Brownian motion with inert drift, but without flux: A model

We study the motion of a Brownian particle which interacts with a stationary obstacle in two dimensions. The Brownian particle acquires drift proportionally to the time spent on the boundary of the obstacle. The system approaches equilibrium, and the equilibrium distribution for the location and drift magnitude has the product form. The distribution for the location is uniform, while the drift distribution depends on the shape of the obstacle, resembling a gamma function for the circular or elliptic obstacle.     

First passage time statistics of Brownian motion with purely time dependent drift and diffusion

Systems where resource availability approaches a critical threshold are common to many engineering and scientific applications and often necessitate the estimation of first passage time statistics of a Brownian motion (Bm) driven by time-dependent drift and diffusion coefficients. Modeling such systems requires solving the associated Fokker-Planck equation subject to an absorbing barrier. Transitional probabilities are derived via the method of images, whose applicability to time dependent problems is shown to be limited to state-independent drift and diffusion coefficients that only depend on time and are proportional to each other. First passage time statistics, such as the survival probabilities and first passage time densities are obtained analytically. The analysis includes the study of different functional forms of the time dependent drift and diffusion, including power-law time dependence and different periodic drivers. As a case study of these theoretical results, a stochastic model of water resources availability in snowmelt dominated regions is presented, where both temperature effects and snow-precipitation input are incorporated.     

Generalized Brownian motion and elasticity

We introduce a family of stochastic processes which are a natural extension of Brownian motion to a tensor form. This allows us to solve a Dirichlet problem of linear elasticity obeying Lamé's equation, [1–(d– 1)]²V(x)+ [·V(x)]=0.     

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.     

Brownian motion near an absorbing sphere

The spherically symmetric solution of the Fokker-Planck equation with absorbing boundary is given in terms of a solution of an equivalent integral equation whose explicit form is found.     

Long-time translation and rotational Brownian motion in two dimensions

The long-time translational and rotational motion of a Brownian particle in two dimensions is studied on the basis of the fluctuation-dissipation theorem and linearized hydrodynamics. The long-time motion follows from the low frequency behavior of the mobility matrix. The coefficient of the long-time tail for the translational motion turns out to be independent of shape and size of the body, in agreement with mode-coupling theory. For rotational Brownian motion the coefficient of the long-time tail is found to depend on the shape of the body. This result is in conflict with a recent prediction from mode-coupling theory, and indicates that the mode-coupling calculation should be revised.This article is dedicated in friendship to Prof. Matthieu Ernst on the occasion of his 60th birthday.     

Asymptotic laws for the winding angles of planar Brownian motion

Using constrained path integrals, we study the winding angle distribution of a two-dimensional Brownian motion around a given point. By a careful analysis of the spectral properties of some Schrödinger-like Hamiltonians, we obtain a generalization of the Messulam-Yor law. Various limiting cases are considered.     

Brownian motion of a nonlinear oscillator

Starting with the Langevin equation for a nonlinear oscillator (the Duffing oscillator) undergoing ordinary Brownian motion, we derive linear transport laws for the motion of the average position and velocity of the oscillator. The resulting linear equations are valid for only small deviations of average values from thermal equilibrium. They contain a renormalized oscillator frequency and a renormalized and non-Markovian friction coefficient, both depending on the nonlinear part of the original equation of motion. Numerical computations of the position correlation function and its spectral density are presented. The spectral density compares favorably with experimental results obtained by Morton using an analog computer method.Technical Note BN-674. Research supported in part by NSF grant GP-12591, and in part by PHS Research Grant No. MG16426-02 from the National Institute of General Medical Sciences.     

Brownian motion on a manifold

The question of the existence and correct form of equations describing Brownian motion on a manifold cannot be answered by mathematics alone, but requires a study of the underlying physics. As in classical mechanics, manifolds enter through the transformation of variables needed to account for the presence of constraints. The constraints are either due to a physical agency that forces the motion to remain on a manifold, or they represent conserved quantities of the equation of motion themselves. Also the Brownian motion is described either by a Smoluchowski diffusion equation or by a Kramers equation. The four cases lead to the following conclusions, (i) Smoluchowski diffusion with a conserved quantity reduces to a diffusion equation on the manifold; (ii) The same is true for diffusion with a physical constraint in three dimensions, but in more dimensions it may happen thatno autonomous equation on the manifold results; (iii) A Kramers equation with a conserved quantity reduces to an equation on the manifold, but in general not of the form of a Kramers equation; (iv) The Kramers equation with a physical constraint reduces to an autonomous Kramers equation on the manifold only for a special shape of that constraint. Throughout, only a certain type of physical constraints has been envisaged, and global questions are ignored. Finally, the customary heuristic construction of a Fokker-Planck equation for a mechanical system on a manifold is demonstrated for the case of Brownian rotation of a rigid body, and its shortcomings are emphasized.     

Kubo-Einstein relation for quantum Brownian motion in a periodic potential

Using a previously derived general formalism for a dissipative quantum particle in a boson bath, we prove that when the damping is Ohmic, the Kubo-Einstein relation between the diffusion constant and the linear mobilityD=kTM holds to all orders in V₀ for a periodic potentialV(x)=V ₀ cos(k)₀ x).     

海杂波的分段分数布朗运动模型

主要研究了分段分数布朗运动(PFBM)模型在雷达海杂波分形建模中的应用.由于自然界和人造系统中研究对象不具有数学上完美的分形特性, 从而研究对象的分形特性无法在整个尺度区间上成立, 传统上, 海杂波的单一分形模型仅利用无标度区间内海杂波的自相似信息进行参数估计, 并没有考虑海杂波在无标度区间以外的尺度下所包含的信息.分段分数布朗运动从频域角度对海杂波频谱进行分段描述, 对应到时域即从粗略尺度和精细尺度两方面描述海杂波时间序列.结合海杂波产生的物理背景, 该模型可以为海杂波时间序列在粗略尺度和精细尺度下表现出的不同粗糙度提供机理性解释.在此基础上, 还研究了具有不同多普勒频率的运动目标对海杂波的影响, 结果表明运动目标对粗略尺度和精细尺度下海杂波的影响程度是不同的.     

A mechanical model of Brownian motion in half-space

We consider the motion of a heavy mass in an ideal gas in a semi-infinite system, with elastic collisions at the boundary. The motion is determined by elastic collisions. We prove in the Brownian motion limit the convergence of the position and velocity process of the heavy particle to a diffusion process in which velocity and position remain coupled.     

Decoherence of quantum Brownian motion in noncommutative space

The decoherence of a harmonic oscillator under two-dimensional quantum Brownian motion on a noncommutative plane is investigated. The interaction with the environment is considered by two separate models so-called coupled and uncoupled. The two-dimensional master equation and its noncommutative counterpart are derived for both employed models. The rate of the linear entropy (predictability sieve) is chosen as a criterion to investigate the purity in the presence of the space noncommutativity. Besides, a two-dimensional charged harmonic oscillator on a plane which is imposed by a perpendicular magnetic field is introduced as a realization of our model. Therefore, our approach provides a formalism to investigate the influence of the magnetic field on the decoherence of the pure states. We show that in the high magnetic field limit the rate of the decoherence will be decreased.     

Brownian motion of suspended particles in an anisotropic medium

Some experimental results in the case of anisotropic media are at variance with the Einstein-Stokes formula of Brownian motion and the Singwi-Sjölander model. The disagreement comes from the anchor effect. The concept of “pseudoparticle” is introduced in order to retain the Singwi-Sjölander formalism. Mössbauer spectroscopy can be used to estimate the thickness of the anchored liquid-crystalline molecular textures forming on the surface of Brownian particles.     

A dynamical theory of Brownian motion for the Rayleigh gas

A dynamical theory of the Brownian motion is worked out for the Rayleigh gas and open problems of this theory are surveyed.     

Brownian motion and correlation in particle image velocimetry

In particle image velocimetry applications involving either low velocities or small seed particles, Brownian motion can be significant. This paper addresses the effects of Brownian motion. First, general equations describing cross-correlation particle image velocimetry are derived that include Brownian motion. When light-sheet illumination particle image velocimetry (PIV) is used Brownian motion diminishes the signal strength. A parameter describing this effect is introduced, and a weighting function describing the contribution to the measured velocity as a function of position is derived. The latter is unaffected by Brownian motion. Microscopic PIV Brownian motion also diminishes the signal strength. The weighting function for microscopic PIV is found to depend on Brownian motion, thus affecting an important experimental parameter, the depth of correlation. For both light-sheet illumination and microscopic PIV, a major consequence of Brownian motion is the spreading of the correlation signal peak. Because the magnitude of the spreading is dependent on temperature, PIV can, in principle, be used to simultaneously measure velocity and temperature. The location of the signal peak provides the velocity data, while the spreading of the peak yields temperature.     

Brownian motion of a classical particle in quantum environment

The Klein–Kramers equation, governing the Brownian motion of a classical particle in a quantum environment under the action of an arbitrary external potential, is derived. Quantum temperature and friction operators are introduced and at large friction the corresponding Smoluchowski equation is obtained. Introducing the Bohm quantum potential, this Smoluchowski equation is extended to describe the Brownian motion of a quantum particle in quantum environment.     

Brownian motion at short time scales

Brownian motion has played important roles in many different fields of science since its origin was first explained by Albert Einstein in 1905. Einstein's theory of Brownian motion, however, is only applicable at long time scales. At short time scales, Brownian motion of a suspended particle is not completely random, due to the inertia of the particle and the surrounding fluid. Moreover, the thermal force exerted on a particle suspended in a liquid is not a white noise, but is colored. Recent experimental developments in optical trapping and detection have made this new regime of Brownian motion accessible. This review summarizes related theories and recent experiments on Brownian motion at short time scales, with a focus on the measurement of the instantaneous velocity of a Brownian particle in a gas and the observation of the transition from ballistic to diffusive Brownian motion in a liquid.     

Brownian motion of massive skyrmions in magnetic thin films

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.