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 ...
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...
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.
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
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.
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.
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.
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...
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.
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.
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...
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...
Brownian Motion in Minkowski Space
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 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...
Operator Fractional Brownian Motion and Martingale Differences
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.
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 $\\hbar$.
Dimensional Properties of Fractional Brownian Motion
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.
Brownian motion meets Riemann curvature
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
Combinatorial fractal Brownian motion model
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.
The open quantum Brownian motions
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)
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
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.
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.
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.
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...
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.
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...
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.
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.
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
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....
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.
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...
Frustrated Brownian Motion of Nonlocal Solitary Waves
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.
From Lagrangian to Brownian motion
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
Intrinsic and extrinsic measurement for Brownian motion
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)
Speckle Patterns and 2-Dimensional Brownian Motion
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)
The Stepping Motion of Brownian Particle Derived by Nonequilibrium Fluctuation
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.
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.
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.
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.
The relativistic Brownian motion: Interdisciplinary applications
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.
Langevin theory of anomalous Brownian motion made simple
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.
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.
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…
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)
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.
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,...
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...
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
From Brownian motion to power of fluctuations
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.
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.
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.
QUANTUM STOCHASTIC PROCESSES: BOSON AND FERMION BROWNIAN MOTION
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.
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 $R^3$.
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...
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.
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...
RESEARCH NOTES On the support of super-Brownian motion with super-Brownian immigration
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 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
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.
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 ...
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.
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 $1.015$.
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...
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
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.
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.
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 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.
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.
On the Generalized Brownian Motion and its Applications in Finance
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.
ABSOLUTE CONTINUITY FOR INTERACTING MEASURE-VALUED BRANCHING BROWNIAN MOTIONS
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
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
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
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
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.
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.
Fractional Brownian Motion and Sheet as White Noise Functionals
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.
Anomalous Brownian motion and viscoelasticity of the ear's mechanoelectrical transducer
Andor-Ardó, Daniel; Kozlov, Andrei; Hudspeth, A. J.
2009-03-01
The Brownian motion of a particle in a complex environment is known to display anomalous power-law scaling in which the mean squared displacement is proportional to a fractional power of time. Using laser interferometry and analytical methods of microrheology, we examine nanometer-scale thermal motions of hair bundles in the internal ear and show that these cellular organelles undergo fractional Brownian motion. This anomalous scaling is caused by viscoelasticity of the gating springs, elements that transmit energy in a sound to the mechanosensitive ion channels. These results demonstrate a connection between rheology and auditory physiology, and indicate that statistical properties of the thermal noise in the ear can be determined by dynamics of a small number of key molecules.
Human behavioral regularity, fractional Brownian motion, and exotic phase transition
Li, Xiaohui; Yang, Guang; An, Kenan; Huang, Jiping
2016-08-01
The mix of competition and cooperation (C&C) is ubiquitous in human society, which, however, remains poorly explored due to the lack of a fundamental method. Here, by developing a Janus game for treating C&C between two sides (suppliers and consumers), we show, for the first time, experimental and simulation evidences for human behavioral regularity. This property is proved to be characterized by fractional Brownian motion associated with an exotic transition between periodic and nonperiodic phases. Furthermore, the periodic phase echoes with business cycles, which are well-known in reality but still far from being well understood. Our results imply that the Janus game could be a fundamental method for studying C&C among humans in society, and it provides guidance for predicting human behavioral activity from the perspective of fractional Brownian motion.
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 at fast time scales and thermal noise imaging
Huang, Rongxin
This dissertation presents experimental studies on Brownian motion at fast time scales, as well as our recent developments in Thermal Noise Imaging which uses thermal motions of microscopic particles for spatial imaging. As thermal motions become increasingly important in the studies of soft condensed matters, the study of Brownian motion is not only of fundamental scientific interest but also has practical applications. Optical tweezers with a fast position-sensitive detector provide high spatial and temporal resolution to study Brownian motion at fast time scales. A novel high bandwidth detector was developed with a temporal resolution of 30 ns and a spatial resolution of 1 A. With this high bandwidth detector, Brownian motion of a single particle confined in an optical trap was observed at the time scale of the ballistic regime. The hydrodynamic memory effect was fully studied with polystyrene particles of different sizes. We found that the mean square displacements of different sized polystyrene particles collapse into one master curve which is determined by the characteristic time scale of the fluid inertia effect. The particle's inertia effect was shown for particles of the same size but different densities. For the first time the velocity autocorrelation function for a single particle was shown. We found excellent agreement between our experiments and the hydrodynamic theories that take into account the fluid inertia effect. Brownian motion of a colloidal particle can be used to probe three-dimensional nano structures. This so-called thermal noise imaging (TNI) has been very successful in imaging polymer networks with a resolution of 10 nm. However, TNI is not efficient at micrometer scale scanning since a great portion of image acquisition time is wasted on large vacant volume within polymer networks. Therefore, we invented a method to improve the efficiency of large scale scanning by combining traditional point-to-point scanning to explore large vacant
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.
Brownian motion in the treasury bill futures market
Dale, Charles
1981-01-01
This paper analyzes prices and volumes in the T-bill futures market using a physical analogy called Brownian Motion. The results are similar to those obtained in previous studies of stock markets. For prices, the T-bill futures market failed to exhibit the presence of resistance and support levels, indicating that chartists could not profit by looking for such levels. For low volumes, T-bill futures exhibited lognormal behavior patterns, meaning that new investors are attracted to markets ...
Unusual Brownian motion of photons in open absorbing media
Zhao, Li-Yi; Tian, Chu-Shun; Zhang, Zhao-Qing; Zhang, Xiang-Dong
2013-01-01
Very recent experiments have discovered that localized light in strongly absorbing media displays intriguing diffusive phenomena. Here we develop a first-principles theory of light propagation in open media with arbitrary absorption strength and sample length. We show analytically that photons in localized open absorbing media exhibit unusual Brownian motion. Specifically, wave transport follows the diffusion equation with the diffusion coefficient exhibiting spatial resolution. Most striking...
From ballistic to Brownian vortex motion in complex oscillatory media
Davidsen, Jörn; Erichsen, Ronaldo; Kapral, Raymond; Chaté, Hugues
2004-01-01
We show that the breaking of the rotation symmetry of spiral waves in two-dimensional complex (period-doubled or chaotic) oscillatory media by synchronization defect lines (SDL) is accompanied by an intrinsic drift of the pattern. Single vortex motion changes from ballistic flights at a well-defined angle from the SDL to Brownian-like diffusion when the turbulent character of the medium increases. It gives rise, in non-turbulent multi-spiral regimes, to a novel ``vortex liquid''.
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.
Non-Markovian weak coupling limit of quantum Brownian motion
Maniscalco, Sabrina; Piilo, Jyrki; Suominen, Kalle-Antti
2008-01-01
We derive and solve analytically the non-Markovian master equation for harmonic quantum Brownian motion proving that, for weak system-reservoir couplings and high temperatures, it can be recast in the form of the master equation for a harmonic oscillator interacting with a squeezed thermal bath. This equivalence guarantees preservation of positivity of the density operator during the time evolution and allows one to establish a connection between the dynamics of Schr\\"odinger cat states in sq...
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...
Stochastic oscillator with random mass: New type of Brownian motion
Gitterman, M.
2014-02-01
The model of a stochastic oscillator subject to additive random force, which includes the Brownian motion, is widely used for analysis of different phenomena in physics, chemistry, biology, economics and social science. As a rule, by the appropriate choice of units one assumes that the particle’s mass is equal to unity. However, for the case of an additional multiplicative random force, the situation is more complicated. As we show in this review article, for the cases of random frequency or random damping, the mass cannot be excluded from the equations of motion, and, for example, besides the restriction of the size of Brownian particle, some restrictions exist also of its mass. In addition to these two types of multiplicative forces, we consider the random mass, which describes, among others, the Brownian motion with adhesion. The fluctuations of mass are modeled as a dichotomous noise, and the first two moments of coordinates show non-monotonic dependence on the parameters of oscillator and noise. In the presence of an additional periodic force an oscillator with random mass is characterized by the stochastic resonance phenomenon, when the appearance of noise increases the input signal.
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...
Strict Concavity of the Half Plane Intersection Exponent for Planar Brownian Motion
Lawler, Gregory
2000-01-01
The intersection exponents for planar Brownian motion measure the exponential decay of probabilities of nonintersection of paths. We study the intersection exponent $\\xi(\\lambda_1,\\lambda_2)$ for Brownian motion restricted to a half plane which by conformal invariance is the same as Brownian motion restricted to an infinite strip. We show that $\\xi$ is a strictly concave function. This result is used in another paper to establish a universality result for conformally invariant intersection ex...
When does fractional Brownian motion not behave as a continuous function with bounded variation?
Azmoodeh, Ehsan; Tikanmäki, Heikki; Valkeila, Esko
2010-01-01
If we compose a smooth function g with fractional Brownian motion B with Hurst index H > 1/2, then the resulting change of variables formula [or It/^o- formula] has the same form as if fractional Brownian motion would be a continuous function with bounded variation. In this note we prove a new integral representation formula for the running maximum of a continuous function with bounded variation. Moreover we show that the analogue to fractional Brownian motion fails.
Self-intersection local times and collision local times of bifractional Brownian motions
2009-01-01
In this paper, we consider the local time and the self-intersection local time for a bifractional Brownian motion, and the collision local time for two independent bifractional Brownian motions. We mainly prove the existence and smoothness of the self-intersection local time and the collision local time, through the strong local nondeterminism of bifractional Brownian motion, L2 convergence and Chaos expansion.
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.
Active Brownian motion of an asymmetric rigid particle
Mammadov, Gulmammad
2012-01-01
Individual movements of a rod-like self-propelled particle on a flat substrate are quantified. Biological systems that fit into this description may be the Gram-negative delta-proteobacterium Myxococcus xanthus, Gram-negative bacterium Escherichia coli, and Mitochondria. There are also non-living analogues such as vibrated polar granulates and self-driven anisotropic colloidal particles. For that we study the Brownian motion of an asymmetric rod-like rigid particle self-propelled at a fixed speed along its long axis in two dimensions. The motion of such a particle in a uniform external potential field is also considered. The theoretical model presented here is anticipated to better describe individual cell motion as well as intracellular transport in 2D than previous models.
Random Functions via Dyson Brownian Motion: Progress and Problems
Wang, Gaoyuan
2016-01-01
We develope a computationally efficient extension of the Dyson Brownian Motion (DBM) algorithm to generate random function in C2 locally. We further explain that random functions generated via DBM show an unstable growth as the traversed distance increases. This feature restricts the use of such functions considerably if they are to be used to model globally defined ones. The latter is the case if one used random functions to model landscapes in string theory. We provide a concrete example, based on a simple axionic potential often used in cosmology, to highlight this problem and also offer an ad hoc modification of DBM that suppresses this growth to some degree.
Permutation entropy of fractional Brownian motion and fractional Gaussian noise
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
Slowdown in branching Brownian motion with inhomogeneous variance
Maillard, Pascal; Zeitouni, Ofer
2013-01-01
We consider a model of Branching Brownian Motion with time-inhomogeneous variance of the form \\sigma(t/T), where \\sigma is a strictly decreasing function. Fang and Zeitouni (2012) showed that the maximal particle's position M_T is such that M_T-v_\\sigma T is negative of order T^{-1/3}, where v_\\sigma is the integral of the function \\sigma over the interval [0,1]. In this paper, we refine we refine this result and show the existence of a function m_T, such that M_T-m_T converges in law, as T\\t...
Random functions via Dyson Brownian Motion: progress and problems
Wang, Gaoyuan; Battefeld, Thorsten
2016-09-01
We develope a computationally efficient extension of the Dyson Brownian Motion (DBM) algorithm to generate random function in C2 locally. We further explain that random functions generated via DBM show an unstable growth as the traversed distance increases. This feature restricts the use of such functions considerably if they are to be used to model globally defined ones. The latter is the case if one uses random functions to model landscapes in string theory. We provide a concrete example, based on a simple axionic potential often used in cosmology, to highlight this problem and also offer an ad hoc modification of DBM that suppresses this growth to some degree.
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.
Theory of Brownian motion with the Alder-Wainwright effect
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
Occupation times for planar and higher dimensional Brownian motion
We consider a planar Brownian motion starting from O at time t = 0 and stopped at t. Denoting by T the time spent in a wedge of apex O and angle Θ, we develop a method to compute systematically the moments of T for general Θ values. We apply it to obtain analytically the second and third moments for a general wedge angle and, also, the fourth moment for the quadrant (Θ = π/2). We compare our results with numerical simulations. Finally, with standard perturbation theory, we establish a general formula for the second moment of an orthant occupation time
Occupation times for planar and higher dimensional Brownian motion
Desbois, Jean [CNRS, University Paris Sud, UMR8626, LPTMS, ORSAY CEDEX, F-91405 (France)
2007-03-09
We consider a planar Brownian motion starting from O at time t = 0 and stopped at t. Denoting by T the time spent in a wedge of apex O and angle {theta}, we develop a method to compute systematically the moments of T for general {theta} values. We apply it to obtain analytically the second and third moments for a general wedge angle and, also, the fourth moment for the quadrant ({theta} = {pi}/2). We compare our results with numerical simulations. Finally, with standard perturbation theory, we establish a general formula for the second moment of an orthant occupation time.
Role of Brownian Motion Hydrodynamics on Nanofluid Thermal Conductivity
W Evans, J Fish, P Keblinski
2005-11-14
We use a simple kinetic theory based analysis of heat flow in fluid suspensions of solid nanoparticles (nanofluids) to demonstrate that the hydrodynamics effects associated with Brownian motion have a minor effect on the thermal conductivity of the nanofluid. Our conjecture is supported by the results of molecular dynamics simulations of heat flow in a model nanofluid with well-dispersed particles. Our findings are consistent with the predictions of the effective medium theory as well as with recent experimental results on well dispersed metal nanoparticle suspensions.
Cavity-enhanced optical detection of carbon nanotube Brownian motion
Stapfner, S; Hunger, D; Weig, E M; Reichel, J; Favero, I
2012-01-01
Optical cavities with small mode volume are well-suited to detect the vibration of sub-wavelength sized objects. Here we employ a fiber-based, high-finesse optical microcavity to detect the Brownian motion of a freely suspended carbon nanotube at room temperature under vacuum. The optical detection resolves deflections of the oscillating tube down to 50pm/Hz^1/2. A full vibrational spectrum of the carbon nanotube is obtained and confirmed by characterization of the same device in a scanning electron microscope. Our work successfully extends the principles of high-sensitivity optomechanical detection to molecular scale nanomechanical systems.
Brownian motion and Harmonic functions on Sol(p,q)
Brofferio, Sara; Salvatori, Maura; Woess, Wolfgang
2012-01-01
The Lie group Sol(p,q) is the semidirect product induced by the action of the real numbers R on the plane R^2 which is given by (x,y) --> (exp{p z} x, exp{-q z} y), where z is in R. Viewing Sol(p,q) as a 3-dimensional manifold, it carries a natural Riemannian metric and Laplace-Beltrami operator. We add a linear drift term in the z-variable to the latter, and study the associated Brownian motion with drift. We derive a central limit theorem and compute the rate of escape. Also, we introduce t...
Brownian Motion and Harmonic Functions on Rotationally Symmetric Manifolds
March, Peter
1986-01-01
We consider Brownian motion $X$ on a rotationally symmetric manifold $M_g = (\\mathbb{R}^n, ds^2), ds^2 = dr^2 + g(r)^2 d\\theta^2$. An integral test is presented which gives a necessary and sufficient condition for the nontriviality of the invariant $\\sigma$-field of $X$, hence for the existence of nonconstant bounded harmonic functions on $M_g$. Conditions on the sectional curvatures are given which imply the convergence or the divergence of the test integral.
Functionals of Brownian motion, localization and metric graphs
Comtet, Alain [Laboratoire de Physique Theorique et Modeles Statistiques, UMR 8626 du CNRS, Universite Paris-Sud, Bat. 100, F-91405 Orsay Cedex (France); Institut Henri Poincare, 11 rue Pierre et Marie Curie, F-75005 Paris (France); Desbois, Jean [Laboratoire de Physique Theorique et Modeles Statistiques, UMR 8626 du CNRS, Universite Paris-Sud, Bat. 100, F-91405 Orsay Cedex (France); Texier, Christophe [Laboratoire de Physique Theorique et Modeles Statistiques, UMR 8626 du CNRS, Universite Paris-Sud, Bat. 100, F-91405 Orsay Cedex (France); Laboratoire de Physique des Solides, UMR 8502 du CNRS, Universite Paris-Sud, Bat. 510, F-91405 Orsay Cedex (France)
2005-09-16
We review several results related to the problem of a quantum particle in a random environment. In an introductory part, we recall how several functionals of Brownian motion arise in the study of electronic transport in weakly disordered metals (weak localization). Two aspects of the physics of the one-dimensional strong localization are reviewed: some properties of the scattering by a random potential (time delay distribution) and a study of the spectrum of a random potential on a bounded domain (the extreme value statistics of the eigenvalues). Then we mention several results concerning the diffusion on graphs, and more generally the spectral properties of the Schroedinger operator on graphs. The interest of spectral determinants as generating functions characterizing the diffusion on graphs is illustrated. Finally, we consider a two-dimensional model of a charged particle coupled to the random magnetic field due to magnetic vortices. We recall the connection between spectral properties of this model and winding functionals of planar Brownian motion. (topical review)
Functionals of Brownian motion, localization and metric graphs
We review several results related to the problem of a quantum particle in a random environment. In an introductory part, we recall how several functionals of Brownian motion arise in the study of electronic transport in weakly disordered metals (weak localization). Two aspects of the physics of the one-dimensional strong localization are reviewed: some properties of the scattering by a random potential (time delay distribution) and a study of the spectrum of a random potential on a bounded domain (the extreme value statistics of the eigenvalues). Then we mention several results concerning the diffusion on graphs, and more generally the spectral properties of the Schroedinger operator on graphs. The interest of spectral determinants as generating functions characterizing the diffusion on graphs is illustrated. Finally, we consider a two-dimensional model of a charged particle coupled to the random magnetic field due to magnetic vortices. We recall the connection between spectral properties of this model and winding functionals of planar Brownian motion. (topical review)
Ergodic Properties of Fractional Brownian-Langevin Motion
Deng, Weihua
2008-01-01
We investigate the time average mean square displacement $\\overline{\\delta^2}(x(t))=\\int_0^{t-\\Delta}[x(t^\\prime+\\Delta)-x(t^\\prime)]^2 dt^\\prime/(t-\\Delta)$ for fractional Brownian and Langevin motion. Unlike the previously investigated continuous time random walk model $\\overline{\\delta^2}$ converges to the ensemble average $ \\sim t^{2 H}$ in the long measurement time limit. The convergence to ergodic behavior is however slow, and surprisingly the Hurst exponent $H=3/4$ marks the critical point of the speed of convergence. When $H^2\\sim k(H) \\cdot\\Delta\\cdot t^{-1}$, when $H=3/4$, ${EB} \\sim (9/16)(\\ln t) \\cdot\\Delta \\cdot t^{-1}$, and when $3/4
An Efficient Method to Study Nondiffusive Motion of Brownian Particles
Lisý Vladimír
2016-01-01
Full Text Available The experimental access to short timescales has pointed to the inadequacy of the standard Langevin theory of the Brownian motion (BM in fluids. The hydrodynamic theory of the BM describes well the observed motion of the particles; however, the published approach should be improved in several points. In particular, it leads to incorrect correlation properties of the thermal noise driving the particles. In our contribution we present an efficient method, which is applicable to linear generalized Langevin equations describing the BM of particles with any kind of memory and apply it to interpret the experiments where nondiffusive BM of particles was observed. It is shown that the applicability of the method is much broader, allowing, among all, to obtain efficient solutions of various problems of anomalous BM.
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…
Unusual Brownian motion of photons in open absorbing media
Zhao, Li-Yi; Zhang, Zhao-Qing; Zhang, Xiang-Dong
2013-01-01
Very recent experiments have discovered that localized light in strongly absorbing media displays intriguing diffusive phenomena. Here we develop a first-principles theory of light propagation in open media with arbitrary absorption strength and sample length. We show analytically that photons in localized open absorbing media exhibit unusual Brownian motion. Specifically, wave transport follows the diffusion equation with the diffusion coefficient exhibiting spatial resolution. Most strikingly, despite that the system is controlled by two parameters -- the ratio of the localization (absorption) length to the sample length -- the spatially resolved diffusion coefficient displays novel single parameter scaling: it depends on the space via the returning probability. Our analytic predictions for this diffusion coefficient are confirmed by numerical simulations. In the strong absorption limit they agree well with the experimental results.
Optimal dividends in the Brownian motion risk model with interest
Fang, Ying; Wu, Rong
2009-07-01
In this paper, we consider a Brownian motion risk model, and in addition, the surplus earns investment income at a constant force of interest. The objective is to find a dividend policy so as to maximize the expected discounted value of dividend payments. It is well known that optimality is achieved by using a barrier strategy for unrestricted dividend rate. However, ultimate ruin of the company is certain if a barrier strategy is applied. In many circumstances this is not desirable. This consideration leads us to impose a restriction on the dividend stream. We assume that dividends are paid to the shareholders according to admissible strategies whose dividend rate is bounded by a constant. Under this additional constraint, we show that the optimal dividend strategy is formed by a threshold strategy.
Brownian Motion as a Limit to Physical Measuring Processes
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...
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.
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.
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)
Self-intersection local times and collision local times of bifractional Brownian motions
JIANG YiMing; WANG YongJin
2009-01-01
In this paper, we consider the local time and the self-intersection local time for a bifrac-tional Brownish motion, and the collision local time for two independent bifractional Brownian motions. We mainly prove the existence and smoothness of the self-intersection local time and the collision local time, through the strong local nondeterminism of bifractional Brownian motion, L2 convergence and Chaos expansion.
Role of Brownian motion on the thermal conductivity enhancement of nanofluids
Gupta, Amit; Kumar, Ranganathan
2007-11-01
This study involves Brownian dynamics simulations of a real nanofluid system in which the interparticle potential is determined based on Debye length and surface interaction of the fluid and the solid. This paper shows that Brownian motion can increase the thermal conductivity of the nanofluid by 6% primarily due to "random walk" motion and not only through diffusion. This increase is limited by the maximum concentration for each particle size and is below that predicted by the effective medium theory. Beyond the maximum limit, particle aggregates begin to form. Brownian motion contribution stays as a constant beyond a certain particle diameter.
Oscillation of harmonic functions for subordinate Brownian motion and its applications
Kim, Panki; Lee, Yunju
2012-01-01
In this paper, we establish an oscillation estimate of nonnegative harmonic functions for a pure-jump subordinate Brownian motion. The infinitesimal generator of such subordinate Brownian motion is an integro-differential operator. As an application, we give a probabilistic proof of the following form of relative Fatou theorem for such subordinate Brownian motion X in bounded kappa-fat open set; if u is a positive harmonic function with respect to X in a bounded kappa-fat open set D and h is ...
An exactly solvable model for Brownian motion : III. Motion of a heavy mass in a linear chain
Ullersma, P.
1966-01-01
The theory on Brownian motion, developed in previous papers1) 2) is applied to a linear chain with harmonic coupling between nearest neighbours. All masses are equal except for one which is heavy compared to the others. This heavy particle behaves as a Brownian particle, which is not subject to an e
Magnetic fields and Brownian motion on the 2-sphere
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 → ∞
On the weak convergence of super-Brownian motion with immigration
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.
Moderate deviations for the quenched mean of the super-Brownian motion with random immigration
2008-01-01
Moderate deviations for the quenched mean of the super-Brownian motion with random immigration are proved for 3≤d≤6, which fills in the gap between central limit theorem(CLT)and large deviation principle(LDP).
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.
Persistence of fractional Brownian motion with moving boundaries and applications
We consider various problems related to the persistence probability of fractional Brownian motion (FBM), which is the probability that the FBM X stays below a certain level until time T. Recently, Oshanin et al (2012, arXiv:1209.3313v2) have studied a physical model, where persistence properties of FBM are shown to be related to scaling properties of a quantity JN, called the steady-state current. It turns out that for this analysis, it is important to determine persistence probabilities of FBM with a moving boundary. We show that one can add a boundary of logarithmic order to an FBM without changing the polynomial rate of decay of the corresponding persistence probability, which proves a result needed in Oshanin et al (2013 Phys. Rev. Lett. at press (arXiv:1209.3313v2)). Moreover, we complement their findings by considering the continuous-time version of JT. Finally, we use the results for moving boundaries in order to improve estimates by Molchan (1999 Commun. Math. Phys. 205 97–111) concerning the persistence properties of other quantities of interest, such as the time when an FBM reaches its maximum on the time interval (0, 1) or the last zero in the interval (0, 1). (paper)
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...
100 years of Einstein's theory of Brownian motion: from pollen grains to protein trains
Chowdhury, Debashish
2005-01-01
Experimental verification of the theoretical predictions made by Albert Einstein in his paper, published in 1905, on the molecular mechanisms of Brownian motion established the existence of atoms. In the last 100 years discoveries of many facets of the ubiquitous Brownian motion has revolutionized our fundamental understanding of the role of {\\it thermal fluctuations} in the exotic structures and complex dynamics exhibited by soft matter like, for example, colloids, gels, etc. The domain of B...
Recurrence and transience for normally reflected Brownian motion in warped product manifolds
de Lima, Levi Lopes
2016-01-01
We establish an integral test describing the exact cut-off between recurrence and transience for normally reflected Brownian motion in certain unbounded domains in a class of warped product manifolds. Besides extending a previous result by Pinsky \\cite{P1}, who treated the case in which the ambient space is flat, our result recovers the classical test for the standard Brownian motion in model spaces. Moreover, it allows us to discuss the recurrence/transience dichotomy for certain generalized...
On exponential stability for stochastic differential equations disturbed by G-Brownian motion
Fei, Weiyin; Fei, Chen
2013-01-01
We first introduce the calculus of Peng's G-Brownian motion on a sublinear expectation space $(\\Omega, {\\cal H}, \\hat{\\mathbb{E}})$. Then we investigate the exponential stability of paths for a class of stochastic differential equations disturbed by a G-Brownian motion in the sense of quasi surely (q.s.). The analyses consist in G-Lyapunov function and some special inequalities. Various sufficient conditions are obtained to ensure the stability of strong solutions. In particular, by means of ...
Weak solutions to stochastic differential equations driven by fractional Brownian motion
Šnupárková, Jana
2009-01-01
Roč. 59, č. 4 (2009), s. 879-907. ISSN 0011-4642 R&D Projects: GA ČR GA201/07/0237 Institutional research plan: CEZ:AV0Z10750506 Keywords : fractional Brownian motion * weak solutions Subject RIV: BA - General Mathematics Impact factor: 0.306, year: 2009 http://library.utia.cas.cz/separaty/2009/SI/snuparkova-weak solutions to stochastic differential equations driven by fractional brownian motion. pdf
On Brownian motion of asymmetric particles - An application as molecular motor
Reichelsdorfer, Martin
2010-01-01
Asymmetric Brownian particles are able to conduct directed average motion under non-equilibrium conditions. This can be used for the construction of molecular motors. The present paper is concerned with the extension and analysis of a biologically inspired model. Due to the often quite abstract formulation, the considerations are also relevant in the broader context of general Brownian motion of asymmetric objects. The model consists of an asymmetric two-dimensional body, which moves along a ...
Volpe, Giorgio; Volpe, Giovanni; Gigan, Sylvain
2014-01-01
The motion of particles in random potentials occurs in several natural phenomena ranging from the mobility of organelles within a biological cell to the diffusion of stars within a galaxy. A Brownian particle moving in the random optical potential associated to a speckle, i.e., a complex interference pattern generated by the scattering of coherent light by a random medium, provides an ideal mesoscopic model system to study such phenomena. Here, we derive a theory for the motion of a Brownian ...
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
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.
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.
Brownian Motion After Einstein and Smoluchowski: Some New Applications and New Experiments
The first half of this review describes the development in mathematical models of Brownian motion after Einstein's and Smoluchowski's seminal papers and current applications to optical tweezers. This instrument of choice among single-molecule biophysicists is also an instrument of such precision that it requires an understanding of Brownian motion beyond Einstein's and Smoluchowski's for its calibration, and can measure effects not present in their theories. This is illustrated with some applications, current and potential. It is also shown how addition of a controlled forced motion on the nano-scale of the thermal motion of the tweezed object can improve the calibration of the instrument in general, and make calibration possible also in complex surroundings. The second half of the present re- view, starting with Sect. 9, describes the co-evolution of biological motility models with models of Brownian motion, including recent results for how to derive cell-type-specific motility models from experimental cell trajectories. (author)
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
From mechanical motion to Brownian motion, thermodynamics and particle transport theory
The motion of a particle in a medium is dealt with either as a problem of mechanics or as a transport process in non-equilibrium statistical physics. The two kinds of approach are often unrelated as they are taught in different textbooks. The aim of this paper is to highlight the link between the mechanical and statistical treatments of particle motion in a medium, starting from the well-studied case of Brownian motion. First, deterministic dynamics is supplemented with stochastic elements accounting for the thermal agitation of the host medium: it is the approach of Langevin, which has been rephrased and extended by Kramers. It handles time-independent and time-dependent stochastic motions as well. In that approach, the host medium is not affected by the guest particles and the latter do not interact with each other. Both limitations are shown to be overcome in thermodynamics, which however is restricted to equilibrium situations, i.e. stochastic motions with no net current. When equilibrium is slightly perturbed, we show how thermodynamic and kinetic concepts supersede mechanical concepts to describe particle transport. The description includes multicomponent transport. The discussions of stochastic dynamics and of thermodynamics are led at the undergraduate level; the treatment of multicomponent transport introduces graduate-level concepts
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) 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...
Coupling of lever arm swing and biased Brownian motion in actomyosin.
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.
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.
ZHANG Jia-Lin; YU Hong-Wei
2005-01-01
@@ We show that the velocity and position dispersions of a test particle with a nonzero constant classical velocity undergoing Brownian motion caused by electromagnetic vacuum fluctuations in a space with plane boundaries can be obtained from those of the static case by Lorentz transformation. We explicitly derive the Lorentz transformations relating the dispersions of the two cases and then apply them to the case of the Brownian motion of a test particle with a constant classical velocity parallel to the boundary between two conducting planes. Our results show that the influence of a nonzero initial velocity is negligible for nonrelativistic test particles.
Brownian Motion in wedges, last passage time and the second arc-sine law
Comtet, Alain; Desbois, Jean
2003-01-01
We consider a planar Brownian motion starting from $O$ at time $t=0$ and stopped at $t=1$ and a set $F= \\{OI_i ; i=1,2,..., n\\}$ of $n$ semi-infinite straight lines emanating from $O$. Denoting by $g$ the last time when $F$ is reached by the Brownian motion, we compute the probability law of $g$. In particular, we show that, for a symmetric $F$ and even $n$ values, this law can be expressed as a sum of $\\arcsin $ or $(\\arcsin)^2 $ functions. The original result of Levy is recovered as the spe...
Anyonic partition functions and windings of planar Brownian motion
The computation of the N-cycle Brownian paths contribution FN(α) to the N-anyon partition function is addressed. A detailed numerical analysis based on a random walk on a lattice indicates that FN0(α)=product k=1N-1[1-(N/k)α]. In the paramount three-anyon case, one can show that F3(α) is built by linear states belonging to the bosonic, fermionic, and mixed representations of S3
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...
Itô's formula with respect to fractional Brownian motion and its application
W. Dai
1996-01-01
Full Text Available Fractional Brownian motion (FBM with Hurst index 1/2
Holographic Brownian Motion in Three-Dimensional Gödel Black Hole
By using the AdS/CFT correspondence and Gödel black hole background, we study the dynamics of heavy quark under a rotating plasma. In that case we follow Atmaja (2013) about Brownian motion in BTZ black hole. In this paper we receive some new results for the case of α2l2≠1. In this case, we must redefine the angular velocity of string fluctuation. We obtain the time evolution of displacement square and angular velocity and show that it behaves as a Brownian particle in non relativistic limit. In this plasma, it seems that relating the Brownian motion to physical observables is rather a difficult work. But our results match with Atmaja work in the limit α2l2→1
Volpe, Giorgio; Volpe, Giovanni; Gigan, Sylvain
2014-01-01
The motion of particles in random potentials occurs in several natural phenomena ranging from the mobility of organelles within a biological cell to the diffusion of stars within a galaxy. A Brownian particle moving in the random optical potential associated to a speckle pattern, i.e., a complex interference pattern generated by the scattering of coherent light by a random medium, provides an ideal model system to study such phenomena. Here, we derive a theory for the motion of a Brownian particle in a speckle field and, in particular, we identify its universal characteristic timescale. Based on this theoretical insight, we show how speckle light fields can be used to control the anomalous diffusion of a Brownian particle and to perform some basic optical manipulation tasks such as guiding and sorting. Our results might broaden the perspectives of optical manipulation for real-life applications. PMID:24496461
Anyonic Partition Functions and Windings of Planar Brownian Motion
Desbois, Jean; Heinemann, Christine; Ouvry, Stéphane
1994-01-01
The computation of the $N$-cycle brownian paths contribution $F_N(\\alpha)$ to the $N$-anyon partition function is adressed. A detailed numerical analysis based on random walk on a lattice indicates that $F_N^{(0)}(\\alpha)= \\prod_{k=1}^{N-1}(1-{N\\over k}\\alpha)$. In the paramount $3$-anyon case, one can show that $F_3(\\alpha)$ is built by linear states belonging to the bosonic, fermionic, and mixed representations of $S_3$.
Wang, Jun; Liang, Jin-Rong; Lv, Long-Jin; Qiu, Wei-Yuan; Ren, Fu-Yao
2012-02-01
In this paper, we study the problem of continuous time option pricing with transaction costs by using the homogeneous subdiffusive fractional Brownian motion (HFBM) Z(t)=X(Sα(t)), 0Black-Scholes equation and the Black-Scholes formula for the fair prices of European option, the turnover and transaction costs of replicating strategies. We also give the total transaction costs.
Time-changed geometric fractional Brownian motion and option pricing with transaction costs
Gu, Hui; Liang, Jin-Rong; Zhang, Yun-Xiu
2012-08-01
This paper deals with the problem of discrete time option pricing by a fractional subdiffusive Black-Scholes model. The price of the underlying stock follows a time-changed geometric fractional Brownian motion. By a mean self-financing delta-hedging argument, the pricing formula for the European call option in discrete time setting is obtained.
Paul H. Bezandry
2012-09-01
Full Text Available This article concerns the existence of almost periodic solutions to a class of abstract stochastic evolution equations driven by fractional Brownian motion in a separable real Hilbert space. Under some sufficient conditions, we establish the existence and uniqueness of a pth-mean almost periodic mild solution to those stochastic differential equations.
Toufik Guendouzi,
2014-03-01
Full Text Available In this paper we investigate the existence and stability of quadratic-mean almost periodic mild solutions to stochastic delay functional differential equations driven by fractional Brownian motion with Hurst parameter H > 1/2 , under some suitable assumptions, by means of semigroup of operators and fixed point method.
Comments on Application of the Fractional Brownian Motion in an Airlift Reactor
Drahoš, Jiří; Punčochář, Miroslav
2002-01-01
Roč. 57, č. 10 (2002), s. 1831-1832. ISSN 0009-2509 R&D Projects: GA ČR GA104/97/S002 Institutional research plan: CEZ:AV0Z4072921 Keywords : application * Brownian motion * airlift reactor Subject RIV: CF - Physical ; Theoretical Chemistry Impact factor: 1.224, year: 2002
Stochastic optimal control problem with infinite horizon driven by G-Brownian motion
Hu, Mingshang; Wang, Falei
2016-01-01
The present paper considers a stochastic optimal control problem, in which the cost function is defined through a backward stochastic differential equation with infinite horizon driven by G-Brownian motion. Then we study the regularities of the value function and establish the dynamic programming principle. Moreover, we prove that the value function is the uniqueness viscosity solution of the related HJBI equation.
Some scaled limit theorems for an immigration super-Brownian motion
2008-01-01
In this paper,the small time limit behaviors for an immigration super-Brownian motion are studied,where the immigration is determined by Lebesgue measure.We first prove a functional central limit theorem,and then study the large and moderate deviations associated with this central tendency.
The oscillation of the occupation time process of super- Brownian motion on Sierpinski gasket
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
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.
Pricing Perpetual American Put Option in theMixed Fractional Brownian Motion
2015-01-01
Under the assumption of the underlying asset is driven by the mixed fractional Brownian motion, we obtain the mixed fractionalBlack-Scholes partial differential equation by fractional Ito formula, and the pricing formula of perpetual American put option bythis partial differential equation theory.
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.
Bertrand, Pierre R; Guillin, Arnaud
2010-01-01
We investigate here the Central Limit Theorem of the Increment Ratio Statistic of a multifractional Brownian motion, leading to a CLT for the time varying Hurst index. The proofs are quite simple relying on Breuer-Major theorems and an original freezing of time strategy. A simulation study shows the goodness of fit of this estimator.
Time integration for particle Brownian motion determined through fluctuating hydrodynamics
Delmotte, Blaise
2015-01-01
Fluctuating hydrodynamics has been successfully combined with several computational methods to rapidly compute the correlated random velocities of Brownian particles. In the overdamped limit where both particle and fluid inertia are ignored, one must also account for a Brownian drift term in order to successfully update the particle positions. In this paper, we introduce and study a midpoint time integration scheme we refer to as the drifter-corrector (DC) that resolves the drift term for fluctuating hydrodynamics-based methods even when constraints are imposed on the fluid flow to obtain higher-order corrections to the particle hydrodynamic interactions. We explore this scheme in the context of the fluctuating force-coupling method (FCM) where the constraint is imposed on the rate-of-strain averaged over the volume occupied by the particle. For the DC, the constraint need only be imposed once per time step, leading to a significant reduction in computational cost with respect to other schemes. In fact, for f...
Brownian motion after Einstein and Smoluchowski: Some new applications and new experiments
Dávid, Selmeczi; Tolic-Nørrelykke, S.F.; Schäffer, E.;
2007-01-01
The first half of this review describes the development in mathematical models of Brownian motion after Einstein's and Smoluchowski's seminal papers and current applications to optical tweezers. This instrument of choice among single-molecule biophysicists is also an instrument of such precision...... that it requires an understanding of Brownian motion beyond Einstein's and Smoluchowski's for its calibration, and can measure effects not present in their theories. This is illustrated with some applications, current and potential. It is also shown how addition of a controlled forced motion on the...... nano-scale of the thermal motion of the tweezed object can improve the calibration of the instrument in general, and make calibration possible also in complex surroundings. The second half of the present review, starting with Sect. 9, describes the co-evolution of biological motility models with models...
Exploiting the color of Brownian motion for high-frequency microrheology of Newtonian fluids
Domínguez-García, Pablo; Mor, Flavio M.; Forró, László; Jeney, Sylvia
2013-09-01
Einstein's stochastic description of the random movement of small objects in a fluid, i.e. Brownian motion, reveals to be quite different, when observed on short timescales. The limitations of Einstein's theory with respect to particle inertia and hydrodynamic memory yield to the apparition of a colored frequency-dependent component in the spectrum of the thermal forces, which is called "the color of Brownian motion". The knowledge of the characteristic timescales of the motion of a trapped microsphere motion in a Newtonian fluid allowed to develop a high-resolution calibration method for optical interferometry. Well-calibrated correlation quantities, such as the mean square displacement or the velocity autocorrelation function, permit to study the mechanical properties of fluids at high frequencies. These properties are estimated by microrheological calculations based on the theoretical relations between the complex mobility of the beads and the rheological properties of a complex fluid.
Dynamics of 2D Stochastic non-Newtonian fluids driven by fractional Brownian motion
Li, Jin; Huang, Jianhua
2011-01-01
A 2D Stochastic incompressible non-Newtonian fluids driven by fractional Bronwnian motion with Hurst parameter $H \\in (1/2,1)$ is studied. The Wiener-type stochastic integrals are introduced for infinite-dimensional fractional Brownian motion. Four groups of assumptions, including the requirement of Nuclear operator or Hilbert-Schmidt operator, are discussed. The existence and regularity of stochastic convolution for the corresponding additive linear stochastic equation are obtained under eac...
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 ...
Molecular dynamics test of the Brownian description of Na(+) motion in water
Wilson, M. A.; Pohorille, A.; Pratt, L. R.
1985-01-01
The present paper provides the results of molecular dynamics calculations on a Na(+) ion in aqueous solution. Attention is given to the sodium-oxygen and sodium-hydrogen radial distribution functions, the velocity autocorrelation function for the Na(+) ion, the autocorrelation function of the force on the stationary ion, and the accuracy of Brownian motion assumptions which are basic to hydrodynamic models of ion dyanmics in solution. It is pointed out that the presented calculations provide accurate data for testing theories of ion dynamics in solution. The conducted tests show that it is feasible to calculate Brownian friction constants for ions in aqueous solutions. It is found that for Na(+) under the considered conditions the Brownian mobility is in error by only 60 percent.
Mirror coupling of reflecting Brownian motion and an application to Chavel's conjecture
Pascu, Mihai N
2010-01-01
In a series of papers, Burdzy et. al. introduced the \\emph{mirror coupling} of reflecting Brownian motions in a smooth bounded domain $D\\subset \\mathbb{R}^{d}$, and used it to prove certain properties of eigenvalues and eigenfunctions of the Neumann Laplaceian on $D$. In the present paper we show that the construction of the mirror coupling can be extended to the case when the two Brownian motions live in different domains $D_{1},D_{2}\\subset \\mathbb{R}^{d}$. As an application of the construction, we derive a unifying proof of the two main results concerning the validity of Chavel's conjecture on the domain monotonicity of the Neumann heat kernel, due to I. Chavel (\\cite{Chavel}), respectively W. S. Kendall (\\cite{Kendall}).
On Nonlinear Quantum Mechanics, Brownian Motion, Weyl Geometry and Fisher Information
Castro C.
2006-01-01
Full Text Available A new nonlinear Schrödinger equation is obtained explicitly from the (fractal Brownian motion of a massive particle with a complex-valued diffusion constant. Real-valued energy plane-wave solutions and solitons exist in the free particle case. One remarkable feature of this nonlinear Schrödinger equation based on a (fractal Brownian motion model, over all the other nonlinear QM models, is that the quantummechanical energy functional coincides precisely with the field theory one. We finalize by showing why a complex momentum is essential to fully understand the physical implications of Weyl’s geometry in QM, along with the interplay between Bohm’s Quantum potential and Fisher Information which has been overlooked by several authors in the past.
K. C. Lee
2013-02-01
Full Text Available Multifractional Brownian motions have become popular as flexible models in describing real-life signals of high-frequency features in geoscience, microeconomics, and turbulence, to name a few. The time-changing Hurst exponent, which describes regularity levels depending on time measurements, and variance, which relates to an energy level, are two parameters that characterize multifractional Brownian motions. This research suggests a combined method of estimating the time-changing Hurst exponent and variance using the local variation of sampled paths of signals. The method consists of two phases: initially estimating global variance and then accurately estimating the time-changing Hurst exponent. A simulation study shows its performance in estimation of the parameters. The proposed method is applied to characterization of atmospheric stability in which descriptive statistics from the estimated time-changing Hurst exponent and variance classify stable atmosphere flows from unstable ones.
Lee, K. C.
2013-02-01
Multifractional Brownian motions have become popular as flexible models in describing real-life signals of high-frequency features in geoscience, microeconomics, and turbulence, to name a few. The time-changing Hurst exponent, which describes regularity levels depending on time measurements, and variance, which relates to an energy level, are two parameters that characterize multifractional Brownian motions. This research suggests a combined method of estimating the time-changing Hurst exponent and variance using the local variation of sampled paths of signals. The method consists of two phases: initially estimating global variance and then accurately estimating the time-changing Hurst exponent. A simulation study shows its performance in estimation of the parameters. The proposed method is applied to characterization of atmospheric stability in which descriptive statistics from the estimated time-changing Hurst exponent and variance classify stable atmosphere flows from unstable ones.
The application of fractional derivatives in stochastic models driven by fractional Brownian motion
Longjin, Lv; Ren, Fu-Yao; Qiu, Wei-Yuan
2010-11-01
In this paper, in order to establish connection between fractional derivative and fractional Brownian motion (FBM), we first prove the validity of the fractional Taylor formula proposed by Guy Jumarie. Then, by using the properties of this Taylor formula, we derive a fractional Itô formula for H∈[1/2,1), which coincides in form with the one proposed by Duncan for some special cases, whose formula is based on the Wick Product. Lastly, we apply this fractional Itô formula to the option pricing problem when the underlying of the option contract is supposed to be driven by a geometric fractional Brownian motion. The case that the drift, volatility and risk-free interest rate are all dependent on t is also discussed.
Quantal Brownian Motion from RPA dynamics: The master and Fokker-Planck equations
From the purely quantal RPA description of the damped harmonic oscillator and of the corresponding Brownian Motion within the full space (phonon subspace plus reservoir), a master equation (as well as a Fokker-Planck equation) for the reduced density matrix (for the reduced Wigner function, respectively) within the phonon subspace is extracted. The RPA master equation agrees with the master equation derived by the time-dependent perturbative approaches which utilize Tamm-Dancoff Hilbert spaces and invoke the rotating wave approximation. Since the RPA yields a full, as well as a contracted description, it can account for both the kinetic and the unperturbed oscillator momenta. The RPA description of the quantal Brownian Motion contrasts with the descriptions provided by the time perturbative approaches whether they invoke or not the rotating wave approximation. The RPA description also contrasts with the phenomenological phase space quantization. (orig.)
Impurity driven Brownian motion of solitons in elongated Bose-Einstein Condensates
Aycock, L M; Genkina, D; Lu, H -I; Galitski, V; Spielman, I B
2016-01-01
Solitons, spatially-localized, mobile excitations resulting from an interplay between nonlinearity and dispersion, are ubiquitous in physical systems from water channels and oceans to optical fibers and Bose-Einstein condensates (BECs). For the first time, we observed and controlled the Brownian motion of solitons. We launched long-lived dark solitons in highly elongated $^{87}\\rm{Rb}$ BECs and showed that a dilute background of impurity atoms in a different internal state dramatically affects the soliton. With no impurities and in one-dimension (1-D), these solitons would have an infinite lifetime, a consequence of integrability. In our experiment, the added impurities scatter off the much larger soliton, contributing to its Brownian motion and decreasing its lifetime. We describe the soliton's diffusive behavior using a quasi-1-D scattering theory of impurity atoms interacting with a soliton, giving diffusion coefficients consistent with experiment.
Brownian motion in wedges, last passage time and the second arc-sine law
Comtet, Alain [Laboratoire de Physique Theorique et Modeles Statistiques. Universite Paris-Sud, Bat. 100, F-91405 Orsay Cedex (France); Desbois, Jean [Laboratoire de Physique Theorique et Modeles Statistiques. Universite Paris-Sud, Bat. 100, F-91405 Orsay Cedex (France)
2003-05-02
We consider a planar Brownian motion starting from O at time t = 0 and stopped at t = 1 and a set F = OI{sub i}; i = 1, 2, ..., n of n semi-infinite straight lines emanating from O. Denoting by g the last time when F is reached by the Brownian motion, we compute the probability law of g. In particular, we show that, for a symmetric F and even n values, this law can be expressed as a sum of arcsin or (arcsin){sup 2} functions. The original result of Levy is recovered as the special case n = 2. A relation with the problem of reaction-diffusion of a set of three particles in one dimension is discussed. (letter to the editor)
Brownian motion in wedges, last passage time and the second arc-sine law
We consider a planar Brownian motion starting from O at time t = 0 and stopped at t = 1 and a set F = OIi; i = 1, 2, ..., n of n semi-infinite straight lines emanating from O. Denoting by g the last time when F is reached by the Brownian motion, we compute the probability law of g. In particular, we show that, for a symmetric F and even n values, this law can be expressed as a sum of arcsin or (arcsin)2 functions. The original result of Levy is recovered as the special case n = 2. A relation with the problem of reaction-diffusion of a set of three particles in one dimension is discussed. (letter to the editor)
Random variables as pathwise integrals with respect to fractional Brownian motion
Mishura, Yuliya; Valkeila, Esko
2011-01-01
We show that a pathwise stochastic integral with respect to fractional Brownian motion with an adapted integrand $g$ can have any prescribed distribution, moreover, we give both necessary and sufficient conditions when random variables can be represented in this form. We also prove that any random variable is a value of such integral in some improper sense. We discuss some applications of these results, in particular, to fractional Black--Scholes model of financial market.
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 ...
Boundary behavior of a constrained Brownian motion between reflecting-repellent walls
Lépingle, Dominique
2009-01-01
International audience Stochastic variational inequalities provide a unified treatment for stochastic differential equations living in a closed domain with normal reflection and (or) singular repellent drift. When the domain is a polyhedron, we prove that the reflected-repelled Brownian motion does not hit the non-smooth part of the boundary. A sufficient condition for non-hitting a face of the polyhedron is derived from the one-dimensional case. A complete answer to the question of attain...
Optimal stochastic control and optimal consumption and portfolio with G-Brownian motion
Fei, Weiyin; Fei, Chen
2013-01-01
By the calculus of Peng's G-sublinear expectation and G-Brownian motion on a sublinear expectation space $(\\Omega, {\\cal H}, \\hat{\\mathbb{E}})$, we first set up an optimality principle of stochastic control problem. Then we investigate an optimal consumption and portfolio decision with a volatility ambiguity by the derived verification theorem. Next the two-fund separation theorem is explicitly obtained. And an illustrative example is provided.
Maximum likelihood drift estimation for the mixing of two fractional Brownian motions
Mishura, Yuliya
2015-01-01
We construct the maximum likelihood estimator (MLE) of the unknown drift parameter $\\theta\\in \\mathbb{R}$ in the linear model $X_t=\\theta t+\\sigma B^{H_1}(t)+B^{H_2}(t),\\;t\\in[0,T],$ where $B^{H_1}$ and $B^{H_2}$ are two independent fractional Brownian motions with Hurst indices $\\frac12
无
2000-01-01
The solutions of the following bilinearstochastic differential equation are stud-ied (X) where Atk, Bt are (deterministic)continuous matrix-valued functions of t and w1(t),..., wm(t) are m independent standard Brownian motions. Conditions are given such thatthe solution is positive if the initial condition is positive.The equation the most probable path must satisfy is also derived and applied to a mathematicalfinance problem.
X-Ray photon correlation spectroscopy study of Brownian motion of gold colloids in glycerol
We report x-ray photon correlation spectroscopy studies of the static structure factor and dynamic correlation function of a gold colloid dispersed in the viscous liquid glycerol. We find a diffusion coefficient for Brownian motion of the gold colloid which agrees well with that extrapolated from measurements made with visible light, but which was determined on an optically opaque sample and in a wave-vector range inaccessible to visible light
Fractional Brownian Motion Approximation Based on Fractional Integration of a White Noise
Chechkin, A. V.; Gonchar, V. Yu.
1999-01-01
We study simple approximations to fractional Gaussian noise and fractional Brownian motion. The approximations are based on spectral properties of the noise. They allow one to consider the noise as the result of fractional integration/differentiation of a white Gaussian noise. We study correlation properties of the approximation to fractional Gaussian noise and point to the peculiarities of persistent and anti-persistent behaviors. We also investigate self-similar properties of the approximat...
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...
Brownian Motion of Stiff Filaments in a Crowded Environment
Fakhri, Nikta; MacKintosh, Frederick C.; Lounis, Brahim; Cognet, Laurent; Pasquali, Matteo
2010-12-01
The thermal motion of stiff filaments in a crowded environment is highly constrained and anisotropic; it underlies the behavior of such disparate systems as polymer materials, nanocomposites, and the cell cytoskeleton. Despite decades of theoretical study, the fundamental dynamics of such systems remains a mystery. Using near-infrared video microscopy, we studied the thermal diffusion of individual single-walled carbon nanotubes (SWNTs) confined in porous agarose networks. We found that even a small bending flexibility of SWNTs strongly enhances their motion: The rotational diffusion constant is proportional to the filament-bending compliance and is independent of the network pore size. The interplay between crowding and thermal bending implies that the notion of a filament’s stiffness depends on its confinement. Moreover, the mobility of SWNTs and other inclusions can be controlled by tailoring their stiffness.
Brownian motion of massive skyrmions in magnetic thin films
Troncoso, Roberto E., E-mail: r.troncoso.c@gmail.com [Centro para el Desarrollo de la Nanociencia y la Nanotecnología, CEDENNA, Avda. Ecuador 3493, Santiago 9170124 (Chile); Núñez, Álvaro S., E-mail: alnunez@dfi.uchile.cl [Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Casilla 487-3, Santiago (Chile)
2014-12-15
We report on the thermal effects on the motion of current-driven massive magnetic skyrmions. The reduced equation for the motion of skyrmion has the form of a stochastic generalized Thiele’s equation. We propose an ansatz for the magnetization texture of a non-rigid single skyrmion that depends linearly with the velocity. By using this ansatz it is found that the skyrmion mass tensor is closely related to intrinsic skyrmion parameters, such as Gilbert damping, skyrmion-charge and dissipative force. We have found an exact expression for the average drift velocity as well as the mean-square velocity of the skyrmion. The longitudinal and transverse mobility of skyrmions for small spin-velocity of electrons is also determined and found to be independent of the skyrmion mass.
Differential Dynamic Microscopy to characterize Brownian motion and bacteria motility
Germain, David; Leocmach, Mathieu; Gibaud, Thomas
2015-01-01
We have developed a lab work module where we teach undergraduate students how to quantify the dynamics of a suspension of microscopic particles, measuring and analyzing the motion of those particles at the individual level or as a group. Differential Dynamic Microscopy (DDM) is a relatively recent technique that precisely does that and constitutes an alternative method to more classical techniques such as dynamics light scattering (DLS) or video particle tracking (VPT). DDM consists in imagin...
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...
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...
Particle mobility between two planar elastic membranes: Brownian motion and membrane deformation
Daddi-Moussa-Ider, Abdallah; Gekle, Stephan
2016-01-01
We study the motion of a solid particle immersed in a Newtonian fluid and confined between two parallel elastic membranes possessing shear and bending rigidity. The hydrodynamic mobility depends on the frequency of the particle motion due to the elastic energy stored in the membrane. Unlike the single-membrane case, a coupling between shearing and bending exists. The commonly used approximation of superposing two single-membrane contributions is found to give reasonable results only for motions in the parallel, but not in the perpendicular direction. We also compute analytically the membrane deformation resulting from the motion of the particle, showing that the presence of the second membrane reduces deformation. Using the fluctuation-dissipation theorem we compute the Brownian motion of the particle, finding a long-lasting subdiffusive regime at intermediate time scales. We finally assess the accuracy of the employed point-particle approximation via boundary-integral simulations for a truly extended particl...
Quantum Brownian motion in a bath of parametric oscillators: A model for system-field interactions
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
Nanoparticle Brownian motion and hydrodynamic interactions in the presence of flow fields.
Uma, B; Swaminathan, T N; Radhakrishnan, R; Eckmann, D M; Ayyaswamy, P S
2011-07-01
We consider the Brownian motion of a nanoparticle in an incompressible Newtonian fluid medium (quiescent or fully developed Poiseuille flow) with the fluctuating hydrodynamics approach. The formalism considers situations where both the Brownian motion and the hydrodynamic interactions are important. The flow results have been modified to account for compressibility effects. Different nanoparticle sizes and nearly neutrally buoyant particle densities are also considered. Tracked particles are initially located at various distances from the bounding wall to delineate wall effects. The results for thermal equilibrium are validated by comparing the predictions for the temperatures of the particle with those obtained from the equipartition theorem. The nature of the hydrodynamic interactions is verified by comparing the velocity autocorrelation functions and mean square displacements with analytical and experimental results where available. The equipartition theorem for a Brownian particle in Poiseuille flow is verified for a range of low Reynolds numbers. Numerical predictions of wall interactions with the particle in terms of particle diffusivities are consistent with results, where available. PMID:21918592
General quantum Brownian motion with initially correlated and nonlinearly coupled environment
The dynamics of an open quantum system exhibiting the quantum Brownian motion is analyzed when the coupling between the system and its environment is nonlinear, and the system and the reservoir are initially correlated. For couplings quadratic in the environment variables, the influence functional for the system is obtained perturbatively up to second order in the coupling constant, and then the propagator is explicitly evaluated when the particle is under the influence of a harmonic potential and an additional anharmonic potential, the so-called washboard potential. As an application of the propagator, the master equation and the Wigner equation are obtained for the quantum Brownian particle moving in a harmonic potential for the generalized correlated initial condition, and then for the specific case of the simplified 'thermal' initial condition. The system is shown to obey the corresponding fluctuation-dissipation theorem
Brownian motion of massive black hole binaries and the final parsec problem
Bortolas, E.; Gualandris, A.; Dotti, M.; Spera, M.; Mapelli, M.
2016-09-01
Massive black hole binaries (BHBs) are expected to be one of the most powerful sources of gravitational waves in the frequency range of the pulsar timing array and of forthcoming space-borne detectors. They are believed to form in the final stages of galaxy mergers, and then harden by slingshot ejections of passing stars. However, evolution via the slingshot mechanism may be ineffective if the reservoir of interacting stars is not readily replenished, and the binary shrinking may come to a halt at roughly a parsec separation. Recent simulations suggest that the departure from spherical symmetry, naturally produced in merger remnants, leads to efficient loss cone refilling, preventing the binary from stalling. However, current N-body simulations able to accurately follow the evolution of BHBs are limited to very modest particle numbers. Brownian motion may artificially enhance the loss cone refilling rate in low-N simulations, where the binary encounters a larger population of stars due its random motion. Here we study the significance of Brownian motion of BHBs in merger remnants in the context of the final parsec problem. We simulate mergers with various particle numbers (from 8k to 1M) and with several density profiles. Moreover, we compare simulations where the BHB is fixed at the centre of the merger remnant with simulations where the BHB is free to random walk. We find that Brownian motion does not significantly affect the evolution of BHBs in simulations with particle numbers in excess of one million, and that the hardening measured in merger simulations is due to collisionless loss cone refilling.
Brownian motion of massive black hole binaries and the final parsec problem
Bortolas, E.; Gualandris, A.; Dotti, M.; Spera, M.; Mapelli, M.
2016-06-01
Massive black hole binaries (BHBs) are expected to be one of the most powerful sources of gravitational waves (GWs) in the frequency range of the pulsar timing array and of forthcoming space-borne detectors. They are believed to form in the final stages of galaxy mergers, and then harden by slingshot ejections of passing stars. However, evolution via the slingshot mechanism may be ineffective if the reservoir of interacting stars is not readily replenished, and the binary shrinking may come to a halt at roughly a parsec separation. Recent simulations suggest that the departure from spherical symmetry, naturally produced in merger remnants, leads to efficient loss cone refilling, preventing the binary from stalling. However, current N-body simulations able to accurately follow the evolution of BHBs are limited to very modest particle numbers. Brownian motion may artificially enhance the loss cone refilling rate in low-N simulations, where the binary encounters a larger population of stars due its random motion. Here we study the significance of Brownian motion of BHBs in merger remnants in the context of the final parsec problem. We simulate mergers with various particle numbers (from 8k to 1M) and with several density profiles. Moreover, we compare simulations where the BHB is fixed at the centre of the merger remnant with simulations where the BHB is free to random walk. We find that Brownian motion does not significantly affect the evolution of BHBs in simulations with particle numbers in excess of one million, and that the hardening measured in merger simulations is due to collisionless loss cone refilling.
Observing Brownian motion and measuring temperatures in vibration-fluidized granular matter
Understanding the behaviour of granular media, either at rest or moving under external driving, is a difficult task, although it is important and of very practical interest. Describing the motion of each individual grain is complicated, not only because of the large number of grains, but also because the mechanisms of interaction at the grain level involve complex contact forces. One would like to have, in fact, a description in terms of a few macroscopic quantities. Since a granular medium resembles a liquid or a gas when strongly vibrated or when flowing out of a container, a natural approach is to adopt usual equilibrium statistical mechanics tools in order to test if such a macroscopic description is possible. In other words, an interesting question is to investigate whether one can model granular media, when close to a liquid-like state for example, using viscosity, temperature, and so on, as one does for normal liquids. With this aim in view, we have developed a non-equilibrium version of the classical Brownian motion experiment. In particular, we have observed the motion of a torsion oscillator immersed in an externally vibrated granular medium of glass spheres, and have collected evidence that the motion is Brownian-like. An approximate fluctuation-dissipation relation holds, and we can define temperature-like and viscosity-like parameters
Generalized Langevin Theory Of The Brownian Motion And The Dynamics Of Polymers In Solution
The review deals with a generalization of the Rouse and Zimm bead-spring models of the dynamics of flexible polymers in dilute solutions. As distinct from these popular theories, the memory in the polymer motion is taken into account. The memory naturally arises as a consequence of the fluid and bead inertia within the linearized Navier-Stokes hydrodynamics. We begin with a generalization of the classical theory of the Brownian motion, which forms the basis of any theory of the polymer dynamics. The random force driving the Brownian particles is not the white one as in the Langevin theory, but “colored”, i.e., statistically correlated in time, and the friction force on the particles depends on the history of their motion. An efficient method of solving the resulting generalized Langevin equations is presented and applied to the solution of the equations of motion of polymer beads. The memory effects lead to several peculiarities in the time correlation functions used to describe the dynamics of polymer chains. So, the mean square displacement of the polymer coils contains algebraic long-time tails and at short times it is ballistic. It is shown how these features reveal in the experimentally observable quantities, such as the dynamic structure factors of the scattering or the viscosity of polymer solutions. A phenomenological theory is also presented that describes the dependence of these quantities on the polymer concentration in solution. (author)
Berestycki, Julien; Harris, John W; Harris, Simon C; Roberts, Matthew I
2012-01-01
We consider a branching particle system where each particle moves as an independent Brownian motion and breeds at a rate proportional to its distance from the origin raised to the power $p$, for $p\\in[0,2)$. The asymptotic behaviour of the right-most particle for this system is already known; in this article we give large deviations probabilities for particles following "difficult" paths, growth rates along "easy" paths, the total population growth rate, and we derive the optimal paths which particles must follow to achieve this growth rate.
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
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 ...
Brownian motion, random walks on trees, and harmonic measure on polynomial Julia sets
Emerson, Nathaniel D.
2006-01-01
We consider the harmonic measure on a disconnected polynomial Julia set in terms of Brownian motion. We show that the harmonic measure of any connected component of such a Julia set is zero. Associated to the polynomial is a combinatorial model, the tree with dynamics. We define a measure on the tree, which is a combinatorial version on harmonic measure. We show that this measure is isomorphic to the harmonic measure on the Julia set. The measure induces a random walk on the tree, which is is...
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...
The pricing of credit default swaps under a generalized mixed fractional Brownian motion
He, Xinjiang; Chen, Wenting
2014-06-01
In this paper, we consider the pricing of the CDS (credit default swap) under a GMFBM (generalized mixed fractional Brownian motion) model. As the name suggests, the GMFBM model is indeed a generalization of all the FBM (fractional Brownian motion) models used in the literature, and is proved to be able to effectively capture the long-range dependence of the stock returns. To develop the pricing mechanics of the CDS, we firstly derive a sufficient condition for the market modeled under the GMFBM to be arbitrage free. Then under the risk-neutral assumption, the CDS is fairly priced by investigating the two legs of the cash flow involved. The price we obtained involves elementary functions only, and can be easily implemented for practical purpose. Finally, based on numerical experiments, we analyze quantitatively the impacts of different parameters on the prices of the CDS. Interestingly, in comparison with all the other FBM models documented in the literature, the results produced from the GMFBM model are in a better agreement with those calculated from the classical Black-Scholes model.
High-resolution detection of Brownian motion for quantitative optical tweezers experiments.
Grimm, Matthias; Franosch, Thomas; Jeney, Sylvia
2012-08-01
We have developed an in situ method to calibrate optical tweezers experiments and simultaneously measure the size of the trapped particle or the viscosity of the surrounding fluid. The positional fluctuations of the trapped particle are recorded with a high-bandwidth photodetector. We compute the mean-square displacement, as well as the velocity autocorrelation function of the sphere, and compare it to the theory of Brownian motion including hydrodynamic memory effects. A careful measurement and analysis of the time scales characterizing the dynamics of the harmonically bound sphere fluctuating in a viscous medium directly yields all relevant parameters. Finally, we test the method for different optical trap strengths, with different bead sizes and in different fluids, and we find excellent agreement with the values provided by the manufacturers. The proposed approach overcomes the most commonly encountered limitations in precision when analyzing the power spectrum of position fluctuations in the region around the corner frequency. These low frequencies are usually prone to errors due to drift, limitations in the detection, and trap linearity as well as short acquisition times resulting in poor statistics. Furthermore, the strategy can be generalized to Brownian motion in more complex environments, provided the adequate theories are available. PMID:23005790
Supercritical super-Brownian motion with a general branching mechanism and travelling waves
Kyprianou, A E; Murillo-Salas, A; Ren, Y -X
2010-01-01
We consider the classical problem of existence, uniqueness and asymptotics of monotone solutions to the travelling wave equation associated to the parabolic semi-group equation of a super-Brownian motion with a general branching mechanism. Whilst we are strongly guided by the probabilistic reasoning of Kyprianou (2004) for branching Brownian motion, the current paper offers a number of new insights. Our analysis incorporates the role of Seneta-Heyde norming which, in the current setting, draws on classical work of Grey (1974). We give a pathwise explanation of Evans' immortal particle picture (the spine decomposition) which uses the Dynkin-Kuznetsov N-measure as a key ingredient. Moreover, in the spirit of Neveu's stopping lines we make repeated use of Dynkin's exit measures. Additional complications arise from the general nature of the branching mechanism. As a consequence of the analysis we also offer an exact X(log X)^2 moment dichotomy for the almost sure convergence of the so-called derivative martingale...
Limitation of the Least Square Method in the Evaluation of Dimension of Fractal Brownian Motions
Qiao, Bingqiang; Zeng, Houdun; Li, Xiang; Dai, Benzhong
2015-01-01
With the standard deviation for the logarithm of the re-scaled range $\\langle |F(t+\\tau)-F(t)|\\rangle$ of simulated fractal Brownian motions $F(t)$ given in a previous paper \\cite{q14}, the method of least squares is adopted to determine the slope, $S$, and intercept, $I$, of the log$(\\langle |F(t+\\tau)-F(t)|\\rangle)$ vs $\\rm{log}(\\tau)$ plot to investigate the limitation of this procedure. It is found that the reduced $\\chi^2$ of the fitting decreases with the increase of the Hurst index, $H$ (the expectation value of $S$), which may be attributed to the correlation among the re-scaled ranges. Similarly, it is found that the errors of the fitting parameters $S$ and $I$ are usually smaller than their corresponding standard deviations. These results show the limitation of using the simple least square method to determine the dimension of a fractal time series. Nevertheless, they may be used to reinterpret the fitting results of the least square method to determine the dimension of fractal Brownian motions more...
In this article we present methods for measuring hindered Brownian motion in the confinement of complex 3D geometries using digital video microscopy. Here we discuss essential features of automated 3D particle tracking as well as diffusion data analysis. By introducing local mean squared displacement-vs-time curves, we are able to simultaneously measure the spatial dependence of diffusion coefficients, tracking accuracies and drift velocities. Such local measurements allow a more detailed and appropriate description of strongly heterogeneous systems as opposed to global measurements. Finite size effects of the tracking region on measuring mean squared displacements are also discussed. The use of these methods was crucial for the measurement of the diffusive behavior of spherical polystyrene particles (505 nm diameter) in a microfluidic chip. The particles explored an array of parallel channels with different cross sections as well as the bulk reservoirs. For this experiment we present the measurement of local tracking accuracies in all three axial directions as well as the diffusivity parallel to the channel axis while we observed no significant flow but purely Brownian motion. Finally, the presented algorithm is suitable also for tracking of fluorescently labeled particles and particles driven by an external force, e.g., electrokinetic or dielectrophoretic forces
Brownian motion of massive black hole binaries and the final parsec problem
Bortolas, E; Dotti, M; Spera, M; Mapelli, M
2016-01-01
Massive black hole binaries (BHBs) are expected to be one of the most powerful sources of gravitational waves (GWs) in the frequency range of the pulsar timing array and of forthcoming space-borne detectors. They are believed to form in the final stages of galaxy mergers, and then harden by slingshot ejections of passing stars. However, evolution via the slingshot mechanism may be ineffective if the reservoir of interacting stars is not readily replenished, and the binary shrinking may come to a halt at roughly a parsec separation. Recent simulations suggest that the departure from spherical symmetry, naturally produced in merger remnants, leads to efficient loss cone refilling, preventing the binary from stalling. However, current N-body simulations able to accurately follow the evolution of BHBs are limited to very modest particle numbers. Brownian motion may artificially enhance the loss cone refilling rate in low-N simulations, where the binary encounters a larger population of stars due its random motion...
T. Turiv
2015-06-01
Full Text Available As recently reported [Turiv T. et al., Science, 2013, Vol. 342, 1351], fluctuations in the orientation of the liquid crystal (LC director can transfer momentum from the LC to a colloid, such that the diffusion of the colloid becomes anomalous on a short time scale. Using video microscopy and single particle tracking, we investigate random thermal motion of colloidal particles in a nematic liquid crystal for the time scales shorter than the expected time of director fluctuations. At long times, compared to the characteristic time of the nematic director relaxation we observe typical anisotropic Brownian motion with the mean square displacement (MSD linear in time τ and inversly proportional to the effective viscosity of the nematic medium. At shorter times, however, the dynamics is markedly nonlinear with MSD growing more slowly (subdiffusion or faster (superdiffusion than τ. These results are discussed in the context of coupling of colloidal particle's dynamics to the director fluctuation dynamics.
We calculate the spectra of the correlation functions for fields with arbitrary spatial dependence as seen by Brownian particles in bounded geometries from knowledge of the spectra of the conditional probability density functions in the infinite domain. Our results show a significant difference for the spectra for 1D, 2D and 3D motions. To our knowledge this is the first demonstration of the influence of dimensionality on the form of the correlation functions. Our results also show the different power dependence on frequency for the ballistic and diffusive cases and the treatment of the crossover is unique. -- Highlights: ► Spectrum of the correlation function for fields with arbitrary spatial dependence. ► Valid for the diffusive and ballistic motions and the problematic crossover region. ► Calculation of the spectra in 1, 2 and 3 dimensions. ► Application to gradient induced relaxation in nmr, and to the search for new forces.
Quantum harmonic Brownian motion in a general environment: A modified phase-space approach
Yeh, L. [Univ. of California, Berkeley, CA (United States). Dept. of Physics]|[Lawrence Berkeley Lab., CA (United States)
1993-06-23
After extensive investigations over three decades, the linear-coupling model and its equivalents have become the standard microscopic models for quantum harmonic Brownian motion, in which a harmonically bound Brownian particle is coupled to a quantum dissipative heat bath of general type modeled by infinitely many harmonic oscillators. The dynamics of these models have been studied by many authors using the quantum Langevin equation, the path-integral approach, quasi-probability distribution functions (e.g., the Wigner function), etc. However, the quantum Langevin equation is only applicable to some special problems, while other approaches all involve complicated calculations due to the inevitable reduction (i.e., contraction) operation for ignoring/eliminating the degrees of freedom of the heat bath. In this dissertation, the author proposes an improved methodology via a modified phase-space approach which employs the characteristic function (the symplectic Fourier transform of the Wigner function) as the representative of the density operator. This representative is claimed to be the most natural one for performing the reduction, not only because of its simplicity but also because of its manifestation of geometric meaning. Accordingly, it is particularly convenient for studying the time evolution of the Brownian particle with an arbitrary initial state. The power of this characteristic function is illuminated through a detailed study of several physically interesting problems, including the environment-induced damping of quantum interference, the exact quantum Fokker-Planck equations, and the relaxation of non-factorizable initial states. All derivations and calculations axe shown to be much simplified in comparison with other approaches. In addition to dynamical problems, a novel derivation of the fluctuation-dissipation theorem which is valid for all quantum linear systems is presented.
Quantum harmonic Brownian motion in a general environment: A modified phase-space approach
After extensive investigations over three decades, the linear-coupling model and its equivalents have become the standard microscopic models for quantum harmonic Brownian motion, in which a harmonically bound Brownian particle is coupled to a quantum dissipative heat bath of general type modeled by infinitely many harmonic oscillators. The dynamics of these models have been studied by many authors using the quantum Langevin equation, the path-integral approach, quasi-probability distribution functions (e.g., the Wigner function), etc. However, the quantum Langevin equation is only applicable to some special problems, while other approaches all involve complicated calculations due to the inevitable reduction (i.e., contraction) operation for ignoring/eliminating the degrees of freedom of the heat bath. In this dissertation, the author proposes an improved methodology via a modified phase-space approach which employs the characteristic function (the symplectic Fourier transform of the Wigner function) as the representative of the density operator. This representative is claimed to be the most natural one for performing the reduction, not only because of its simplicity but also because of its manifestation of geometric meaning. Accordingly, it is particularly convenient for studying the time evolution of the Brownian particle with an arbitrary initial state. The power of this characteristic function is illuminated through a detailed study of several physically interesting problems, including the environment-induced damping of quantum interference, the exact quantum Fokker-Planck equations, and the relaxation of non-factorizable initial states. All derivations and calculations axe shown to be much simplified in comparison with other approaches. In addition to dynamical problems, a novel derivation of the fluctuation-dissipation theorem which is valid for all quantum linear systems is presented
Híjar, Humberto
2015-02-01
We study the Brownian motion of a particle bound by a harmonic potential and immersed in a fluid with a uniform shear flow. We describe this problem first in terms of a linear Fokker-Planck equation which is solved to obtain the probability distribution function for finding the particle in a volume element of its associated phase space. We find the explicit form of this distribution in the stationary limit and use this result to show that both the equipartition law and the equation of state of the trapped particle are modified from their equilibrium form by terms increasing as the square of the imposed shear rate. Subsequently, we propose an alternative description of this problem in terms of a generalized Langevin equation that takes into account the effects of hydrodynamic correlations and sound propagation on the dynamics of the trapped particle. We show that these effects produce significant changes, manifested as long-time tails and resonant peaks, in the equilibrium and nonequilibrium correlation functions for the velocity of the Brownian particle. We implement numerical simulations based on molecular dynamics and multiparticle collision dynamics, and observe a very good quantitative agreement between the predictions of the model and the numerical results, thus suggesting that this kind of numerical simulations could be used as complement of current experimental techniques. PMID:25768490
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.
PENG ShiGe
2009-01-01
This is a survey on normal distributions and the related central limit theorem under sublinear expectation. We also present Brownian motion under sublinear expectations and the related stochastic calculus of Ito's type. The results provide new and robust tools for the problem of probability model uncertainty arising in financial risk, statistics and other industrial problems.
2009-01-01
This is a survey on normal distributions and the related central limit theorem under sublinear expectation.We also present Brownian motion under sublinear expectations and the related stochastic calculus of It?’s type.The results provide new and robust tools for the problem of probability model uncertainty arising in financial risk,statistics and other industrial problems.
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.
Gayo Willy
2016-01-01
Full Text Available Philippine Stock Exchange Composite Index (PSEi is the main stock index of the Philippine Stock Exchange (PSE. PSEi is computed using a weighted mean of the top 30 publicly traded companies in the Philippines, called component stocks. It provides a single value by which the performance of the Philippine stock market is measured. Unfortunately, these weights, which may vary for every trading day, are not disclosed by the PSE. In this paper, we propose a model of forecasting the PSEi by estimating the weights based on historical data and forecasting each component stock using Monte Carlo simulation based on a Geometric Brownian Motion (GBM assumption. The model performance is evaluated and its forecast compared is with the results using a direct GBM forecast of PSEi over different forecast periods. Results showed that the forecasts using WGBM will yield smaller error compared to direct GBM forecast of PSEi.
Intermittency in an interacting generalization of the geometric Brownian motion model
We propose a minimal interacting generalization of the geometric Brownian motion model, which turns out to be formally equivalent to a model describing the dynamics of networks of analogue neurons. For sufficiently strong interactions, such systems may have many meta-stable states. Transitions between meta-stable states are associated with macroscopic reorganizations of the system, which can be triggered by random external forcing. Such a system will exhibit intermittent dynamics within a large part of its parameter space. We propose market dynamics as a possible application of this model, in which case random external forcing would correspond to the arrival of important information. The emergence of a model of interacting prices of the type considered here can be argued to follow naturally from a general argument based on integrating out all non-price degrees of freedom from the dynamics of a hypothetical complete description of economic dependences
The dynamics of a complex system is usually recorded in the form of time series, which can be studied through its visibility graph from a complex network perspective. We investigate the visibility graphs extracted from fractional Brownian motions and multifractal random walks, and find that the degree distributions exhibit power-law behaviors, in which the power-law exponent α is a linear function of the Hurst index H of the time series. We also find that the degree distribution of the visibility graph is mainly determined by the temporal correlation of the original time series with minor influence from the possible multifractal nature. As an example, we study the visibility graphs constructed from three Chinese stock market indexes and unveil that the degree distributions have power-law tails, where the tail exponents of the visibility graphs and the Hurst indexes of the indexes are close to the α∼H linear relationship.
Brownian motion properties of optoelectronic random bit generators based on laser chaos.
Li, Pu; Yi, Xiaogang; Liu, Xianglian; Wang, Yuncai; Wang, Yongge
2016-07-11
The nondeterministic property of the optoelectronic random bit generator (RBG) based on laser chaos are experimentally analyzed from two aspects of the central limit theorem and law of iterated logarithm. The random bits are extracted from an optical feedback chaotic laser diode using a multi-bit extraction technique in the electrical domain. Our experimental results demonstrate that the generated random bits have no statistical distance from the Brownian motion, besides that they can pass the state-of-the-art industry-benchmark statistical test suite (NIST SP800-22). All of them give a mathematically provable evidence that the ultrafast random bit generator based on laser chaos can be used as a nondeterministic random bit source. PMID:27410852
A Brownian motion technique to simulate gasification and its application to C/C composite ablation
Ablation of carbon-carbon composites (C/C) results in a heterogeneous surface recession mainly due to some gasification processes (oxidation, sublimation) possibly coupled to bulk mass transfer. In order to simulate and analyse the material/environment interactions during ablation, a Brownian motion simulation method featuring special Random Walk rules close to the wall has been implemented to efficiently simulate mass transfer in the low Peclet number regime. A sticking probability law adapted to this kind of Random Walk has been obtained for first-order heterogeneous reactions. In order to simulate the onset of surface roughness, the interface recession is simultaneously handled in 3D using a Simplified Marching Cube discretization. This tool is validated by comparison to analytical models. Then, its ability to provide reliable and accurate solutions of ablation phenomena in 3D is illustrated. (authors)
Verrier, Nicolas; Fournel, Thierry
2015-01-01
In-line digital holography is a valuable tool for sizing, locating and tracking micro- or nano-objects in a volume. When a parametric imaging model is available, Inverse Problems approaches provide a straightforward estimate of the object parameters by fitting data with the model, thereby allowing accurate reconstruction. As recently proposed and demonstrated, combining pixel super-resolution techniques with Inverse Problems approaches improves the estimation of particle size and 3D-position. Here we demonstrate the accurate tracking of colloidal particles in Brownian motion. Particle size and 3D-position are jointly optimized from video holograms acquired with a digital holographic microscopy set up based on a "low-end" microscope objective ($\\times 20$, $\\rm NA\\ 0.5$). Exploiting information redundancy makes it possible to characterize particles with a standard deviation of 15 nm in size and a theoretical resolution of 2 x 2 x 5 nm$^3$ for position under additive white Gaussian noise assumption.
Effect of solvent on directional drift in Brownian motion of particle/molecule with broken symmetry
Kong, FanDong; Sheng, Nan; Wan, RongZheng; Hu, GuoHui; Fang, HaiPing
2016-08-01
The directional drifting of particles/molecules with broken symmetry has received increasing attention. Through molecular dynamics simulations, we investigate the effects of various solvents on the time-dependent directional drifting of a particle with broken symmetry. Our simulations show that the distance of directional drift of the asymmetrical particle is reduced while the ratio of the drift to the mean displacement of the particle is enhanced with increasing mass, size, and interaction strength of the solvent atoms in a short time range. Among the parameters considered, solvent atom size is a particularly influential factor for enhancing the directional drift of asymmetrical particles, while the effects of the interaction strength and the mass of the solvent atoms are relatively weaker. These findings are of great importance to the understanding and control of the Brownian motion of particles in various physical, chemical, and biological processes within finite time spans.
Super-Brownian motion: Lp-convergence of martingales through the pathwise spine decomposition
Murillo-Salas, A E Kyprianou A
2011-01-01
Evans (1992) described the semi-group of a superprocess with quadratic branching mechanism under a martingale change of measure in terms of the semi-group of an immortal particle and the semigroup of the superprocess prior to the change of measure. This result, commonly referred to as the spine decomposition, alludes to a pathwise decomposition in which independent copies of the original process `immigrate' along the path of the immortal particle. For branching particle diffusions the analogue of this decomposition has already been demonstrated in the pathwise sense, see for example Hardy and Harris (2009). The purpose of this short note is to exemplify a new {\\it pathwise} spine decomposition for supercritical super-Brownian motion with general branching mechanism (cf. Kyprianou et al. (2010)) by studying $L^p$ convergence of naturally underlying additive martingales in the spirit of analogous arguments for branching particle diffusions due to Hardys and Harris (2009). Amongst other ingredients, the Dynkin-K...
Inference on the hurst parameter and the variance of diffusions driven by fractional Brownian motion
Berzin, Corinne; León, José R
2014-01-01
This book is devoted to a number of stochastic models that display scale invariance. It primarily focuses on three issues: probabilistic properties, statistical estimation and simulation of the processes considered. It will be of interest to probability specialists, who will find here an uncomplicated presentation of statistics tools, and to those statisticians who wants to tackle the most recent theories in probability in order to develop Central Limit Theorems in this context; both groups will also benefit from the section on simulation. Algorithms are described in great detail, with a focus on procedures that is not usually found in mathematical treatises. The models studied are fractional Brownian motions and processes that derive from them through stochastic differential equations. Concerning the proofs of the limit theorems, the “Fourth Moment Theorem” is systematically used, as it produces rapid and helpful proofs that can serve as models for the future. Readers will also find elegant and new proof...
Ni, Xiao-Hui; Jiang, Zhi-Qiang; Zhou, Wei-Xing
2009-10-01
The dynamics of a complex system is usually recorded in the form of time series, which can be studied through its visibility graph from a complex network perspective. We investigate the visibility graphs extracted from fractional Brownian motions and multifractal random walks, and find that the degree distributions exhibit power-law behaviors, in which the power-law exponent α is a linear function of the Hurst index H of the time series. We also find that the degree distribution of the visibility graph is mainly determined by the temporal correlation of the original time series with minor influence from the possible multifractal nature. As an example, we study the visibility graphs constructed from three Chinese stock market indexes and unveil that the degree distributions have power-law tails, where the tail exponents of the visibility graphs and the Hurst indexes of the indexes are close to the α∼H linear relationship.
Stochastic shell models driven by a multiplicative fractional Brownian-motion
Bessaih, Hakima; Garrido-Atienza, María J.; Schmalfuss, Björn
2016-04-01
We prove existence and uniqueness of the solution of a stochastic shell-model. The equation is driven by an infinite dimensional fractional Brownian-motion with Hurst-parameter H ∈(1 / 2 , 1) , and contains a non-trivial coefficient in front of the noise which satisfies special regularity conditions. The appearing stochastic integrals are defined in a fractional sense. First, we prove the existence and uniqueness of variational solutions to approximating equations driven by piecewise linear continuous noise, for which we are able to derive important uniform estimates in some functional spaces. Then, thanks to a compactness argument and these estimates, we prove that these variational solutions converge to a limit solution, which turns out to be the unique pathwise mild solution associated to the shell-model with fractional noise as driving process.
Minchew, Candace L.; Didenko, Vladimir V.
2014-01-01
We describe a new type of bio-nanomachine which runs on thermal noise. The machine is solely powered by the random motion of water molecules in its environment and does not ever require re-fuelling. The construct, which is made of DNA and vaccinia virus topoisomerase protein, can detect DNA damage by employing fluorescence. It uses Brownian motion as a cyclic motor to continually separate and bring together two types of fluorescent hairpins participating in FRET. This bio-molecular oscillator is a fast and specific sensor of 5′OH double-strand DNA breaks present in phagocytic phase of apoptosis. The detection takes 30 s in solution and 3 min in cell suspensions. The phagocytic phase is critical for the effective execution of apoptosis as it ensures complete degradation of the dying cells’ DNA, preventing release of pathological, viral and tumor DNA and self-immunization. The construct can be used as a smart FRET probe in studies of cell death and phagocytosis. PMID:25268504
From Mechanical Motion to Brownian Motion, Thermodynamics and Particle Transport Theory
Bringuier, E.
2008-01-01
The motion of a particle in a medium is dealt with either as a problem of mechanics or as a transport process in non-equilibrium statistical physics. The two kinds of approach are often unrelated as they are taught in different textbooks. The aim of this paper is to highlight the link between the mechanical and statistical treatments of particle…
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.
Randrup, Jorgen; Moller, Peter
2011-01-01
Although nuclear fission can be understood qualitatively as an evolution of the nuclear shape, a quantitative description has proven to be very elusive. In particular, until now, there exists no model with demonstrated predictive power for the fission fragment mass yields. Exploiting the expected strongly damped character of nuclear dynamics, we treat the nuclear shape evolution in analogy with Brownian motion and perform random walks on five-dimensional fission potential-energy surfaces whic...
Xie, Wen-Jie
2010-01-01
Nonlinear time series analysis aims at understanding the dynamics of stochastic or chaotic processes. In recent years, quite a few methods have been proposed to transform a single time series to a complex network so that the dynamics of the process can be understood by investigating the topological properties of the network. We study the topological properties of horizontal visibility graphs constructed from fractional Brownian motions with different Hurst index $H\\in(0,1)$. Special attention has been paid to the impact of Hurst index on the topological properties. It is found that the clustering coefficient $C$ decreases when $H$ increases. We also found that the mean length $L$ of the shortest paths increases exponentially with $H$ for fixed length $N$ of the original time series. In addition, $L$ increases linearly with respect to $N$ when $H$ is close to 1 and in a logarithmic form when $H$ is close to 0. Although the occurrence of different motifs changes with $H$, the motif rank pattern remains unchange...
Numerically pricing American options under the generalized mixed fractional Brownian motion model
Chen, Wenting; Yan, Bowen; Lian, Guanghua; Zhang, Ying
2016-06-01
In this paper, we introduce a robust numerical method, based on the upwind scheme, for the pricing of American puts under the generalized mixed fractional Brownian motion (GMFBM) model. By using portfolio analysis and applying the Wick-Itô formula, a partial differential equation (PDE) governing the prices of vanilla options under the GMFBM is successfully derived for the first time. Based on this, we formulate the pricing of American puts under the current model as a linear complementarity problem (LCP). Unlike the classical Black-Scholes (B-S) model or the generalized B-S model discussed in Cen and Le (2011), the newly obtained LCP under the GMFBM model is difficult to be solved accurately because of the numerical instability which results from the degeneration of the governing PDE as time approaches zero. To overcome this difficulty, a numerical approach based on the upwind scheme is adopted. It is shown that the coefficient matrix of the current method is an M-matrix, which ensures its stability in the maximum-norm sense. Remarkably, we have managed to provide a sharp theoretic error estimate for the current method, which is further verified numerically. The results of various numerical experiments also suggest that this new approach is quite accurate, and can be easily extended to price other types of financial derivatives with an American-style exercise feature under the GMFBM model.
Detection of two-sided alternatives in a Brownian motion model
Hadjiliadis, Olympia
2007-01-01
This work examines the problem of sequential detection of a change in the drift of a Brownian motion in the case of two-sided alternatives. Applications to real life situations in which two-sided changes can occur are discussed. Traditionally, 2-CUSUM stopping rules have been used for this problem due to their asymptotically optimal character as the mean time between false alarms tends to $\\infty$. In particular, attention has focused on 2-CUSUM harmonic mean rules due to the simplicity in calculating their first moments. In this paper, we derive closed-form expressions for the first moment of a general 2-CUSUM stopping rule. We use these expressions to obtain explicit upper and lower bounds for it. Moreover, we derive an expression for the rate of change of this first moment as one of the threshold parameters changes. Based on these expressions we obtain explicit upper and lower bounds to this rate of change. Using these expressions we are able to find the best 2-CUSUM stopping rule with respect to the exten...