Brownian motion in a fluid in elongational flow

Brownian motion of a spherical particle in stationary elongational flow is studied. We derive the Langevin equation together with the fluctuation-dissipation theorem for the particle from nonequilibrium fluctuating hydrodynamics to linear order in the elongation-rate-dependent inverse penetration depths. We then analyze how the velocity autocorrelation function as well as the mean square displacement are modified by the elongational flow. We find that for times small compared to the inverse elongation rate the behavior is similar to that found in the absence of the elongational flow. Upon approaching times comparable to the inverse elongation rate the behavior changes and one passes into a time domain where it becomes fundamentally different. In particular, we discuss the modification of thet ^–3/2 long-time tail of the velocity autocorrelation function and comment on the resulting contribution to the mean square displacement. The possibility of defining a diffusion coefficient in both time domains is discussed.     

Collective Transport of Coupled Brownian Motors with Low Randomness

The transport properties of coupled Brownian motors in rocking ratchet are investigated via solving Langevin equation. By means of velocity, diffusion coefficient, and their ratio （Peclet number）, different features from a single particle have been found. In the regime of low-to-moderate D, the average velocity of elastically coupled Brownian motors is larger than that of a single Brownian particles; the Peclet number of elastically coupled Brownian motors is peaked functions of intensity of noise D but the Peclet number of a single Brownian motor decreases monotonously with the increase of a single Brownian motor. The results exhibit an interesting cooperative behavior between coupled particles subjected to a rocking force, which can generate directed transport with low randomness or high transport coherence in symmetrical periodic potential.     

Coupling effect of Brownian motion and laminar shear flow on colloid coagulation:a Brownian dynamics simulation study

Simultaneous orthokinetic and perikinetic coagulations(SOPCs) are studied for small and large Peclet numbers(P e) 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 P 1/2 e,while with high Peclet numbers,the ratio of overall coagulation rate to the rate of pure orthokinetic coagulation is proportional to P 1/2 e.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).     

Performance characteristics of an irreversible thermally driven Brownian microscopic heat engine

Brownian particles moving in a spatially asymmetric but periodic potential (ratchet), with an external load force and connected to an alternating hot and cold reservoir, are modeled as a microscopic heat engine, referred to as the Brownian heat engine. The heat flow via both the potential energy and the kinetic energy of the particles are considered simultaneously. The forward and backward particle currents are determined using an Arrhenius' factor. Expressions for the power output and efficiency are derived analytically. The maximum power output and efficiency are calculated. It is expounded that the Brownian heat engine is always irreversible and its efficiency cannot approach the efficiency η_C of the Carnot heat engine even in quasistatic limit. The influence of the main parameters such as the load, the barrier height of the potential, the asymmetry of the potential and the temperature ratio of the heat reservoirs on the performance of the Brownian heat engine is discussed in detail. It is found that the Brownian heat engines may be controlled to operate in different regions through variation of some parameters.     

Brownian motion under a time-dependent periodic potential proportional to the square of the position of a particle and classical fluctuation squeezing

A general stationary case of a Brownian particle with a time-dependent periodic potential proportional to the square of the position of the particle is treated. Even though the vigorous change of the time-dependent proportionality coefficient is applied, there are cases where the fluctuation of the particle decreases in contrary to our intuition, which is called classical fluctuation squeezing. We obtain time-average variances analytically for general cases of an arbitrary change in the coefficient and find conditions favorable for classical fluctuation squeezing. We introduce an asymmetric function behaves like trigonometric cosine one and consider its behavior explicitly.     

Lattice Boltzmann simulation of the two-dimensional flow of a magnetic suspension between two parallel walls under a non-uniform magnetic field

We have developed a lattice Boltzmann method based on fluctuation hydrodynamics that is applicable to the flow problem of a particle suspension. In this method, we have introduced the viscosity-modifying method, rather than the velocity-scaling method, in which a modified viscosity is used for generating random forces in lattice Boltzmann simulations. The viscosity-modifying method is found to be applicable to the simulation of a magnetic particle suspension. We have applied this method to the two-dimensional Poiseuille flow of a magnetic suspension between two parallel walls in order to investigate the behavior of magnetic particles in a non-uniform applied magnetic field. From the results of the snapshots, the pair correlation function between the magnetic pole and the magnetic particles and the averaged local particle velocity and magnetization distributions, it was observed that the behavior of the magnetic particles changes significantly depending upon which factor dominates the phenomenon in the balance between the magnetic particle–particle interaction, the non-uniform applied magnetic field and the translational and rotational Brownian motion.     

Brownian excursions on combs

In this paper we use techniques from Ito excursion theory to analyze Brownian motion on generalized combs. Ito excursion theory is a little-known area of probability theory and we therefore present a brief introduction for the uninitiated. A general method for analyzing transport along the backbone of the comb is demonstrated and the specific case of a comb whose teeth are scaling branching trees is examined. We then present a recursive method for evaluating the distribution of the first passage times on hierarchical combs.     

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.     

The kinetic boundary layer for the Fokker-Planck equation with absorbing boundary

The Fokker-Planck equation for the distribution of position and velocity of a Brownian particle is a particularly simple linear transport equation. Its normal solutions and an apparently complete set of stationary boundary layer solutions can be determined explicitly. By a numerical algorithm we select linear combinations of them that approximately fulfill the boundary condition for a completely absorbing plane wall, and that approach a linearly increasing position space density far from the wall. Various aspects of these approximate solutions are discussed. In particular we find that the extrapolated asymptotic density reaches zero at a distance x_M beyond the wall. We find x_M=1.46 in units of the velocity persistence length of the Brownian particle. This study was motivated by certain problems in the theory of diffusion-controlled reactions, and the results might be used to test approximate theories employed in that field.     

长时相干效应与分形布朗运动

 我们研究了阻尼布朗粒子，在具有幂律长时相干C(t)~t^-β（0<β<1，1<β<2）的无规涨落力作用下的运动情况。我们发现它是作分形布朗运动，而不是作普通的布朗运动，而且，找出了分形布朗运动的有效Fokker-Planck方程，以及相应的精确解。于是第一次把长时相干效应和分形布朗运动建立了定量的联系。     

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.     

Charged Brownian particles: Kramers and Smoluchowski equations and the hydrothermodynamical picture

We consider a charged Brownian gas under the influence of external and non-uniform electric, magnetic and mechanical fields, immersed in a non-uniform bath temperature. With the collision time as an expansion parameter, we study the solution to the associated Kramers equation, including a linear reactive term. To the first order we obtain the asymptotic (overdamped) regime, governed by transport equations, namely: for the particle density, a Smoluchowski-reactive like equation; for the particle’s momentum density, a generalized Ohm’s-like equation; and for the particle’s energy density, a Maxwell-Cattaneo-like equation. Defining a nonequilibrium temperature as the mean kinetic energy density, and introducing Boltzmann’s entropy density via the one particle distribution function, we present a complete thermohydrodynamical picture for a charged Brownian gas. We probe the validity of the local equilibrium approximation, Onsager relations, variational principles associated to the entropy production, and apply our results to: carrier transport in semiconductors, hot carriers and Brownian motors. Finally, we outline a method to incorporate non-linear reactive kinetics and a mean field approach to interacting Brownian particles.     

Fractional Brownian Motions and Enhanced Diffusion in a Unidirectional Wave-Like Turbulence

We study transport in random undirectional wave-like velocity fields with nonlinear dispersion relations. For this simple model, we have several interesting findings: (1) In the absence of molecular diffusion the entire family of fractional Brownian motions (FBMs), persistent or anti-persistent, can arise in the scaling limit. (2) The infrared cutoff may alter the scaling limit depending on whether the cutoff exceeds certain critical value or not. (3) Small, but nonzero, molecular diffusion can drastically change the scaling limit. As a result, some regimes stay intact; some (persistent) FBM regimes become non-Gaussian and some other FBM regimes become Brownian motions with enhanced diffusion coefficients. Moreover, in the particular regime where the scaling limit is a Brownian motion in the absence of molecular diffusion, the vanishing molecular diffusion limit of the enhanced diffusion coefficient is strictly larger than the diffusion coefficient with zero molecular diffusion. This is the first such example that we are aware of to demonstrate rigorously a nonperturbative effect of vanishing molecular diffusion on turbulent diffusion coefficient.     

Spherical particle Brownian motion in viscous medium as non-Markovian random process

The Brownian motion of a spherical particle in an infinite medium is described by the conventional methods and integral transforms considering the entrainment of surrounding particles of the medium by the Brownian particle. It is demonstrated that fluctuations of the Brownian particle velocity represent a non-Markovian random process. The features of Brownian motion in short time intervals and in small displacements are considered.     

系统非对称性及记忆性对布朗马达输运行为的影响

在对分数阶布朗马达输运现象研究的基础上,引入了描述系统势场对称性的参数(简称对称性参数),并详细分析了该参数及记忆性参数(分数阶阶数)对粒子输运状态的影响.仿真结果表明,分数阶阶数和对称性参数的共同作用会使得布朗粒子形成定向输运反向流,反向后达到最大平均流速所对应的阶数与外加驱动力频率无关联,但会随对称性参数的增加而单调递增.     

Brownian motion and thermophoresis effects on unsteady stagnation point flow of Eyring-Powell nanofluid: a Galerkin approach

This article concerns the analysis of an unsteady stagnation point flow of Eyring-Powell nanofluid over a stretching sheet. The influence of thermophoresis and Brownian motion is also considered in transport equations. The nonlinear ODE set is obtained from the governing nonlinear equations via suitable transformations. The numerical experiments are performed using the Galerkin scheme. A tabular form comparison analysis of outcomes attained via the Galerkin approach and numerical scheme (RK-4) is available to show the credibility of the Galerkin method. The numerical exploration is carried out for various governing parameters, namely, Brownian motion, steadiness, thermophoresis, stretching ratio, velocity slip, concentration slip, thermal slip, and fluid parameters, and Hartmann, Prandtl and Schmidt numbers. The velocity of fluid enhances with an increase in fluid and magnetic parameters for the case of opposing, but the behavior is reversed for assisting cases. The Brownian motion and thermophoresis parameters cause an increase in temperature for both cases (assisting and opposing). The Brownian motion parameter provides a drop-in concentration while an increase is noticed for the thermophoresis parameter. All the outcomes and the behavior of emerging parameters are illustrated graphically. The comparison analysis and graphical plots endorse the appropriateness of the Galerkin method. It is concluded that said method could be extended to other problems of a complex nature.     

Scaling Limits of a Tagged Particle in the Exclusion Process with Variable Diffusion Coefficient

We prove a law of large numbers and a central limit theorem for a tagged particle in a symmetric simple exclusion process in ℤ with variable diffusion coefficient. The scaling limits are obtained from a similar result for the current through −1/2 for a zero-range process with bond disorder. For the CLT, we prove convergence to a fractional Brownian motion of Hurst exponent 1/4.     

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.     

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.     

少体硬球系统的动力学与统计研究

研究了少数几个封闭于箱子中的硬球组成的系统的动力学与统计行为.着重研究单粒子位形 空间的碰撞分布.计算表明，硬球的半径较小时，单粒子统计分布函数在空间主要是均匀分 布；随着半径的增大，均匀分布部分逐渐减小.当硬球半径与箱子尺寸比值超过临界值时， 单粒子分布函数呈现双峰形式.还利用少体硬球系统模拟布朗运动.研究表明，当硬球系统作 为介质时，系统不存在扩散过程；发现大粒子的平均平方位移与时间是平方关系，说明大粒 子在硬球介质中的输运是弹道输运过程. 关键词： 硬球 动力学 布朗运动 遍历