from the air to an initially stationary sphere as it acquires Brownian motion. Individual particle trajectories vary wildly, but a clear relaxation process emerges from an ensemble average over tens of thousands of trajectories. The translational and rotational energies in equilibrium Brownian motion are determined. Empirical relationships are obtained for the mean translational and rotational relaxation times, the mean initial power input to the particle, the mean rates of energy transfer between the particle and air, and the diffusivity. These relationships are functions of the ratio of the particle mass to an average air molecule mass and the Knudsen number, which is the ratio of the mean free path in the air to the particle diameter. The ratio of the molecular radius to the particle radius also enters as a correction factor. The implications of Brownian relaxation for the second law of thermodynamics are discussed.
Lectures from Markov processes to Brownian motion
Chung, Kai Lai
1982-01-01
This book evolved from several stacks of lecture notes written over a decade and given in classes at slightly varying levels. In transforming the over lapping material into a book, I aimed at presenting some of the best features of the subject with a minimum of prerequisities and technicalities. (Needless to say, one man's technicality is another's professionalism. ) But a text frozen in print does not allow for the latitude of the classroom; and the tendency to expand becomes harder to curb without the constraints of time and audience. The result is that this volume contains more topics and details than I had intended, but I hope the forest is still visible with the trees. The book begins at the beginning with the Markov property, followed quickly by the introduction of option al times and martingales. These three topics in the discrete parameter setting are fully discussed in my book A Course In Probability Theory (second edition, Academic Press, 1974). The latter will be referred to throughout this book ...
Minimal Cost of a Brownian Risk without Ruin
Luo, Shangzhen
2011-01-01
In this paper, we study a risk process modeled by a Brownian motion with drift (the diffusion approximation model). The insurance entity can purchase reinsurance to lower its risk and receive cash injections at discrete times to avoid ruin. Proportional reinsurance and excess-of-loss reinsurance are considered. The objective is to find the optimal reinsurance and cash injection strategy that minimizes the total cost to keep the company's surplus process non-negative, i.e. without ruin, where the cost function is defined as the total discounted value of the injections. The optimal solution is found explicitly by solving the according quasi-variational inequalities (QVIs).
Conditional limit theorems for regulated fractional Brownian motion
Awad, Hernan; 10.1214/09-AAP605
2009-01-01
We consider a stationary fluid queue with fractional Brownian motion input. Conditional on the workload at time zero being greater than a large value $b$, we provide the limiting distribution for the amount of time that the workload process spends above level $b$ over the busy cycle straddling the origin, as $b\\to\\infty$. Our results can be interpreted as showing that long delays occur in large clumps of size of order $b^{2-1/H}$. The conditional limit result involves a finer scaling of the queueing process than fluid analysis, thereby departing from previous related literature.
Quantum Brownian motion in a bath of parametric oscillators a model for system-field interactions
Hu, B L; Andrew Matacz
1993-01-01
The quantum Brownian motion paradigm provides a unified framework where one can see the interconnection of some basic quantum statistical processes like decoherence, dissipation, particle creation, noise and fluctuation. We treat the case where the Brownian particle is coupled linearly to a bath of time dependent quadratic oscillators. While the bath mimics a scalar field, the motion of the Brownian particle modeled by a single oscillator could be used to depict the behavior of a particle detector, a quantum field mode or the scale factor of the universe. An important result of this paper is the derivation of the influence functional encompassing the noise and dissipation kernels in terms of the Bogolubov coefficients. This method enables one to trace the source of statistical processes like decoherence and dissipation to vacuum fluctuations and particle creation, and in turn impart a statistical mechanical interpretation of quantum field processes. With this result we discuss the statistical mechanical origi...
Diffusion of torqued active Brownian particles
Sevilla, Francisco J.
An analytical approach is used to study the diffusion of active Brownian particles that move at constant speed in three-dimensional space, under the influence of passive (external) and active (internal) torques. The Smoluchowski equation for the position distribution of the particles is obtained from the Kramer-Fokker-Planck equation corresponding to Langevin equations for active Brownian particles subject to torques. In addition of giving explicit formulas for the mean square-displacement, the non-Gaussian behavior is analyzed through the kurtosis of the position distribution that exhibits an oscillatory behavior in the short-time limit. FJS acknowledges support from PAPIIT-UNAM through the grant IN113114
A Brownian model for recurrent volcanic eruptions: an application to Miyakejima volcano (Japan)
Garcia-Aristizabal, Alexander; Marzocchi, Warner; Fujita, Eisuke
2012-03-01
The definition of probabilistic models as mathematical structures to describe the response of a volcanic system is a plausible approach to characterize the temporal behavior of volcanic eruptions and constitutes a tool for long-term eruption forecasting. This kind of approach is motivated by the fact that volcanoes are complex systems in which a completely deterministic description of the processes preceding eruptions is practically impossible. To describe recurrent eruptive activity, we apply a physically motivated probabilistic model based on the characteristics of the Brownian passage-time (BPT) distribution; the physical process defining this model can be described by the steady rise of a state variable from a ground state to a failure threshold; adding Brownian perturbations to the steady loading produces a stochastic load-state process (a Brownian relaxation oscillator) in which an eruption relaxes the load state to begin a new eruptive cycle. The Brownian relaxation oscillator and Brownian passage-time distribution connect together physical notions of unobservable loading and failure processes of a point process with observable response statistics. The Brownian passage-time model is parameterized by the mean rate of event occurrence, μ, and the aperiodicity about the mean, α. We apply this model to analyze the eruptive history of Miyakejima volcano, Japan, finding a value of 44.2 (±6.5 years) for the μ parameter and 0.51 (±0.01) for the (dimensionless) α parameter. The comparison with other models often used in volcanological literature shows that this physically motivated model may be a good descriptor of volcanic systems that produce eruptions with a characteristic size. BPT is clearly superior to the Exponential distribution, and the fit to the data is comparable to other two-parameters models. Nonetheless, being a physically motivated model, it provides an insight into the macro-mechanical processes driving the system.
Generalized Einstein Relation for Brownian Motion in Tilted Periodic Potential
Sakaguchi, Hidetsugu
2006-01-01
A generalized Einstein relation is studied for Brownian motion in a tilted potential. The exact form of the diffusion constant of the Brownian motion is compared with the generalized Einstein relation. The generalized Einstein relation is a good approximation in a parameter range where the Brownian motion exhibits stepwise motion.
Kolmogorov complexity and the geometry of Brownian motion
Fouche, Willem L.
2014-01-01
In this paper, we continue the study of the geometry of Brownian motions which are encoded by Kolmogorov-Chaitin random reals (complex oscillations). We unfold Kolmogorov-Chaitin complexity in the context of Brownian motion and specifically to phenomena emerging from the random geometric patterns generated by a Brownian motion.
Magnetization direction in the Heisenberg model exhibiting fractional Brownian motion
DEFF Research Database (Denmark)
Zhang, Zhengping; Mouritsen, Ole G.; Zuckermann, Martin J.
1993-01-01
ferromagnetic phase characterizing fractional Brownian motion, whereas a value H congruent-to 0. 5, reflecting ordinary Brownian motion, applies in the paramagnetic phase. A field-induced crossover from fractional to ordinary Brownian motion has been observed in the ferromagnetic phase....
International Nuclear Information System (INIS)
Single-file diffusion behaves as normal diffusion at small time and as subdiffusion at large time. These properties can be described in terms of fractional Brownian motion with variable Hurst exponent or multifractional Brownian motion. We introduce a new stochastic process called Riemann–Liouville step fractional Brownian motion which can be regarded as a special case of multifractional Brownian motion with a step function type of Hurst exponent tailored for single-file diffusion. Such a step fractional Brownian motion can be obtained as a solution of the fractional Langevin equation with zero damping. Various kinds of fractional Langevin equations and their generalizations are then considered in order to decide whether their solutions provide the correct description of the long and short time behaviors of single-file diffusion. The cases where the dissipative memory kernel is a Dirac delta function, a power-law function and a combination of these functions are studied in detail. In addition to the case where the short time behavior of single-file diffusion behaves as normal diffusion, we also consider the possibility of a process that begins as ballistic motion
Altintas, Ersin; Böhringer, Karl F.; Fujita, Hiroyuki
2008-11-01
Nanosystems operating in liquid media may suffer from random Brownian motion due to thermal fluctuations. Biomolecular motors exploit these random fluctuations to generate a controllable directed movement. Inspired by nature, we proposed and realized a nano-system based on Brownian motion of nanobeads for linear transport in microfluidic channels. The channels limit the degree-of-freedom of the random motion of beads into one dimension, which was rectified by a three-phase dielectrophoretic ratchet biasing the spatial probability distribution of the nanobead towards the transportation direction. We micromachined the proposed device and experimentally traced the rectified motion of nanobeads and observed the shift in the bead distribution as a function of applied voltage. We identified three regions of operation; (1) a random motion region, (2) a Brownian motor region, and (3) a pure electric actuation region. Transportation in the Brownian motor region required less applied voltage compared to the pure electric transport.
Permutation entropy of fractional Brownian motion and fractional Gaussian noise
International Nuclear Information System (INIS)
We have worked out theoretical curves for the permutation entropy of the fractional Brownian motion and fractional Gaussian noise by using the Bandt and Shiha [C. Bandt, F. Shiha, J. Time Ser. Anal. 28 (2007) 646] theoretical predictions for their corresponding relative frequencies. Comparisons with numerical simulations show an excellent agreement. Furthermore, the entropy-gap in the transition between these processes, observed previously via numerical results, has been here theoretically validated. Also, we have analyzed the behaviour of the permutation entropy of the fractional Gaussian noise for different time delays
Theory of Brownian motion with the Alder-Wainwright effect
International Nuclear Information System (INIS)
The Stokes-Boussinesq-Langevin equation, which describes the time evolution of Brownian motion with the Alder-Wainwright effect, can be treated in the framework of the theory of KMO-Langevin equations which describe the time evolution of a real, stationary Gaussian process with T-positivity (reflection positivity) originating in axiomatic quantum field theory. After proving the fluctuation-dissipation theorems for KMO-Langevin equations, the authors obtain an explicit formula for the deviation from the classical Einstein relation that occurs in the Stokes-Boussinesq-Langevin equation with a white noise as its random force. The authors interested in whether or not it can be measured experimentally
Planar aggregation and the coalescing Brownian flow
Norris, James; Turner, Amanda
2008-01-01
We study a scaling limit associated to a model of planar aggregation. The model is obtained by composing certain independent random conformal maps. The evolution of harmonic measure on the boundary of the cluster is shown to converge to the coalescing Brownian flow.
Brownian shape motion: Fission fragment mass distributions
Sierk Arnold J.; Randrup Jørgen; Möller Peter
2012-01-01
It was recently shown that remarkably accurate fission-fragment mass distributions can be obtained by treating the nuclear shape evolution as a Brownian walk on previously calculated five-dimensional potential-energy surfaces; the current status of this novel method is described here.
Brownian shape motion: Fission fragment mass distributions
Directory of Open Access Journals (Sweden)
Sierk Arnold J.
2012-02-01
Full Text Available It was recently shown that remarkably accurate fission-fragment mass distributions can be obtained by treating the nuclear shape evolution as a Brownian walk on previously calculated five-dimensional potential-energy surfaces; the current status of this novel method is described here.
Brownian particles in supramolecular polymer solutions
Gucht, van der J.; Besseling, N.A.M.; Knoben, W.; Bouteiller, L.; Cohen Stuart, M.A.
2003-01-01
The Brownian motion of colloidal particles embedded in solutions of hydrogen-bonded supramolecular polymers has been studied using dynamic light scattering. At short times, the motion of the probe particles is diffusive with a diffusion coefficient equal to that in pure solvent. At intermediate time
Chaos, Dissipation and Quantal Brownian Motion
Cohen, Doron
1999-01-01
Energy absorption by driven chaotic systems, the theory of energy spreading and quantal Brownian motion are considered. In particular we discuss the theory of a classical particle that interacts with quantal chaotic degrees of freedom, and try to relate it to the problem of quantal particle that interacts with an effective harmonic bath.
Engineering Autonomous Chemomechanical Nanomachines Using Brownian Ratchets
Lavella, Gabriel
Nanoscale machines which directly convert chemical energy into mechanical work are ubiquitous in nature and are employed to perform a diverse set of tasks such as transporting molecules, maintaining molecular gradients, and providing motion to organisms. Their widespread use in nature suggests that large technological rewards can be obtained by designing synthetic machines that use similar mechanisms. This thesis addresses the technological adaptation of a specific mechanism known as the Brownian ratchet for the design of synthetic autonomous nanomachines. My efforts were focused more specifically on synthetic chemomechanical ratchets which I deem will be broadly applicable in the life sciences. In my work I have theoretically explored the biophysical mechanisms and energy landscapes that give rise to the ratcheting phenomena and devised devices that operate off these principles. I demonstrate two generations of devices that produce mechanical force/deformation in response to a user specified ligand. The first generation devices, fabricatied using a combination nanoscale lithographic processes and bioconjugation techniques, were used to provide evidence that the proposed ratcheting phenomena can be exploited in synthetic architectures. Second generation devices fabricated using self-assembled DNA/hapten motifs were constructed to gain a precise understanding of ratcheting dynamics and design constraints. In addition, the self-assembled devices enabled fabrication en masse, which I feel will alleviate future experimental hurdles in analysis and facilitate its adaptation to technologies. The product of these efforts is an architecture that has the potential to enable numerous technologies in biosensing and drug delivery. For example, the coupling of molecule-specific actuation to the release of drugs or signaling molecules from nanocapsules or porous materials could be transformative. Such architectures could provide possible avenues to pressing issues in biology and
Assessing relative volatility/intermittency/energy dissipation
DEFF Research Database (Denmark)
Barndorff-Nielsen, Ole E.; Pakkanen, Mikko S.; Schmiegel, Jürgen
2014-01-01
process in particular. This estimation method is motivated by the assessment of relative energy dissipation in empirical data of turbulence, but it is also applicable in other areas. We develop a probabilistic asymptotic theory for realised relative power variations of Brownian semistationary processes......, and introduce inference methods based on the theory. We also discuss how to extend the asymptotic theory to other classes of processes exhibiting stochastic volatility/intermittency. As an empirical application, we study relative energy dissipation in data of atmospheric turbulence.......We introduce the notion of relative volatility/intermittency and demonstrate how relative volatility statistics can be used to estimate consistently the temporal variation of volatility/intermittency when the data of interest are generated by a non-semimartingale, or a Brownian semistationary...
Millen, J; Deesuwan, T; Barker, P; Anders, J
2014-06-01
Einstein realized that the fluctuations of a Brownian particle can be used to ascertain the properties of its environment. A large number of experiments have since exploited the Brownian motion of colloidal particles for studies of dissipative processes, providing insight into soft matter physics and leading to applications from energy harvesting to medical imaging. Here, we use heated optically levitated nanospheres to investigate the non-equilibrium properties of the gas surrounding them. Analysing the sphere's Brownian motion allows us to determine the temperature of the centre-of-mass motion of the sphere, its surface temperature and the heated gas temperature in two spatial dimensions. We observe asymmetric heating of the sphere and gas, with temperatures reaching the melting point of the material. This method offers opportunities for accurate temperature measurements with spatial resolution on the nanoscale, and provides a means for testing non-equilibrium thermodynamics. PMID:24793558
The inviscid Burgers equation with initial data of Brownian type
She, Zhen-Su; Aurell, Erik; Frisch, Uriel
1992-09-01
The solutions to Burgers equation, in the limit of vanishing viscosity, are investigated when the initial velocity is a Brownian motion (or fractional Brownian motion) function, i.e. a Gaussian process with scaling exponent 0< h<1 (type A) or the derivative thereof, with scaling exponent -1< h<0 (type B). Largesize numerical experiments are performed, helped by the fact that the solution is essentially obtained by performing a Legendre transform. The main result is obtained for type A and concerns the Lagrangian function x(a) which gives the location at time t=1 of the fluid particle which started at the location a. It is found to be a complete Devil's staircase. The cumulative probability of Lagrangian shock intervals Δ a (also the distribution of shock amplitudes) follows a ( Δa)- h law for small Δ a. The remaining (regular) Lagrangian locations form a Cantor set of dimension h. In Eulerian coordinates, the shock locations are everywhere dense. The scaling properties of various statistical quantities are also found. Heuristic interpretations are provided for some of these results. Rigorous results for the case of Brownian motion are established in a companion paper by Ya. Sinai. For type B initial velocities (e.g. white noise), there are very few small shocks and shock locations appear to be isolated. Finally, it is shown that there are universality classes of random but smooth (non-scaling) initial velocities such that the long-time large-scale behavior is, after rescaling, the same as for type A or B.
Pitman, Jim
1999-01-01
For a random process $X$ consider the random vector defined by the values of $X$ at times $0 \\lt U_{n,1} \\lt ... \\lt U_{n,n} \\lt 1$ and the minimal values of $X$ on each of the intervals between consecutive pairs of these times, where the $U_{n,i}$ are the order statistics of $n$ independent uniform $(0,1)$ variables, independent of $X$. The joint law of this random vector is explicitly described when $X$ is a Brownian motion. Corresponding results for Brownian bridge, excursion, and meander ...
Anomalous diffusion as modeled by a nonstationary extension of Brownian motion
Cushman, John H.; O'Malley, Daniel; Park, Moongyu
2009-03-01
If the mean-square displacement of a stochastic process is proportional to tβ , β≠1 , then it is said to be anomalous. We construct a family of Markovian stochastic processes with independent nonstationary increments and arbitrary but a priori specified mean-square displacement. We label the family as an extended Brownian motion and show that they satisfy a Langevin equation with time-dependent diffusion coefficient. If the time derivative of the variance of the process is homogeneous, then by computing the fractal dimension it can be shown that the complexity of the family is the same as that of the Brownian motion. For two particles initially separated by a distance x , the finite-size Lyapunov exponent (FSLE) measures the average rate of exponential separation to a distance ax . An analytical expression is developed for the FSLEs of the extended Brownian processes and numerical examples presented. The explicit construction of these processes illustrates that contrary to what has been stated in the literature, a power-law mean-square displacement is not necessarily related to a breakdown in the classical central limit theorem (CLT) caused by, for example, correlation (fractional Brownian motion or correlated continuous-time random-walk schemes) or infinite variance (Levy motion). The classical CLT, coupled with nonstationary increments, can and often does give rise to power-law moments such as the mean-square displacement.
Narrow escape for a stochastically gated Brownian ligand.
Reingruber, Jürgen; Holcman, David
2010-02-17
Molecular activation in cellular microdomains is usually characterized by a forward binding rate, which is the reciprocal of the arrival time of a ligand to a key target. Upon chemical interactions or conformational changes, a Brownian ligand may randomly switch between different states, and when target activation is possible in a specific state only, switching can significantly alter the activation process. The main goal of this paper is to study the mean time for a switching ligand to activate a small substrate, modelled as the time to exit a microdomain through a small absorbing window on the surface. We present the equations for the mean sojourn times the ligand spends in each state, and study the escape process with switching between two states in dimension one and three. When the ligand can exit in only one of the two states, we find that switching always decreases its sojourn time in the state where it can exit. Moreover, the fastest exit is obtained when the ligand diffuses most of the time in the state with the maximal diffusion coefficient, although this may imply that it spends most of the time 'hidden' in the state where it cannot exit. We discuss the physical mechanisms responsible for this apparent paradox. In dimension three we confirm our results with Brownian simulations. Finally, we suggest possible applications in cellular biology. PMID:21389363
Stochastic description of quantum Brownian dynamics
Yan, Yun-An; Shao, Jiushu
2016-08-01
Classical Brownian motion has well been investigated since the pioneering work of Einstein, which inspired mathematicians to lay the theoretical foundation of stochastic processes. A stochastic formulation for quantum dynamics of dissipative systems described by the system-plus-bath model has been developed and found many applications in chemical dynamics, spectroscopy, quantum transport, and other fields. This article provides a tutorial review of the stochastic formulation for quantum dissipative dynamics. The key idea is to decouple the interaction between the system and the bath by virtue of the Hubbard-Stratonovich transformation or Itô calculus so that the system and the bath are not directly entangled during evolution, rather they are correlated due to the complex white noises introduced. The influence of the bath on the system is thereby defined by an induced stochastic field, which leads to the stochastic Liouville equation for the system. The exact reduced density matrix can be calculated as the stochastic average in the presence of bath-induced fields. In general, the plain implementation of the stochastic formulation is only useful for short-time dynamics, but not efficient for long-time dynamics as the statistical errors go very fast. For linear and other specific systems, the stochastic Liouville equation is a good starting point to derive the master equation. For general systems with decomposable bath-induced processes, the hierarchical approach in the form of a set of deterministic equations of motion is derived based on the stochastic formulation and provides an effective means for simulating the dissipative dynamics. A combination of the stochastic simulation and the hierarchical approach is suggested to solve the zero-temperature dynamics of the spin-boson model. This scheme correctly describes the coherent-incoherent transition (Toulouse limit) at moderate dissipation and predicts a rate dynamics in the overdamped regime. Challenging problems
Brownian Thermal Noise in Multilayer Coated Mirrors
Hong, Ting; Gustafson, Eric K; Adhikari, Rana X; Chen, Yanbei
2012-01-01
We analyze the Brownian thermal noise of a multi-layer dielectric coating, used in high-precision optical measurements including interferometric gravitational-wave detectors. We assume the coating material to be isotropic, and therefore study thermal noises arising from shear and bulk losses of the coating materials. We show that coating noise arises not only from layer thickness fluctuations, but also from fluctuations of the interface between the coating and substrate, driven by internal fluctuating stresses of the coating. In addition, the non-zero photoeleastic coefficients of the thin films modifies the influence of the thermal noise on the laser field. The thickness fluctuations of different layers are statistically independent, however, there exists a finite coherence between layers and the substrate-coating interface. Taking into account uncertainties in material parameters, we show that significant uncertainties still exist in estimating coating Brownian noise.
Modeling an efficient Brownian heat engine
Asfaw, Mesfin
2008-09-01
We discuss the effect of subdividing the ratchet potential on the performance of a tiny Brownian heat engine that is modeled as a Brownian particle hopping in a viscous medium in a sawtooth potential (with or without load) assisted by alternately placed hot and cold heat baths along its path. We show that the velocity, the efficiency and the coefficient of performance of the refrigerator maximize when the sawtooth potential is subdivided into series of smaller connected barrier series. When the engine operates quasistatically, we analytically show that the efficiency of the engine can not approach the Carnot efficiency and, the coefficient of performance of the refrigerator is always less than the Carnot refrigerator due to the irreversible heat flow via the kinetic energy.
Brownian motion on random dynamical landscapes
Suñé Simon, Marc; Sancho, José María; Lindenberg, Katja
2016-03-01
We present a study of overdamped Brownian particles moving on a random landscape of dynamic and deformable obstacles (spatio-temporal disorder). The obstacles move randomly, assemble, and dissociate following their own dynamics. This landscape may account for a soft matter or liquid environment in which large obstacles, such as macromolecules and organelles in the cytoplasm of a living cell, or colloids or polymers in a liquid, move slowly leading to crowding effects. This representation also constitutes a novel approach to the macroscopic dynamics exhibited by active matter media. We present numerical results on the transport and diffusion properties of Brownian particles under this disorder biased by a constant external force. The landscape dynamics are characterized by a Gaussian spatio-temporal correlation, with fixed time and spatial scales, and controlled obstacle concentrations.
Brownian thermal noise in multilayer coated mirrors
Hong, Ting; Yang, Huan; Gustafson, Eric K.; Adhikari, Rana X.; Chen, Yanbei
2013-04-01
We analyze the Brownian thermal noise of a multilayer dielectric coating used in high-precision optical measurements, including interferometric gravitational-wave detectors. We assume the coating material to be isotropic, and therefore study thermal noises arising from shear and bulk losses of the coating materials. We show that coating noise arises not only from layer thickness fluctuations, but also from fluctuations of the interface between the coating and substrate, driven by fluctuating shear stresses of the coating. Although thickness fluctuations of different layers are statistically independent, there exists a finite coherence between the layers and the substrate-coating interface. In addition, photoelastic coefficients of the thin layers (so far not accurately measured) further influence the thermal noise, although at a relatively low level. Taking into account uncertainties in material parameters, we show that significant uncertainties still exist in estimating coating Brownian noise.
Coupling of lever arm swing and biased Brownian motion in actomyosin.
Directory of Open Access Journals (Sweden)
Qing-Miao Nie
2014-04-01
Full Text Available An important unresolved problem associated with actomyosin motors is the role of Brownian motion in the process of force generation. On the basis of structural observations of myosins and actins, the widely held lever-arm hypothesis has been proposed, in which proteins are assumed to show sequential structural changes among observed and hypothesized structures to exert mechanical force. An alternative hypothesis, the Brownian motion hypothesis, has been supported by single-molecule experiments and emphasizes more on the roles of fluctuating protein movement. In this study, we address the long-standing controversy between the lever-arm hypothesis and the Brownian motion hypothesis through in silico observations of an actomyosin system. We study a system composed of myosin II and actin filament by calculating free-energy landscapes of actin-myosin interactions using the molecular dynamics method and by simulating transitions among dynamically changing free-energy landscapes using the Monte Carlo method. The results obtained by this combined multi-scale calculation show that myosin with inorganic phosphate (Pi and ADP weakly binds to actin and that after releasing Pi and ADP, myosin moves along the actin filament toward the strong-binding site by exhibiting the biased Brownian motion, a behavior consistent with the observed single-molecular behavior of myosin. Conformational flexibility of loops at the actin-interface of myosin and the N-terminus of actin subunit is necessary for the distinct bias in the Brownian motion. Both the 5.5-11 nm displacement due to the biased Brownian motion and the 3-5 nm displacement due to lever-arm swing contribute to the net displacement of myosin. The calculated results further suggest that the recovery stroke of the lever arm plays an important role in enhancing the displacement of myosin through multiple cycles of ATP hydrolysis, suggesting a unified movement mechanism for various members of the myosin family.
Holographic Brownian Motion at Finite Density
Banerjee, Pinaki
2015-01-01
We study holographic Brownian motion of a heavy charged particle at zero and small (but finite) temperature in presence of finite density. We are primarily interested in the dynamics at (near) zero temperature which is holographically described by motion of a fundamental string in an (near-) extremal RN black hole. We compute analytically the functional form of retarded Green's function and also compare that numerically at leading order in small frequency.
Frustrated Brownian Motion of Nonlocal Solitary Waves
International Nuclear Information System (INIS)
We investigate the evolution of solitary waves in a nonlocal medium in the presence of disorder. By using a perturbational approach, we show that an increasing degree of nonlocality may largely hamper the Brownian motion of self-trapped wave packets. The result is valid for any kind of nonlocality and in the presence of nonparaxial effects. Analytical predictions are compared with numerical simulations based on stochastic partial differential equations.
The quantum brownian particle and memory effects
International Nuclear Information System (INIS)
The Quantum Brownian particle, immersed in a heat bath, is described by a statistical operator whose evolution is ruled by a Generalized Master Equation (GME). The heat bath degrees of freedom are considered to be either white noise or coloured noise correlated,while the GME is considered under either the Markov or Non-Markov approaches. The comparison between these considerations are fully developed and their physical meaning is discussed. (author)
Brownian particles in supramolecular polymer solutions
Gucht, van der, J.; Besseling, N.A.M.; Knoben, W.; Bouteiller, L; Cohen Stuart, M. A.
2003-01-01
The Brownian motion of colloidal particles embedded in solutions of hydrogen-bonded supramolecular polymers has been studied using dynamic light scattering. At short times, the motion of the probe particles is diffusive with a diffusion coefficient equal to that in pure solvent. At intermediate time scales the particles are slowed down as a result of trapping in elastic cages formed by the polymer chains, while at longer times the motion is diffusive again, but with a much smaller diffusion c...
Intrinsic and extrinsic measurement for Brownian motion
International Nuclear Information System (INIS)
Based upon the Smoluchowski equation on curved manifolds, three physical observables are considered for Brownian displacement, namely geodesic displacement s, Euclidean displacement δR, and projected displacement δR⊥. The Weingarten–Gauss equations are used to calculate the mean-square Euclidean displacements in the short-time regime. Our findings show that from an extrinsic point of view the geometry of the space affects the Brownian motion in such a way that the particle’s diffusion is decelerated, contrasting with the intrinsic point of view where dynamics is controlled by the sign of the Gaussian curvature (Castro-Villarreal, 2010 J. Stat. Mech. P08006). Furthermore, it is possible to give exact formulas for 〈δR〉 and 〈δR2〉 on spheres and minimal surfaces, which are valid for all values of time. In the latter case, surprisingly, Brownian motion corresponds to the usual diffusion in flat geometries, albeit minimal surfaces have non-zero Gaussian curvature. Finally, the two-dimensional case is emphasized due to its close relation to surface self-diffusion in fluid membranes. (paper)
Optimum analysis of a Brownian refrigerator.
Luo, X G; Liu, N; He, J Z
2013-02-01
A Brownian refrigerator with the cold and hot reservoirs alternating along a space coordinate is established. The heat flux couples with the movement of the Brownian particles due to an external force in the spatially asymmetric but periodic potential. After using the Arrhenius factor to describe the behaviors of the forward and backward jumps of the particles, the expressions for coefficient of performance (COP) and cooling rate are derived analytically. Then, through maximizing the product of conversion efficiency and heat flux flowing out, a new upper bound only depending on the temperature ratio of the cold and hot reservoirs is found numerically in the reversible situation, and it is a little larger than the so-called Curzon and Ahlborn COP ε(CA)=(1/√[1-τ])-1. After considering the irreversible factor owing to the kinetic energy change of the moving particles, we find the optimized COP is smaller than ε(CA) and the external force even does negative work on the Brownian particles when they jump from a cold to hot reservoir. PMID:23496491
Interacting Brownian Swarms: Some Analytical Results
Directory of Open Access Journals (Sweden)
Guillaume Sartoretti
2016-01-01
Full Text Available We consider the dynamics of swarms of scalar Brownian agents subject to local imitation mechanisms implemented using mutual rank-based interactions. For appropriate values of the underlying control parameters, the swarm propagates tightly and the distances separating successive agents are iid exponential random variables. Implicitly, the implementation of rank-based mutual interactions, requires that agents have infinite interaction ranges. Using the probabilistic size of the swarm’s support, we analytically estimate the critical interaction range below that flocked swarms cannot survive. In the second part of the paper, we consider the interactions between two flocked swarms of Brownian agents with finite interaction ranges. Both swarms travel with different barycentric velocities, and agents from both swarms indifferently interact with each other. For appropriate initial configurations, both swarms eventually collide (i.e., all agents interact. Depending on the values of the control parameters, one of the following patterns emerges after collision: (i Both swarms remain essentially flocked, or (ii the swarms become ultimately quasi-free and recover their nominal barycentric speeds. We derive a set of analytical flocking conditions based on the generalized rank-based Brownian motion. An extensive set of numerical simulations corroborates our analytical findings.
Institute of Scientific and Technical Information of China (English)
王剑君; 廖芳芳
2011-01-01
In this paper a new kind of hybrid process is presented. Under the hypothesis of underlying asset price submitting to multidimensional Fractional Brownian Motions and Poisson Processes, by applying equivalent martingale measure, we obtain the pricing formulas of several kinds of European power options by means of the generalized pricing formula of European contingent claim.%假设标的资产价格服从受多维分数布朗运动和泊松过程共同驱动的一类混合模型，在等价鞅测度下，通过这一模型的欧式未定权益的一般定价公式，求出了几种欧式幂型期权的定价公式．
Energy Technology Data Exchange (ETDEWEB)
Plyukhin, A.V., E-mail: aplyukhin@anselm.edu [Department of Mathematics, Saint Anselm College, Manchester, NH 03102 (United States)
2013-06-03
A model of an autonomous isothermal Brownian motor with an internal propulsion mechanism is considered. The motor is a Brownian particle which is semi-transparent for molecules of surrounding ideal gas. Molecular passage through the particle is controlled by a potential similar to that in the transition rate theory, i.e. characterized by two stationary states with a finite energy difference separated by a potential barrier. The internal potential drop maintains the diode-like asymmetry of molecular fluxes through the particle, which results in the particle's stationary drift.
From Lagrangian to Brownian motion
International Nuclear Information System (INIS)
We present a Lagrangian describing an idealized liquid interacting with a particle immersed in it. We show that the equation describing the motion of the particle as a functional of the initial conditions of the liquid incorporates noise and friction, which are attributed to specific dynamical processes. The equation is approximated to yield a Langevin equation with parameters depending on the Lagrangian and the temperature of the liquid. The origin of irreversibility and dissipation is discussed
On a non-linear transformation between Brownian martingales
Shkolnikov, Mykhaylo
2012-01-01
The paper studies a non-linear transformation between Brownian martingales, which is given by the inverse of the pricing operator in the mathematical finance terminology. Subsequently, the solvability of systems of equations corresponding to such transformations is investigated. The latter give rise to novel monotone pathwise couplings of an arbitrary number of certain diffusion processes with varying diffusion coefficients. In the case that there is an uncountable number of these diffusion processes and that the index set is an interval such couplings can be viewed as models for the growth of one-dimensional random surfaces. With this motivation in mind, we derive the appropriate stochastic partial differential equations for the growth of such surfaces.
Langevin theory of anomalous Brownian motion made simple
International Nuclear Information System (INIS)
During the century from the publication of the work by Einstein (1905 Ann. Phys. 17 549) Brownian motion has become an important paradigm in many fields of modern science. An essential impulse for the development of Brownian motion theory was given by the work of Langevin (1908 C. R. Acad. Sci., Paris 146 530), in which he proposed an 'infinitely more simple' description of Brownian motion than that by Einstein. The original Langevin approach has however strong limitations, which were rigorously stated after the creation of the hydrodynamic theory of Brownian motion (1945). Hydrodynamic Brownian motion is a special case of 'anomalous Brownian motion', now intensively studied both theoretically and in experiments. We show how some general properties of anomalous Brownian motion can be easily derived using an effective method that allows one to convert the stochastic generalized Langevin equation into a deterministic Volterra-type integro-differential equation for the mean square displacement of the particle. Within the Gibbs statistics, the method is applicable to linear equations of motion with any kind of memory during the evolution of the system. We apply it to memoryless Brownian motion in a harmonic potential well and to Brownian motion in fluids, taking into account the effects of hydrodynamic memory. Exploring the mathematical analogy between Brownian motion and electric circuits, which are at nanoscales also described by the generalized Langevin equation, we calculate the fluctuations of charge and current in RLC circuits that are in contact with the thermal bath. Due to the simplicity of our approach it could be incorporated into graduate courses of statistical physics. Once the method is established, it allows bringing to the attention of students and effectively solving a number of attractive problems related to Brownian motion.
Quantum Brownian motion in a Landau level
Cobanera, E.; Kristel, P.; Morais Smith, C.
2016-06-01
Motivated by questions about the open-system dynamics of topological quantum matter, we investigated the quantum Brownian motion of an electron in a homogeneous magnetic field. When the Fermi length lF=ℏ /(vFmeff) becomes much longer than the magnetic length lB=(ℏc /e B ) 1 /2 , then the spatial coordinates X ,Y of the electron cease to commute, [X ,Y ] =i lB2 . As a consequence, localization of the electron becomes limited by Heisenberg uncertainty, and the linear bath-electron coupling becomes unconventional. Moreover, because the kinetic energy of the electron is quenched by the strong magnetic field, the electron has no energy to give to or take from the bath, and so the usual connection between frictional forces and dissipation no longer holds. These two features make quantum Brownian motion topological, in the regime lF≫lB , which is at the verge of current experimental capabilities. We model topological quantum Brownian motion in terms of an unconventional operator Langevin equation derived from first principles, and solve this equation with the aim of characterizing diffusion. While diffusion in the noncommutative plane turns out to be conventional, with the mean displacement squared being proportional to tα and α =1 , there is an exotic regime for the proportionality constant in which it is directly proportional to the friction coefficient and inversely proportional to the square of the magnetic field: in this regime, friction helps diffusion and the magnetic field suppresses all fluctuations. We also show that quantum tunneling can be completely suppressed in the noncommutative plane for suitably designed metastable potential wells, a feature that might be worth exploiting for storage and protection of quantum information.
Polar Functions of Multiparameter Bifractional Brownian Sheets
Institute of Scientific and Technical Information of China (English)
Zhen-long Chen
2009-01-01
Let BH,K={BH,K(t), t ∈RN+} be an (N,d)-bifractional Brownian sheet with Hurst indices for BH,Kare investigated. The relationship between the class of continuous functions satisfying the Lipschitz condition and the class of polar-functions of BH,Kis presented. The Hausdorff dimension of the fixed points and an inequality concerning the Kolmogorov's entropy index for BH,Kare obtained. A question proposed by LeGall about the existence of no-polar, continuous functions statisfying the Holder condition is also solved.
Brownian motion of particles in nematic fluids
Yao, Xuxia; Nayani, Karthik; Park, Jung; Srinivasarao, Mohan
2011-03-01
We studied the brownian motion of both charged and neutral polystyrene particles in two nematic fluids, a thermotropic liquid crystal, E7, and a lyotropic chromonic liquid crystal, Sunset Yellow FCF (SSY). Homogeneous planar alignment of E7 was easliy achieved by using rubbed polyimide film coated on the glass. For SSY planar mondomain, we used the capillary method recently developed in our lab. By tracking a single particle, the direction dependent diffussion coefficients and Stokes drag were measured in the nematic phase and isotropic phase for both systems.
Brownian transport in corrugated channels with inertia
Ghosh, P. K.; Hänggi, P.; Marchesoni, F.; Nori, F.; Schmid, G.
2012-08-01
Transport of suspended Brownian particles dc driven along corrugated narrow channels is numerically investigated in the regime of finite damping. We show that inertial corrections cannot be neglected as long as the width of the channel bottlenecks is smaller than an appropriate particle diffusion length, which depends on the the channel corrugation and the drive intensity. With such a diffusion length being inversely proportional to the damping constant, transport through sufficiently narrow obstructions turns out to be always sensitive to the viscosity of the suspension fluid. The inertia corrections to the transport quantifiers, mobility, and diffusivity markedly differ for smoothly and sharply corrugated channels.
Brownian transport in corrugated channels with inertia
Ghosh, P K; Marchesoni, F; Nori, F; Schmid, G; 10.1103/PhysRevE.86.021112
2012-01-01
The transport of suspended Brownian particles dc-driven along corrugated narrow channels is numerically investigated in the regime of finite damping. We show that inertial corrections cannot be neglected as long as the width of the channel bottlenecks is smaller than an appropriate particle diffusion length, which depends on the the channel corrugation and the drive intensity. Being such a diffusion length inversely proportional to the damping constant, transport through sufficiently narrow obstructions turns out to be always sensitive to the viscosity of the suspension fluid. The inertia corrections to the transport quantifiers, mobility and diffusivity, markedly differ for smoothly and sharply corrugated channels.
Arithmetic area for m planar Brownian paths
Desbois, Jean
2012-01-01
We pursue the analysis made in [1] on the arithmetic area enclosed by m closed Brownian paths. We pay a particular attention to the random variable S{n1,n2, ...,n} (m) which is the arithmetic area of the set of points, also called winding sectors, enclosed n1 times by path 1, n2 times by path 2, ...,nm times by path m. Various results are obtained in the asymptotic limit m->infinity. A key observation is that, since the paths are independent, one can use in the m paths case the SLE information, valid in the 1-path case, on the 0-winding sectors arithmetic area.
Arithmetic area for m planar Brownian paths
Desbois, Jean; Ouvry, Stephane
2012-01-01
We pursue the analysis made in [1] on the arithmetic area enclosed by m closed Brownian paths. We pay a particular attention to the random variable S{n1,n2, ...,n} (m) which is the arithmetic area of the set of points, also called winding sectors, enclosed n1 times by path 1, n2 times by path 2, ...,nm times by path m. Various results are obtained in the asymptotic limit m->infinity. A key observation is that, since the paths are independent, one can use in the m paths case the SLE informatio...
Langevin Theory of Anomalous Brownian Motion Made Simple
Tothova, Jana; Vasziova, Gabriela; Glod, Lukas; Lisy, Vladimir
2011-01-01
During the century from the publication of the work by Einstein (1905 "Ann. Phys." 17 549) Brownian motion has become an important paradigm in many fields of modern science. An essential impulse for the development of Brownian motion theory was given by the work of Langevin (1908 "C. R. Acad. Sci.", Paris 146 530), in which he proposed an…
The Stepping Motion of Brownian Particle Derived by Nonequilibrium Fluctuation
Institute of Scientific and Technical Information of China (English)
ZHAN Yong; ZHAO Tong-Jun; YU Hui; SONG Yan-Li; AN Hai-Long
2003-01-01
The direct motion of Brownian particle is considered as a result of system derived by external nonequilibriumfluctuating. The cooperative effects caused by asymmetric ratchet potential, external rocking force and additive colorednoise drive a Brownian particle in the directed stepping motion. This provides this kind of motion of kinesin along amicrotubule observed in experiments with a reasonable explanation.
Holographic Brownian motion and time scales in strongly coupled plasmas
A. Nata Atmaja; J. de Boer; M. Shigemori
2010-01-01
We study Brownian motion of a heavy quark in field theory plasma in the AdS/CFT setup and discuss the time scales characterizing the interaction between the Brownian particle and plasma constituents. In particular, the mean-free-path time is related to the connected 4-point function of the random fo
Magnetic fields and Brownian motion on the 2-sphere
International Nuclear Information System (INIS)
Using constrained path integrals, we study some statistical properties of Brownian paths on the two dimensional sphere. A generalized Levy's law for the probability P(A) that a closed Brownian path encloses an algebraic area A is obtained. Distributions of scaled variables related to the winding of paths around some fixed point are recovered in the asymptotic regime t → ∞
Tested Demonstrations. Brownian Motion: A Classroom Demonstration and Student Experiment.
Kirksey, H. Graden; Jones, Richard F.
1988-01-01
Shows how video recordings of the Brownian motion of tiny particles may be made. Describes a classroom demonstration and cites a reported experiment designed to show the random nature of Brownian motion. Suggests a student experiment to discover the distance a tiny particle travels as a function of time. (MVL)
Temporal Correlations of the Running Maximum of a Brownian Trajectory
Bénichou, Olivier; Krapivsky, P. L.; Mejía-Monasterio, Carlos; Oshanin, Gleb
2016-08-01
We study the correlations between the maxima m and M of a Brownian motion (BM) on the time intervals [0 ,t1] and [0 ,t2], with t2>t1. We determine the exact forms of the distribution functions P (m ,M ) and P (G =M -m ), and calculate the moments E {(M-m ) k} and the cross-moments E {mlMk} with arbitrary integers l and k . We show that correlations between m and M decay as √{t1/t2 } when t2/t1→∞ , revealing strong memory effects in the statistics of the BM maxima. We also compute the Pearson correlation coefficient ρ (m ,M ) and the power spectrum of Mt, and we discuss a possibility of extracting the ensemble-averaged diffusion coefficient in single-trajectory experiments using a single realization of the maximum process.
Blowup and Conditionings of $\\psi$-super Brownian Exit Measures
Athreya, Siva R
2011-01-01
We extend earlier results on conditioning of super-Brownian motion to general branching rules. We obtain representations of the conditioned process, both as an $h$-transform, and as an unconditioned superprocess with immigration along a branching tree. Unlike the finite-variance branching setting, these trees are no longer binary, and strictly positive mass can be created at branch points. This construction is singular in the case of stable branching. We analyze this singularity first by approaching the stable branching function via analytic approximations. In this context the singularity of the stable case can be attributed to blow up of the mass created at the first branch of the tree. Other ways of approaching the stable case yield a branching tree that is different in law. To explain this anomaly we construct a family of martingales whose backbones have multiple limit laws.
Transient cluster formation in sheared non-Brownian suspensions.
Thøgersen, Kjetil; Dabrowski, Marcin; Malthe-Sørenssen, Anders
2016-02-01
We perform numerical simulations of non-Brownian suspensions in the laminar flow regime to study the scaling behavior of particle clusters and collisions under shear. As the particle fraction approaches the maximum packing fraction, large transient clusters appear in the system. We use methods from percolation theory to discuss the cluster size distribution. We also give a scaling relation for the percolation threshold as well as system size effects through time-dependent fluctuations of this threshold and relate them to system size. System size effects are important close to the maximum packing fraction due to the divergence of the cluster length scale. We then investigate the transient nature of the clusters through characterization of particle collisions and show that collision times exhibit scale-invariant properties. Finally, we show that particle collision times can be modeled as first-passage processes. PMID:26986381
Differential dynamic microscopy to characterize Brownian motion and bacteria motility
Germain, David; Leocmach, Mathieu; Gibaud, Thomas
2016-03-01
We have developed a lab module for undergraduate students, which involves the process of quantifying the dynamics of a suspension of microscopic particles using Differential Dynamic Microscopy (DDM). DDM is a relatively new technique that constitutes an alternative method to more classical techniques such as dynamic light scattering (DLS) or video particle tracking (VPT). The technique consists of imaging a particle dispersion with a standard light microscope and a camera and analyzing the images using a digital Fourier transform to obtain the intermediate scattering function, an autocorrelation function that characterizes the dynamics of the dispersion. We first illustrate DDM in the textbook case of colloids under Brownian motion, where we measure the diffusion coefficient. Then we show that DDM is a pertinent tool to characterize biological systems such as motile bacteria.
Properties of Brownian Image Models in Scale-Space
DEFF Research Database (Denmark)
Pedersen, Kim Steenstrup
2003-01-01
law that apparently governs natural images. Furthermore, the distribution of Brownian images mapped into jet space is Gaussian and an analytical expression can be derived for the covariance matrix of Brownian images in jet space. This matrix is also a good approximation of the covariance matrix...... Brownian images) will be discussed in relation to linear scale-space theory, and it will be shown empirically that the second order statistics of natural images mapped into jet space may, within some scale interval, be modeled by the Brownian image model. This is consistent with the 1/f 2 power spectrum...... of natural images in jet space. The consequence of these results is that the Brownian image model can be used as a least committed model of the covariance structure of the distribution of natural images....
Brownian Warps for Non-Rigid Registration
DEFF Research Database (Denmark)
Nielsen, Mads; Johansen, Peter; Jackson, Andrew D.;
2008-01-01
A Brownian motion model in the group of diffeomorphisms has been introduced as inducing a least committed prior on warps. This prior is source-destination symmetric, fulfills a natural semi-group property for warps, and with probability 1 creates invertible warps. Using this as a least committed ...... images, and show that the obtained warps are also in practice source-destination symmetric and in an example on X-ray spine registration provides extrapolations from landmark point superior to those of spline solutions. Udgivelsesdato: July......A Brownian motion model in the group of diffeomorphisms has been introduced as inducing a least committed prior on warps. This prior is source-destination symmetric, fulfills a natural semi-group property for warps, and with probability 1 creates invertible warps. Using this as a least committed...... prior, we formulate a Partial Differential Equation for obtaining the maximally likely warp given matching constraints derived from the images. We solve for the free boundary conditions, and the bias toward smaller areas in the finite domain setting. Furthermore, we demonstrate the technique on 2D...
Dynamics and Efficiency of Brownian Rotors
Bauer, Wolfgang R
2008-01-01
Brownian rotors play an important role in biological systems and in future nano-technological applications. However the mechanisms determining their dynamics, efficiency and performance remain to be characterized. Here the F0 portion of the F-ATP synthase is considered as a paradigm of a Brownian rotor. In a generic analytical model we analyze the stochastic rotation of F0-like motors as a function of the driving free energy difference and of the free energy profile the rotor is subjected to. The latter is composed of the rotor interaction with its surroundings, of the free energy of chemical transitions, and of the workload. The dynamics and mechanical efficiency of the rotor depends on the magnitude of its stochastic motion driven by the free energy energy difference and its rectification on the reaction-diffusion path. We analyze which free energy profiles provide maximum flow and how their arrangement on the underlying reaction-diffusion path affects rectification and -- by this -- the efficiency.
Vertices of the least concave majorant of Brownian motion with parabolic drift
Groeneboom, Piet
2010-01-01
It was shown in Groeneboom (1983) that the least concave majorant of one-sided Brownian motion without drift can be characterized by a jump process with independent increments, which is the inverse of the process of slopes of the least concave majorant. This result can be used to prove the result of Sparre Andersen (1954) that the number of vertices of the smallest concave majorant of the empirical distribution function of a sample of size n from the uniform distribution on [0,1] is asymptotically normal, with an asymptotic expectation and variance which are both of order log n. A similar (Markovian) inverse jump process was introduced in Groeneboom (1989), in an analysis of the least concave majorant of two-sided Brownian motion with a parabolic drift. This process is quite different from the process for one-sided Brownian motion without drift: the number of vertices in a (corresponding slopes) interval has an expectation proportional to the length of the interval and the variance of the number of vertices i...
Beyond multifractional Brownian motion: new stochastic models for geophysical modelling
Lévy Véhel, J.
2013-09-01
Multifractional Brownian motion (mBm) has proved to be a useful tool in various areas of geophysical modelling. Although a versatile model, mBm is of course not always an adequate one. We present in this work several other stochastic processes which could potentially be useful in geophysics. The first alternative type is that of self-regulating processes: these are models where the local regularity is a function of the amplitude, in contrast to mBm where it is tuned exogenously. We demonstrate the relevance of such models for digital elevation maps and for temperature records. We also briefly describe two other types of alternative processes, which are the counterparts of mBm and of self-regulating processes when the intensity of local jumps is considered in lieu of local regularity: multistable processes allow one to prescribe the local intensity of jumps in space/time, while this intensity is governed by the amplitude for self-stabilizing processes.
Dynamical objectivity in quantum Brownian motion
Tuziemski, J.; Korbicz, J. K.
2015-11-01
Classical objectivity as a property of quantum states —a view proposed to explain the observer-independent character of our world from quantum theory, is an important step in bridging the quantum-classical gap. It was recently derived in terms of spectrum broadcast structures for small objects embedded in noisy photon-like environments. However, two fundamental problems have arisen: a description of objective motion and applicability to other types of environments. Here we derive an example of objective states of motion in quantum mechanics by showing the formation of dynamical spectrum broadcast structures in the celebrated, realistic model of decoherence —Quantum Brownian Motion. We do it for realistic, thermal environments and show their noise-robustness. This opens a potentially new method of studying the quantum-to-classical transition.
Arithmetic area for m planar Brownian paths
International Nuclear Information System (INIS)
We pursue the analysis made in Desbois and Ouvry (2011 J. Stat. Mech. P05024) on the arithmetic area enclosed by m closed Brownian paths. We pay particular attention to the random variable Sn1,n2,...,nm(m), which is the arithmetic area of the set of points, also called winding sectors, enclosed n1 times by path 1, n2 times by path 2,..., and nm times by path m. Various results are obtained in the asymptotic limit m→∞. A key observation is that, since the paths are independent, one can use in the m-path case the SLE information, valid in the one-path case, on the zero-winding sectors arithmetic area
Arithmetic area for m planar Brownian paths
Desbois, Jean; Ouvry, Stéphane
2012-05-01
We pursue the analysis made in Desbois and Ouvry (2011 J. Stat. Mech. P05024) on the arithmetic area enclosed by m closed Brownian paths. We pay particular attention to the random variable Sn1, n2,..., nm(m), which is the arithmetic area of the set of points, also called winding sectors, enclosed n1 times by path 1, n2 times by path 2,..., and nm times by path m. Various results are obtained in the asymptotic limit m\\to \\infty . A key observation is that, since the paths are independent, one can use in the m-path case the SLE information, valid in the one-path case, on the zero-winding sectors arithmetic area.
Extreme fluctuations of active Brownian motion
Pietzonka, Patrick; Kleinbeck, Kevin; Seifert, Udo
2016-05-01
In active Brownian motion, an internal propulsion mechanism interacts with translational and rotational thermal noise and other internal fluctuations to produce directed motion. We derive the distribution of its extreme fluctuations and identify its universal properties using large deviation theory. The limits of slow and fast internal dynamics give rise to a kink-like and parabolic behavior of the corresponding rate functions, respectively. For dipolar Janus particles in two- and three-dimensions interacting with a field, we predict a novel symmetry akin to, but different from, the one related to entropy production. Measurements of these extreme fluctuations could thus be used to infer properties of the underlying, often hidden, network of states.
Hausdorff Dimension of Cut Points for Brownian Motion
Lawler, Gregory
1996-01-01
Let $B$ be a Brownian motion in $R^d$, $d=2,3$. A time $t\\in [0,1]$ is called a cut time for $B[0,1]$ if $B[0,t) \\cap B(t,1] = \\emptyset$. We show that the Hausdorff dimension of the set of cut times equals $1 - \\zeta$, where $\\zeta = \\zeta_d$ is the intersection exponent. The theorem, combined with known estimates on $\\zeta_3$, shows that the percolation dimension of Brownian motion (the minimal Hausdorff dimension of a subpath of a Brownian path) is strictly greater than one in $R^3$.
The Brownian Cactus I. Scaling limits of discrete cactuses
Curien, Nicolas; Miermont, Grégory
2011-01-01
The cactus of a pointed graph is a discrete tree associated with this graph. Similarly, with every pointed geodesic metric space $E$, one can associate an $\\R$-tree called the continuous cactus of $E$. We prove under general assumptions that the cactus of random planar maps distributed according to Boltzmann weights and conditioned to have a fixed large number of vertices converges in distribution to a limiting space called the Brownian cactus, in the Gromov-Hausdorff sense. Moreover, the Brownian cactus can be interpreted as the continuous cactus of the so-called Brownian map.
An excursion approach to maxima of the Brownian bridge
Perman, Mihael; Wellner, Jon A.
2016-01-01
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.
The exit distribution for iterated Brownian motion in cones
Banuelos, Rodrigo; DeBlassie, Dante
2004-01-01
We study the distribution of the exit place of iterated Brownian motion in a cone, obtaining information about the chance of the exit place having large magnitude. Along the way, we determine the joint distribution of the exit time and exit place of Brownian motion in a cone. This yields information on large values of the exit place (harmonic measure) for Brownian motion. The harmonic measure for cones has been studied by many authors for many years. Our results are sharper than any previousl...
Two-sided estimates on the density of Brownian motion with singular drift
Kim, Panki; Song, Renming
2006-01-01
Let μ = μ 1 ⋯ μ d be such that each μ i is a signed measure on \\R d belonging to the Kato class \\K d , 1 . The existence and uniqueness of a continuous Markov process X on \\R d , called a Brownian motion with drift μ , was recently established by Bass and Chen. In this paper we study the potential theory of X ...
Hybrid finite element and Brownian dynamics method for diffusion-controlled reactions
Bauler, Patricia; Huber, Gary A.; McCammon, J. Andrew
2012-01-01
Diffusion is often the rate determining step in many biological processes. Currently, the two main computational methods for studying diffusion are stochastic methods, such as Brownian dynamics, and continuum methods, such as the finite element method. This paper proposes a new hybrid diffusion method that couples the strengths of each of these two methods. The method is derived for a general multidimensional system, and is presented using a basic test case for 1D linear and radially symmetri...
Quantum Brownian motion in a bath of parametric oscillators: A model for system-field interactions
International Nuclear Information System (INIS)
The quantum Brownian motion paradigm provides a unified framework where one can see the interconnection of some basic quantum statistical processes such as decoherence, dissipation, particle creation, noise, and fluctuation. The present paper continues the investigation begun in earlier papers on the quantum Brownian motion in a general environment via the influence functional formalism. Here, the Brownian particle is coupled linearly to a bath of the most general time-dependent quadratic oscillators. This bath of parametric oscillators minics a scalar field, while the motion of the Brownian particle modeled by a single oscillator could be used to depict the behavior of a particle detector, a quantum field mode, or the scale factor of the Universe. An important result of this paper is the derivation of the influence functional encompassing the noise and dissipation kernels in terms of the Bogolubov coefficients, thus setting the stage for the influence functional formalism treatment of problems in quantum field theory in curved spacetime. This method enables one to trace the source of statistical processes such as decoherence and dissipation to vacuum fluctuations and particle creation, and in turn impart a statistical mechanical interpretation of quantum field processes. With this result we discuss the statistical mechanical origin of quantum noise and thermal radiance from black holes and from uniformly accelerated observers in Minkowski space as well as from the de Sitter universe discovered by Hawking, Unruh, and Gibbons and Hawking. We also derive the exact evolution operator and master equation for the reduced density matrix of the system interacting with a parametric oscillator bath in an initial squeezed thermal state. These results are useful for decoherence and back reaction studies for systems and processes of interest in semiclassical cosmology and gravity. Our model and results are also expected to be useful for related problems in quantum optics
Parameter Estimation for Generalized Brownian Motion with Autoregressive Increments
Fendick, Kerry
2011-01-01
This paper develops methods for estimating parameters for a generalization of Brownian motion with autoregressive increments called a Brownian ray with drift. We show that a superposition of Brownian rays with drift depends on three types of parameters - a drift coefficient, autoregressive coefficients, and volatility matrix elements, and we introduce methods for estimating each of these types of parameters using multidimensional times series data. We also cover parameter estimation in the contexts of two applications of Brownian rays in the financial sphere: queuing analysis and option valuation. For queuing analysis, we show how samples of queue lengths can be used to estimate the conditional expectation functions for the length of the queue and for increments in its net input and lost potential output. For option valuation, we show how the Black-Scholes-Merton formula depends on the price of the security on which the option is written through estimates not only of its volatility, but also of a coefficient ...
Random times and enlargements of filtrations in a Brownian setting
Mansuy, Roger
2006-01-01
In November 2004, M. Yor and R. Mansuy jointly gave six lectures at Columbia University, New York. These notes follow the contents of that course, covering expansion of filtration formulae; BDG inequalities up to any random time; martingales that vanish on the zero set of Brownian motion; the Azéma-Emery martingales and chaos representation; the filtration of truncated Brownian motion; attempts to characterize the Brownian filtration. The book accordingly sets out to acquaint its readers with the theory and main examples of enlargements of filtrations, of either the initial or the progressive kind. It is accessible to researchers and graduate students working in stochastic calculus and excursion theory, and more broadly to mathematicians acquainted with the basics of Brownian motion.
Holographic Brownian Motion in Two Dimensional Rotating Fluid
Atmaja, Ardian Nata
2012-01-01
The Brownian motion of a heavy quark under a rotating plasma corresponds to BTZ black hole is studied using holographic method from string theory. The heavy quark represented as the end of string at the boundary of BTZ black hole and the corresponding rotating plasma is two dimensional spacetime. The string fluctuation requires the angular velocity to be equal to the ratio between inner horizon and outer horizon, known as terminal velocity and also related to the zero total force condition. With this angular velocity, the string fluctuation solution has oscillatory modes in time and radial coordinates. We show the displacement square of this solution behaves as a Brownian particle in non-relativistic limit. For relativistic case, we argue that it is more appropriate to compute just the the leading order of low frequency limit of random-random force correlator. The Brownian motion relates this correlator with physical observables: mass of Brownian particle, friction coefficient and temperature of the plasma.
Directed transport of Brownian particles in a changing temperature field
Energy Technology Data Exchange (ETDEWEB)
Grillo, A [DMFCI, Facolta di Ingegneria, Universita di Catania. Viale Andrea Doria 6, 95125 Catania (Italy); Jinha, A [HPL-Faculty of Kinesiology, University of Calgary, 2500 University Drive NW, Calgary, Alberta, T2N 1N4 (Canada); Federico, S [HPL-Faculty of Kinesiology, University of Calgary, 2500 University Drive NW, Calgary, Alberta, T2N 1N4 (Canada); Ait-Haddou, R [HPL-Faculty of Kinesiology, University of Calgary, 2500 University Drive NW, Calgary, Alberta, T2N 1N4 (Canada); Herzog, W [HPL-Faculty of Kinesiology, University of Calgary, 2500 University Drive NW, Calgary, Alberta, T2N 1N4 (Canada); Giaquinta, G [DMFCI, Facolta di Ingegneria, Universita di Catania. Viale Andrea Doria 6, 95125 Catania (Italy)
2008-01-11
We study the interaction of Brownian particles with a changing temperature field in the presence of a one-dimensional periodic adiabatic potential. We show the existence of directed transport through the determination of the overall current of Brownian particles crossing the boundary of the system. With respect to the case of Brownian particles in a thermal bath, we determine a current which exhibits a contribution explicitly related to the presence of a thermal gradient. Beyond the self-consistent calculation of the temperature and probability density distribution of Brownian particles, we evaluate the energy consumption for directed transport to take place. Our description is based on Streater's model, and solutions are obtained by perturbing the system from its initial thermodynamic equilibrium state.
Directed transport of Brownian particles in a changing temperature field
International Nuclear Information System (INIS)
We study the interaction of Brownian particles with a changing temperature field in the presence of a one-dimensional periodic adiabatic potential. We show the existence of directed transport through the determination of the overall current of Brownian particles crossing the boundary of the system. With respect to the case of Brownian particles in a thermal bath, we determine a current which exhibits a contribution explicitly related to the presence of a thermal gradient. Beyond the self-consistent calculation of the temperature and probability density distribution of Brownian particles, we evaluate the energy consumption for directed transport to take place. Our description is based on Streater's model, and solutions are obtained by perturbing the system from its initial thermodynamic equilibrium state
Rotational Brownian Dynamics simulations of clathrin cage formation
International Nuclear Information System (INIS)
The self-assembly of nearly rigid proteins into ordered aggregates is well suited for modeling by the patchy particle approach. Patchy particles are traditionally simulated using Monte Carlo methods, to study the phase diagram, while Brownian Dynamics simulations would reveal insights into the assembly dynamics. However, Brownian Dynamics of rotating anisotropic particles gives rise to a number of complications not encountered in translational Brownian Dynamics. We thoroughly test the Rotational Brownian Dynamics scheme proposed by Naess and Elsgaeter [Macromol. Theory Simul. 13, 419 (2004); Naess and Elsgaeter Macromol. Theory Simul. 14, 300 (2005)], confirming its validity. We then apply the algorithm to simulate a patchy particle model of clathrin, a three-legged protein involved in vesicle production from lipid membranes during endocytosis. Using this algorithm we recover time scales for cage assembly comparable to those from experiments. We also briefly discuss the undulatory dynamics of the polyhedral cage
Symmetry Relations for Trajectories of a Brownian Motor
Astumian, R. Dean
2007-01-01
A Brownian Motor is a nanoscale or molecular device that combines the effects of thermal noise, spatial or temporal asymmetry, and directionless input energy to drive directed motion. Because of the input energy, Brownian motors function away from thermodynamic equilibrium and concepts such as linear response theory, fluctuation dissipation relations, and detailed balance do not apply. The {\\em generalized} fluctuation-dissipation relation, however, states that even under strongly thermodynam...
Fast simulation of Brownian dynamics in a crowded environment
Smith, Stephen; Grima, Ramon
2016-01-01
Brownian dynamics simulations are an increasingly popular tool for understanding spatially-distributed biochemical reaction systems. Recent improvements in our understanding of the cellular environment show that volume exclusion effects are fundamental to reaction networks inside cells. These systems are frequently studied by incorporating inert hard spheres (crowders) into three-dimensional Brownian dynamics simulations, however these methods are extremely slow owing to the sheer number of p...
Brownian motion and gambling: from ratchets to paradoxical games
Parrondo, J M R
2014-01-01
Two losing gambling games, when alternated in a periodic or random fashion, can produce a winning game. This paradox has been inspired by certain physical systems capable of rectifying fluctuations: the so-called Brownian ratchets. In this paper we review this paradox, from Brownian ratchets to the most recent studies on collective games, providing some intuitive explanations of the unexpected phenomena that we will find along the way.
On drift parameter estimation in models with fractional Brownian motion
Kozachenko, Yuriy; Mishura, Yuliya
2011-01-01
We consider a stochastic differential equation involving standard and fractional Brownian motion with unknown drift parameter to be estimated. We investigate the standard maximum likelihood estimate of the drift parameter, two non-standard estimates and three estimates for the sequential estimation. Model strong consistency and some other properties are proved. The linear model and Ornstein-Uhlenbeck model are studied in detail. As an auxiliary result, an asymptotic behavior of the fractional derivative of the fractional Brownian motion is established.
Brownian Motion on a Sphere: Distribution of Solid Angles
Krishna, M. M. G.; Samuel, Joseph; Sinha, Supurna
2000-01-01
We study the diffusion of Brownian particles on the surface of a sphere and compute the distribution of solid angles enclosed by the diffusing particles. This function describes the distribution of geometric phases in two state quantum systems (or polarised light) undergoing random evolution. Our results are also relevant to recent experiments which observe the Brownian motion of molecules on curved surfaces like micelles and biological membranes. Our theoretical analysis agrees well with the...
Bertoin, Jean; Yor, Marc
2012-01-01
We present a two-dimensional extension of an identity in distribution due to Bougerol \\cite{Bou} that involves the exponential functional of a linear Brownian motion. Even though this identity does not extend at the level of processes, we point at further striking relations in this direction.
Byczkowski, Tomasz; Chorowski, Jakub; Graczyk, Piotr; Malecki, Jacek
2011-01-01
The purpose of the paper is to provide integral representations of the Poisson kernel for a half-space and balls for hyperbolic Brownian motion and for the classical Ornstein-Uhlenbeck process. The method of proof is based on Girsanov's theorem and yields more complete results as those based on Feynmann-Kac technique.
On two-dimensional fractional Brownian motion and fractional Brownian random field
Qian, Hong; Raymond, Gary M.; Bassingthwaighte, James B.
1998-01-01
As a generalization of one-dimensional fractional Brownian motion (1dfBm), we introduce a class of two-dimensional, self-similar, strongly correlated random walks whose variance scales with power law N2H (0 < H < 1). We report analytical results on the statistical size and shape, and segment distribution of its trajectory in the limit of large N. The relevance of these results to polymer theory is discussed. We also study the basic properties of a second generalization of 1dfBm, the two-dimen...
Ideal bulk pressure of active Brownian particles
Speck, Thomas; Jack, Robert L.
2016-06-01
The extent to which active matter might be described by effective equilibrium concepts like temperature and pressure is currently being discussed intensely. Here, we study the simplest model, an ideal gas of noninteracting active Brownian particles. While the mechanical pressure exerted onto confining walls has been linked to correlations between particles' positions and their orientations, we show that these correlations are entirely controlled by boundary effects. We also consider a definition of local pressure, which describes interparticle forces in terms of momentum exchange between different regions of the system. We present three pieces of analytical evidence which indicate that such a local pressure exists, and we show that its bulk value differs from the mechanical pressure exerted on the walls of the system. We attribute this difference to the fact that the local pressure in the bulk does not depend on boundary effects, contrary to the mechanical pressure. We carefully examine these boundary effects using a channel geometry, and we show a virial formula for the pressure correctly predicts the mechanical pressure even in finite channels. However, this result no longer holds in more complex geometries, as exemplified for a channel that includes circular obstacles.
Brownian dipole rotator in alternating electric field
Rozenbaum, V. M.; Vovchenko, O. Ye.; Korochkova, T. Ye.
2008-06-01
The study addresses the azimuthal jumping motion of an adsorbed polar molecule in a periodic n -well potential under the action of an external alternating electric field. Starting from the perturbation theory of the Pauli equation with respect to the weak field intensity, explicit analytical expressions have been derived for the time dependence of the average dipole moment as well as the frequency dependences of polarizability and the average angular velocity, the three quantities exhibiting conspicuous stochastic resonance. As shown, unidirectional rotation can arise only provided simultaneous modulation of the minima and maxima of the potential by an external alternating field. For a symmetric potential of hindered rotation, the average angular velocity, if calculated by the second-order perturbation theory with respect to the field intensity, has a nonzero value only at n=2 , i.e., when two azimuthal wells specify a selected axis in the system. Particular consideration is given to the effect caused by the asymmetry of the two-well potential on the dielectric loss spectrum and other Brownian motion parameters. When the asymmetric potential in a system of dipole rotators arises from the average local fields induced by an orientational phase transition, the characteristics concerned show certain peculiarities which enable detection of the phase transition and determination of its parameters.
Brownian dipole rotator in alternating electric field.
Rozenbaum, V M; Vovchenko, O Ye; Korochkova, T Ye
2008-06-01
The study addresses the azimuthal jumping motion of an adsorbed polar molecule in a periodic n -well potential under the action of an external alternating electric field. Starting from the perturbation theory of the Pauli equation with respect to the weak field intensity, explicit analytical expressions have been derived for the time dependence of the average dipole moment as well as the frequency dependences of polarizability and the average angular velocity, the three quantities exhibiting conspicuous stochastic resonance. As shown, unidirectional rotation can arise only provided simultaneous modulation of the minima and maxima of the potential by an external alternating field. For a symmetric potential of hindered rotation, the average angular velocity, if calculated by the second-order perturbation theory with respect to the field intensity, has a nonzero value only at n=2 , i.e., when two azimuthal wells specify a selected axis in the system. Particular consideration is given to the effect caused by the asymmetry of the two-well potential on the dielectric loss spectrum and other Brownian motion parameters. When the asymmetric potential in a system of dipole rotators arises from the average local fields induced by an orientational phase transition, the characteristics concerned show certain peculiarities which enable detection of the phase transition and determination of its parameters. PMID:18643221
From Brownian motion to power of fluctuations
Directory of Open Access Journals (Sweden)
B. Berche
2012-12-01
Full Text Available The year 2012 marks the 140th birth anniversary of Marian Smoluchowski (28.05.1872-5.09.1917, a man who "made ground-breaking contribution to the theory of Brownian motion, the theory of sedimentation, the statistical nature of the Second Law, the theory and practice of density fluctuations (critical opalescence. During his final years of scientific creativity his pioneering theory of coagulation and diffusion-limited reaction rate appeared. These outstanding achievements present true gems which dominate the description of soft matter physics and chemical physics as well as the related areas up till now!" This quotation was taken from the lecture by Peter Hanggi given at international conference Statistical Physics: Modern Trends and Applications that took place in Lviv, Ukraine on July 3-6, 2012 (see conference web-page for more details and was dedicated to the commemoration of Smoluchowski's work. This and forthcoming issues of the Condensed Matter Physics contain papers presented at this conference.
From Brownian Dynamics to Markov Chain: An Ion Channel Example
Chen, Wan
2014-02-27
A discrete rate theory for multi-ion channels is presented, in which the continuous dynamics of ion diffusion is reduced to transitions between Markovian discrete states. In an open channel, the ion permeation process involves three types of events: an ion entering the channel, an ion escaping from the channel, or an ion hopping between different energy minima in the channel. The continuous dynamics leads to a hierarchy of Fokker-Planck equations, indexed by channel occupancy. From these the mean escape times and splitting probabilities (denoting from which side an ion has escaped) can be calculated. By equating these with the corresponding expressions from the Markov model, one can determine the Markovian transition rates. The theory is illustrated with a two-ion one-well channel. The stationary probability of states is compared with that from both Brownian dynamics simulation and the hierarchical Fokker-Planck equations. The conductivity of the channel is also studied, and the optimal geometry maximizing ion flux is computed. © 2014 Society for Industrial and Applied Mathematics.
Studying protein assembly with reversible Brownian dynamics of patchy particles
International Nuclear Information System (INIS)
Assembly of protein complexes like virus shells, the centriole, the nuclear pore complex, or the actin cytoskeleton is strongly determined by their spatial structure. Moreover, it is becoming increasingly clear that the reversible nature of protein assembly is also an essential element for their biological function. Here we introduce a computational approach for the Brownian dynamics of patchy particles with anisotropic assemblies and fully reversible reactions. Different particles stochastically associate and dissociate with microscopic reaction rates depending on their relative spatial positions. The translational and rotational diffusive properties of all protein complexes are evaluated on-the-fly. Because we focus on reversible assembly, we introduce a scheme which ensures detailed balance for patchy particles. We then show how the macroscopic rates follow from the microscopic ones. As an instructive example, we study the assembly of a pentameric ring structure, for which we find excellent agreement between simulation results and a macroscopic kinetic description without any adjustable parameters. This demonstrates that our approach correctly accounts for both the diffusive and reactive processes involved in protein assembly
Modelling Collective Opinion Formation by Means of Active Brownian Particles
Schweitzer, F; Schweitzer, Frank; Holyst, Janusz
1999-01-01
The concept of active Brownian particles is used to model a collective opinion formation process. It is assumed that individuals in community create a two-component communication field that influences the change of opinions of other persons and/or can induce their migration. The communication field is described by a reaction-diffusion equation, meaning that it has a certain lifetime, which models memory effects, further it can spread out in the community. Within our stochastic approach, the opinion change of the individuals is described by a master equation, while the migration is described by a set of Langevin equations, coupled by the communication field. In the mean-field limit which holds for fast communication, we derive a critical population size, above which the community separates into a majority and a minority with opposite opinions. The existence of external support (e.g. from mass media) can change the ratio between minority and majority, until above a critical external support the supported subpop...
Nanofluidic Brownian Ratchet via atomically-stepped surfaces
Rahmani, Amir; Colosqui, Carlos
2015-11-01
Theoretical analysis and fully atomistic molecular dynamics simulations reveal a Brownian ratchet mechanism by which thermal motion can drive the directional displacement of liquids confined in micro- or nanoscale channels and pores. The particular systems discussed in this talk consist of two immiscible liquids confined in a slit-like nanochannel with atomically-stepped surfaces. Mean displacement rates reported in molecular dynamics simulations are in close agreement with theoretical predictions via analytical solution of a Smoluchowski equation for the probability density of the position of the liquid-liquid interface. The direction of the thermally-driven displacement of liquid is determined by the nanostructure surface geometry and thus imbibition or drainage can occur against the direction of action of capillary forces. The studied surface nanostructure with directional asymmetry can control the dynamics of wetting processes such as capillary filling, wicking, and imbibition in porous materials. The proposed physical mechanisms and derived analytical expressions can be applied to design nanofluidic and microfluidic devices for passive handling and separation.
Effect of interfaces on the nearby Brownian motion
Huang, Kai
2016-01-01
Near-boundary Brownian motion is a classic hydrodynamic problem of great importance in a variety of fields, from biophysics to micro-/nanofluidics. However, due to challenges in experimental measurements of near-boundary dynamics, the effect of interfaces on Brownian motion has remained elusive. Here, we report a computational study of this effect using microsecond-long large-scale molecular dynamics simulations and our newly developed Green-Kubo relation for friction at the liquid-solid interface. Our computer experiment unambiguously reveals that the t^(-3/2) long-time decay of the velocity autocorrelation function of a Brownian particle in bulk liquid is replaced by a t^(-5/2) decay near a boundary. We discover a general breakdown of traditional no-slip boundary condition at short time scales and we show that this breakdown has a profound impact on the near-boundary Brownian motion. Our results demonstrate the potential of Brownian-particle based micro-/nano-sonar to probe the local wettability of liquid-s...
Brownian Dynamics of charged particles in a constant magnetic field
Hou, L J; Piel, A; Shukla, P K
2009-01-01
Numerical algorithms are proposed for simulating the Brownian dynamics of charged particles in an external magnetic field, taking into account the Brownian motion of charged particles, damping effect and the effect of magnetic field self-consistently. Performance of these algorithms is tested in terms of their accuracy and long-time stability by using a three-dimensional Brownian oscillator model with constant magnetic field. Step-by-step recipes for implementing these algorithms are given in detail. It is expected that these algorithms can be directly used to study particle dynamics in various dispersed systems in the presence of a magnetic field, including polymer solutions, colloidal suspensions and, particularly complex (dusty) plasmas. The proposed algorithms can also be used as thermostat in the usual molecular dynamics simulation in the presence of magnetic field.
Brownian dynamics simulations with hard-body interactions: Spherical particles
Behringer, Hans; 10.1063/1.4761827
2012-01-01
A novel approach to account for hard-body interactions in (overdamped) Brownian dynamics simulations is proposed for systems with non-vanishing force fields. The scheme exploits the analytically known transition probability for a Brownian particle on a one-dimensional half-line. The motion of a Brownian particle is decomposed into a component that is affected by hard-body interactions and into components that are unaffected. The hard-body interactions are incorporated by replacing the affected component of motion by the evolution on a half-line. It is discussed under which circumstances this approach is justified. In particular, the algorithm is developed and formulated for systems with space-fixed obstacles and for systems comprising spherical particles. The validity and justification of the algorithm is investigated numerically by looking at exemplary model systems of soft matter, namely at colloids in flow fields and at protein interactions. Furthermore, a thorough discussion of properties of other heurist...
Quantum Brownian motion model for the stock market
Meng, Xiangyi; Zhang, Jian-Wei; Guo, Hong
2016-06-01
It is believed by the majority today that the efficient market hypothesis is imperfect because of market irrationality. Using the physical concepts and mathematical structures of quantum mechanics, we construct an econophysical framework for the stock market, based on which we analogously map massive numbers of single stocks into a reservoir consisting of many quantum harmonic oscillators and their stock index into a typical quantum open system-a quantum Brownian particle. In particular, the irrationality of stock transactions is quantitatively considered as the Planck constant within Heisenberg's uncertainty relationship of quantum mechanics in an analogous manner. We analyze real stock data of Shanghai Stock Exchange of China and investigate fat-tail phenomena and non-Markovian behaviors of the stock index with the assistance of the quantum Brownian motion model, thereby interpreting and studying the limitations of the classical Brownian motion model for the efficient market hypothesis from a new perspective of quantum open system dynamics.
How superdiffusion gets arrested: ecological encounters explain shift from Levy to Brownian movement
de Jager, M.; Bartumeus, F.; Kölzsch, A.; Weissing, F.J.; Hengeveld, G.M.; Nolet, B.A.; Herman, P.M.J.; de Koppel, J.
2014-01-01
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 fora
Direct observation of ballistic Brownian motion on a single particle
Huang, Rongxin; Lukic, Branimir; Jeney, Sylvia; Florin, Ernst-Ludwig
2010-01-01
At fast timescales, the self-similarity of random Brownian motion is expected to break down and be replaced by ballistic motion. So far, an experimental verification of this prediction has been out of reach due to a lack of instrumentation fast and precise enough to capture this motion. With a newly developed detector, we have been able to observe the Brownian motion of a single particle in an optical trap with 75 MHz bandwidth and sub-{AA}ngstrom spatial precision. We report the first measur...
Energy and efficiency optimization of a Brownian heat engine
Bekele, Mulugeta; Yalew, Yeneneh
2007-03-01
A simple Brownian heat engine is modeled as a Brownian particle moving in an external sawtooth potential (with or without) load assisted by the thermal kick it gets from alternately placed hot and cold heat reservoirs along its path. We get closed form expression for its current in terms of the parameters characterizing the model. After analyzing the way it consumes energy to do useful work, we also get closed form expressions for its efficiency as well as for its coefficient of performance when the engine performs as a refrigerator. Recently suggested optimization criteria enables us to exhaustively explore and compare the different operating conditions of the engine.
The dimension of the Brownian frontier is greater than 1
Bishop, Christopher J.; Jones, Peter; Pemantle, Robin; Peres, Yuval
1995-01-01
Consider a planar Brownian motion run for finite time. The frontier or ``outer boundary'' of the path is the boundary of the unbounded component of the complement. Burdzy (1989) showed that the frontier has infinite length. We improve this by showing that the Hausdorff dimension of the frontier is strictly greater than 1. (It has been conjectured that the Brownian frontier has dimension $4/3$, but this is still open.) The proof uses Jones's Traveling Salesman Theorem and a self-similar tiling...
The Dimension of the Frontier of Planar Brownian Motion
Lawler, Gregory
1996-01-01
Let $B$ be a two dimensional Brownian motion and let the frontier of $B[0,1]$ be defined as the set of all points in $B[0,1]$ that are in the closure of the unbounded connected component of its complement. We prove that the Hausdorff dimension of the frontier equals $2(1 - \\alpha)$ where $\\alpha$ is an exponent for Brownian motion called the two-sided disconnection exponent. In particular, using an estimate on $\\alpha$ due to Werner, the Hausdorff dimension is greater than $1.015$.
Winding statistics of a Brownian particle on a ring
International Nuclear Information System (INIS)
We consider a Brownian particle moving on a ring. We study the probability distributions of the total number of turns and the net number of counter-clockwise turns the particle makes until time t. Using a method based on the renewal properties of a Brownian walker, we find exact analytical expressions of these distributions. This method serves as an alternative to the standard path integral techniques which are not always easily adaptable for certain observables. For large t, we show that these distributions have Gaussian scaling forms. We also compute large deviation functions associated to these distributions characterizing atypically large fluctuations. We provide numerical simulations in support of our analytical results. (paper)
Fractional Brownian Motion and Sheet as White Noise Functionals
Institute of Scientific and Technical Information of China (English)
Zhi Yuan HUANG; Chu Jin LI; Jian Ping WAN; Ying WU
2006-01-01
In this short note, we show that it is more natural to look the fractional Brownian motion as functionals of the standard white noises, and the fractional white noise calculus developed by Hu and (φ)ksendal follows directly from the classical white noise functional calculus. As examples we prove that the fractional Girsanov formula, the Ito type integrals and the fractional Black-Scholes formula are easy consequences of their classical counterparts. An extension to the fractional Brownian sheet is also briefly discussed.
DNA transport by a micromachined Brownian ratchet device
Bader, J S; Henck, S A; Deem, M W; McDermott, G A; Bustillo, J M; Simpson, J W; Mulhern, G T; Rothberg, J M; Bader, Joel S; Hammond, Richard W.; Henck, Steven A.; Deem, Michael W.; Dermott, Gregory A. Mc; Bustillo, James M.; Simpson, John W.; Mulhern, Gregory T.; Rothberg, Jonathan M.
1999-01-01
We have micromachined a silicon-chip device that transports DNA with aBrownian ratchet that rectifies the Brownian motion of microscopic particles.Transport properties for a DNA 50mer agree with theoretical predictions, andthe DNA diffusion constant agrees with previous experiments. This type ofmicromachine could provide a generic pump or separation component for DNA orother charged species as part of a microscale lab-on-a-chip. A device withreduced feature size could produce a size-based separation of DNA molecules,with applications including the detection of single nucleotide polymorphisms.
Stability theorems for stochastic differential equations driven by G-Brownian motion
Zhang, Defei
2011-01-01
In this paper, stability theorems for stochastic differential equations and backward stochastic differential equations driven by G-Brownian motion are obtained. We show the existence and uniqueness of solutions to forward-backward stochastic differential equations driven by G-Brownian motion. Stability theorem for forward-backward stochastic differential equations driven by G-Brownian motion is also presented.
Noise-to-signal transition of a Brownian particle in the cubic potential: I. general theory
Filip, Radim; Zemánek, Pavel
2016-06-01
The noise-to-signal transitions are very interesting processes in physics, as they might transform environmental noise to useful mechanical effects. We theoretically analyze stochastic noise-to-signal transition of overdamped Brownian motion of a particle in the cubic potential. The particle reaches thermal equilibrium with its environment in the quadratic potential which is suddenly swapped to the cubic potential. We predict a simultaneous increase of both the displacement and signal-to-noise ratio in the cubic potential for the position linearly powered by the temperature of the particle environment. The short-time analysis and numerical simulations fully confirm different dynamical regimes of this noise-to-signal transition.
The Statistics of Burgers Turbulence Initialized with Fractional Brownian Noise Data
Ryan, Reade
The statistics of the solution to the inviscid Burgers equation are investigated when the initial velocity potential is fractional Brownian motion. Using the theory of large deviations for Gaussian processes, we characterize the tails of the probability distribution functions (PDFs) of the velocity, the distance between shocks, and the shock strength. These PDFs are shown to decay like ``stretched'' exponentials of the form . Our method of proof can also be used to extend these results to a much larger class of Gaussian potentials. This work generalizes the results of Avellaneda and E [2, 3] on the inviscid Burgers equation with white-noise initial data.
On the theory of Brownian motion with the Alder-Wainwright effect
Okabe, Yasunori
1986-12-01
The Stokes-Boussinesq-Langevin equation, which describes the time evolution of Brownian motion with the Alder-Wainwright effect, can be treated in the framework of the theory of KMO-Langevin equations which describe the time evolution of a real, stationary Gaussian process with T-positivity (reflection positivity) originating in axiomatic quantum field theory. After proving the fluctuation-dissipation theorems for KMO-Langevin equations, we obtain an explicit formula for the deviation from the classical Einstein relation that occurs in the Stokes-Boussinesq-Langevin equation with a white noise as its random force. We are interested in whether or not it can be measured experimentally.
On Drift Parameter Estimation in Models with Fractional Brownian Motion by Discrete Observations
Directory of Open Access Journals (Sweden)
Yuliya Mishura
2014-06-01
Full Text Available We study a problem of an unknown drift parameter estimation in a stochastic differen- tial equation driven by fractional Brownian motion. We represent the likelihood ratio as a function of the observable process. The form of this representation is in general rather complicated. However, in the simplest case it can be simplified and we can discretize it to establish the a. s. convergence of the discretized version of maximum likelihood estimator to the true value of parameter. We also investigate a non-standard estimator of the drift parameter showing further its strong consistency.
Statistical Inference for Time-changed Brownian Motion Credit Risk Models
T. R. Hurd; Zhuowei Zhou
2011-01-01
We consider structural credit modeling in the important special case where the log-leverage ratio of the firm is a time-changed Brownian motion (TCBM) with the time-change taken to be an independent increasing process. Following the approach of Black and Cox, one defines the time of default to be the first passage time for the log-leverage ratio to cross the level zero. Rather than adopt the classical notion of first passage, with its associated numerical challenges, we accept an alternative ...
Integration formula for Brownian motion on classical compact Lie groups
Dahlqvist, Antoine
2012-01-01
We give new formulas for the moments of the entries of a random matrix whose law is a Brownian motion on a classical compact Lie group. As we let the time go to infinity, these formulas imply the one B. Collins and P. Sniady obtained for Haar measure.
Entropy production of a Brownian ellipsoid in the overdamped limit
Marino, Raffaele; Eichhorn, Ralf; Aurell, Erik
2016-01-01
We analyze the translational and rotational motion of an ellipsoidal Brownian particle from the viewpoint of stochastic thermodynamics. The particle's Brownian motion is driven by external forces and torques and takes place in an heterogeneous thermal environment where friction coefficients and (local) temperature depend on space and time. Our analysis of the particle's stochastic thermodynamics is based on the entropy production associated with single particle trajectories. It is motivated by the recent discovery that the overdamped limit of vanishing inertia effects (as compared to viscous fricion) produces a so-called "anomalous" contribution to the entropy production, which has no counterpart in the overdamped approximation, when inertia effects are simply discarded. Here we show that rotational Brownian motion in the overdamped limit generates an additional contribution to the "anomalous" entropy. We calculate its specific form by performing a systematic singular perturbation analysis for the generating function of the entropy production. As a side result, we also obtain the (well-known) equations of motion in the overdamped limit. We furthermore investigate the effects of particle shape and give explicit expressions of the "anomalous entropy" for prolate and oblate spheroids and for near-spherical Brownian particles.
Option Pricing in a Fractional Brownian Motion Environment
Cipian Necula
2008-01-01
The purpose of this paper is to obtain a fractional Black-Scholes formula for the price of an option for every t in [0,T], a fractional Black-Scholes equation and a risk-neutral valuation theorem if the underlying is driven by a fractional Brownian motion BH (t), 1/2
The Inviscid Burgers Equation with Brownian Initial Velocity
Bertoin, Jean
The law of the (Hopf-Cole) solution of the inviscid Burgers equation with Brownian initial velocity is made explicit. As examples of applications, we investigate the smoothness of the solution, the statistical distribution of the shocks, we determine the exact Hausdorff function of the Lagrangian regular points and investigate the existence of Lagrangian regular points in a fixed Borel set.
Brownian molecular rotors: Theoretical design principles and predicted realizations
Schönborn, Jan Boyke; Herges, Rainer; Hartke, Bernd
2009-01-01
We propose simple design concepts for molecular rotors driven by Brownian motion and external photochemical switching. Unidirectionality and efﬁciency of the motion is measured by explicit simulations. Two different molecular scaffolds are shown to yield viable molecular rotors when decorated with suitable substituents.
Asset pricing puzzles explained by incomplete Brownian equilibria
DEFF Research Database (Denmark)
Christensen, Peter Ove; Larsen, Kasper
We examine a class of Brownian based models which produce tractable incomplete equilibria. The models are based on finitely many investors with heterogeneous exponential utilities over intermediate consumption who receive partially unspanned income. The investors can trade continuously on a finit...... markets. Consequently, our model can simultaneously help explaining the risk-free rate and equity premium puzzles....
Brownian motion, geometry, and generalizations of Picard's little theorem
Goldberg, S. I.; Mueller, C.
1982-01-01
Brownian motion is introduced as a tool in Riemannian geometry to show how useful it is in the function theory of manifolds, as well as the study of maps between manifolds. As applications, a generalization of Picard's little theorem, and a version of it for Riemann surfaces of large genus are given.
Brownian colloidal particles: Ito, Stratonovich, or a different stochastic interpretation
Sancho, J. M.
2011-12-01
Recent experiments on Brownian colloidal particles have been studied theoretically in terms of overdamped Langevin equations with multiplicative white noise using an unconventional stochastic interpretation. Complementary numerical simulations of the same system are well described using the conventional Stratonovich interpretation. Here we address this dichotomy from a more generic starting point: the underdamped Langevin equation and its corresponding Fokker-Planck equation.
On the Generalized Brownian Motion and its Applications in Finance
DEFF Research Database (Denmark)
Høg, Esben; Frederiksen, Per; Schiemert, Daniel
This paper deals with dynamic term structure models (DTSMs) and proposes a new way to handle the limitation of the classical affine models. In particular, the paper expands the exibility of the DTSMs by applying generalized Brownian motions with dependent increments as the governing force of the ...
Fuzzy Itand#244; Integral Driven by a Fuzzy Brownian Motion
Didier Kumwimba Seya; Rostin Mabela Makengo; Marcel Rémon; Walo Omana Rebecca
2015-01-01
In this paper we take into account the fuzzy stochastic integral driven by fuzzy Brownian motion. To define the metric between two fuzzy numbers and to take into account the limit of a sequence of fuzzy numbers, we invoke the Hausdorff metric. First this fuzzy stochastic integral is constructed for fuzzy simple stochastic functions, then the construction is done for fuzzy stochastic integrable functions.
Diffusion of Particle in Hyaluronan Solution, a Brownian Dynamics Simulation
Takasu, Masako; Tomita, Jungo
2004-04-01
Diffusion of a particle in hyaluronan solution is investigated using Brownian dynamics simulation. The slowing down of diffusion is observed, in accordance with the experimental results. The temperature dependence of the diffusion is calculated, and a turnover is obtained when the temperature is increased.
ABSOLUTE CONTINUITY FOR INTERACTING MEASURE-VALUED BRANCHING BROWNIAN MOTIONS
Institute of Scientific and Technical Information of China (English)
ZHAOXUELEI
1997-01-01
The moments and absohite continuity of measure-valued branching Brownian motions with bounded interacting intensity are hivestigated. An estimate of higher order moments is obtained. The ahsolute continuity is verified in the one dimension case. This therehy verifies the conjecture of Méléard and Roelly in [5].
Finite time extinction of super-Brownian motions with catalysts
Dawson, Donald A.; Fleischmann, Klaus; Mueller, Carl
1998-01-01
Consider a catalytic super-Brownian motion $X=X^\\Gamma$ with finite variance branching. Here `catalytic' means that branching of the reactant $X$ is only possible in the presence of some catalyst. Our intrinsic example of a catalyst is a stable random measure $\\Gamma $ on $R$ of index $0< gamma
The valuation of currency options by fractional Brownian motion.
Shokrollahi, Foad; Kılıçman, Adem
2016-01-01
This research aims to investigate a model for pricing of currency options in which value governed by the fractional Brownian motion model (FBM). The fractional partial differential equation and some Greeks are also obtained. In addition, some properties of our pricing formula and simulation studies are presented, which demonstrate that the FBM model is easy to use. PMID:27504243
SOME GEOMETRIC PROPERTIES OF BROWNIAN MOTION ON SIERPINSKI GASKET
Institute of Scientific and Technical Information of China (English)
WUJUN; XIAOYIMIN
1995-01-01
Let {X(t),t≥0} be Brownian motion on Sierpinski gasket,The Hausdorff and packing dimensions of the image of a ompact set are studied,The uniform Hausdorff and packing dimensions of the inverse image are also discussed.
Rotational Brownian Motion on Sphere Surface and Rotational Relaxation
Institute of Scientific and Technical Information of China (English)
Ekrem Aydner
2006-01-01
The spatial components of the autocorrelation function of noninteracting dipoles are analytically obtained in terms of rotational Brownian motion on the surface of a unit sphere using multi-level jumping formalism based on Debye's rotational relaxation model, and the rotational relaxation functions are evaluated.
Occupation times distribution for Brownian motion on graphs
Desbois, J
2002-01-01
Considering a Brownian motion on a general graph, we study the joint law for the occupation times on all the bonds. In particular, we show that the Laplace transform of this distribution can be expressed as the ratio of two determinants. We give two formulations, with arc or vertex matrices, for this result and discuss a simple example. (letter to the editor)
Occupation times distribution for Brownian motion on graphs
International Nuclear Information System (INIS)
Considering a Brownian motion on a general graph, we study the joint law for the occupation times on all the bonds. In particular, we show that the Laplace transform of this distribution can be expressed as the ratio of two determinants. We give two formulations, with arc or vertex matrices, for this result and discuss a simple example. (letter to the editor)
Occupation times distribution for Brownian motion on graphs
Energy Technology Data Exchange (ETDEWEB)
Desbois, Jean [Laboratoire de Physique Theorique et Modeles Statistiques, Universite Paris-Sud, Bat. 100, F-91405 Orsay (France)
2002-11-22
Considering a Brownian motion on a general graph, we study the joint law for the occupation times on all the bonds. In particular, we show that the Laplace transform of this distribution can be expressed as the ratio of two determinants. We give two formulations, with arc or vertex matrices, for this result and discuss a simple example. (letter to the editor)
Brownian motion and the parabolicity of minimal graphs
Neel, Robert W.
2008-01-01
We prove that minimal graphs (other than planes) are parabolic in the sense that any bounded harmonic function is determined by its boundary values. The proof relies on using the coupling introduced in the author's earlier paper "A martingale approach to minimal surfaces" to show that Brownian motion on such a minimal graph almost surely strikes the boundary in finite time.
A generalized Brownian motion model for turbulent relative particle dispersion
Shivamoggi, B. K.
2016-08-01
There is speculation that the difficulty in obtaining an extended range with Richardson-Obukhov scaling in both laboratory experiments and numerical simulations is due to the finiteness of the flow Reynolds number Re in these situations. In this paper, a generalized Brownian motion model has been applied to describe the relative particle dispersion problem in more realistic turbulent flows and to shed some light on this issue. The fluctuating pressure forces acting on a fluid particle are taken to be a colored noise and follow a stationary process and are described by the Uhlenbeck-Ornstein model while it appears plausible to take their correlation time to have a power-law dependence on Re, thus introducing a bridge between the Lagrangian quantities and the Eulerian parameters for this problem. This ansatz is in qualitative agreement with the possibility of a connection speculated earlier by Corrsin [26] between the white-noise representation for the fluctuating pressure forces and the large-Re assumption in the Kolmogorov [4] theory for the 3D fully developed turbulence (FDT) as well as a similar argument of Monin and Yaglom [23] and a similar result of Sawford [13] and Borgas and Sawford [24]. It also provides an insight into the result that the Richardson-Obukhov scaling holds only in the infinite-Re limit and disappears otherwise. This ansatz further provides a determination of the Richardson-Obukhov constant g as a function of Re, with an asymptotic constant value in the infinite-Re limit. It is shown to lead to full agreement, in the small-Re limit as well, with the Batchelor-Townsend [27] scaling for the rate of change of the mean square interparticle separation in 3D FDT, hence validating its soundness further.
International Nuclear Information System (INIS)
Simultaneous orthokinetic and perikinetic coagulations (SOPCs) are studied for small and large Peclet numbers (Pe) using Brownian dynamics simulation. The results demonstrate that the contributions of the Brownian motion and the shear flow to the overall coagulation rate are basically not additive. At the early stages of coagulation with small Peclet numbers, the ratio of overall coagulation rate to the rate of pure perikinetic coagulation is proportional to Pe1/2, while with high Peclet numbers, the ratio of overall coagulation rate to the rate of pure orthokinetic coagulation is proportional to Pe−1/2. Moreover, our results show that the aggregation rate generally changes with time for the SOPC, which is different from that for pure perikinetic and pure orthokinetic coagulations. By comparing the SOPC with pure perikinetic and pure orthokinetic coagulations, we show that the redistribution of particles due to Brownian motion can play a very important role in the SOPC. In addition, the effects of redistribution in the directions perpendicular and parallel to the shear flow direction are different. This perspective explains the behavior of coagulation due to the joint effects of the Brownian motion (perikinetic) and the fluid motion (orthokinetic). (electromagnetism, optics, acoustics, heat transfer, classical mechanics, and fluid dynamics)
双分数布朗运动下再装期权定价模型%Reload option pricing model in bi-fractional Brownian motion environment
Institute of Scientific and Technical Information of China (English)
薛红; 吴江增
2015-01-01
Underlying asset process follows the stochastic differential equation driven by bi-fractional Brownian motion.The financial market mathematical model is built by the stochas-tic analysis for bi-fractional Brownian motion.Using the actuarial approach, the pricingfor-mula of reload option in bi-fractional Brownian motion environment is obtained.%在标的资产服从双分数布朗运动驱动的随机微分方程,借助双分数布朗运动随机分析理论,建立双分数布朗运动环境下金融市场数学模型,运用保险精算方法,得到了双分数布朗运动环境下再装期权定价公式.
Application of GPU processing for Brownian particle simulation
Cheng, Way Lee; Sheharyar, Ali; Sadr, Reza; Bouhali, Othmane
2015-01-01
Reports on the anomalous thermal-fluid properties of nanofluids (dilute suspension of nano-particles in a base fluid) have been the subject of attention for 15 years. The underlying physics that govern nanofluid behavior, however, is not fully understood and is a subject of much dispute. The interactions between the suspended particles and the base fluid have been cited as a major contributor to the improvement in heat transfer reported in the literature. Numerical simulations are instrumental in studying the behavior of nanofluids. However, such simulations can be computationally intensive due to the small dimensions and complexity of these problems. In this study, a simplified computational approach for isothermal nanofluid simulations was applied, and simulations were conducted using both conventional CPU and parallel GPU implementations. The GPU implementations significantly improved the computational performance, in terms of the simulation time, by a factor of 1000-2500. The results of this investigation show that, as the computational load increases, the simulation efficiency approaches a constant. At a very high computational load, the amount of improvement may even decrease due to limited system memory.
D'Auria, Bernardo
2011-01-01
In this paper we study a reflected Markov-modulated Brownian motion with a two sided reflection in which the drift, diffusion coefficient and the two boundaries are (jointly) modulated by a finite state space irreducible continuous time Markov chain. The goal is to compute the stationary distribution of this Markov process, which in addition to the complication of having a stochastic boundary can also include jumps at state change epochs of the underlying Markov chain because of the boundary changes. We give the general theory and then specialize to the case where the underlying Markov chain has two states. Moreover, motivated by an application of optimal dividend strategies, we consider the case where the lower barrier is zero and the upper barrier is subject to control. In this case we generalized earlier results from the case of a reflected Brownian motion to the Markov modulated case.