Uphill anomalous transport in a deterministic system with speed-dependent friction coefficient

We investigate the transport of a deterministic Brownian particle theoretically, which moves in simple onedimensional, symmetric periodic potentials under the influence of both a time periodic and a static biasing force. The physical system employed contains a friction coefficient that is speed-dependent. Within the tailored parameter regime, the absolute negative mobility, in which a particle can travel in the direction opposite to a constant applied force, is observed.This behavior is robust and can be maximized at two regimes upon variation of the characteristic factor of friction coefficient. Further analysis reveals that this uphill motion is subdiffusion in terms of localization(diffusion coefficient with the form D(t) ~t~(-1) at long times). We also have observed the non-trivially anomalous subdiffusion which is significantly deviated from the localization; whereas most of the downhill motion evolves chaotically, with the normal diffusion.     

Quantum Brownian motion in a one-dimensional disordered system

We study the diffusion of a quantum Brownian particle in a one-dimensional periodic potential with substitutional disorder. The particle is coupled to a dissipative environment, which induces a frictional force proportional to the velocity. The dynamics for arbitrary temperature is studied by using Feynman's influence-functional theory. We calculate the mobility to lowest order in the disorder and strength of the periodic potential. It is shown that for weak dissipation the linear mobility, which vanishes atT=0 due to localization effects, may exhibit a maximum and a subsequent minimum with increasing temperature. The relation to the diffusion of heavy particles in metals or doped semiconductors is briefly discussed.     

Derivation of diffusion coefficient of a Brownian particle in tilted periodic potential from the coordinate moments

In this research, diffusion of an overdamped Brownian particle in the tilted periodic potential is investigated. Using the one-dimensional hopping model, the formulations of the mean velocity VN and effective diffusion coefficient DN of the Brownian particle have been obtained [B. Derrida, J. Stat. Phys. 31 (1983) 433]. Based on the relation between the effective diffusion coefficient and the moments of the mean first passage time, the formulation of effective diffusion coefficient Deff of the Brownian particle also has been obtained [P. Reimann, et al., Phys. Rev. E 65 (2002) 031104]. In this research, we'll give another analytical expression of the effective diffusion coefficient Deff from the moments of the particle's coordinate.     

Artificial sieves for quasimassless particles

Brownian motion on a symmetric vibrated periodic substrate is shown to be extremely sensitive to the particle mass even in the regime of large damping. This phenomenon is the most apparent for high vibration frequencies, a condition of technological interest, which is investigated here both analytically and numerically. When plotted versus the damping constant, both the particle mobility and the diffusion coefficient develop sharp (correlated) peaks, thus suggesting efficient schemes for separating submicron particles according to their mass or geometry.     

Brownian diffusion in a nonuniform gas

An analysis is made of the effects on the diffusion of Brownian particles whose Knudsen number is large compared to unity, of nonuniformities in the host gas. As examples, in one type of nonuniformity of the host gas, the Chapman-Enskog velocity distribution function for the gas molecules is used; in the other, the host gas is a free-molecule Couette flow. In both cases, a new force on the Brownian particles appears. Two techniques are used (extending Kramers' method and utilizing the Chapman-Enskog method) to transform the new Fokker-Planck equation into generalized Smoluchowski and convective diffusion equations. In these equations, the diffusion coefficient appears as a second-order tensor. Thus, it is demonstrated that Brownian diffusion in a nonuniform gas is anisotropic.The work of Slinn was financially supported in part by Battelle Memorial Institute and in part by U.S. Atomic Energy Commission Contract AT(45-1)-1830. The work of Shen was supported in part by U.S. Air Force Office of Scientific Research Contract 49(638)-1346.     

Stochastic diffusion in a periodic potential

This paper deals with two equations for classical stochastic diffusion in a potential. First, the full Fokker-Planck equation in phase-space for a Brownian particle in a periodic potential and linearly coupled to an external field is considered. The solution discussed previously by the author and co-worker is improved upon. An alternative approximation is introduced. Then, the simpler Smoluchowski equation, which is derivable from the Fokker-Planck equation under suitable conditions, is solved using Hill's determinant method. Finally a WKB-type method is proposed to solve the Smoluchowski equation for a general class of potentials.     

Adiabatic Brownian ratchets with the inclusion of inertia

Inertial corrections to the drift velocity of a Brownian particle have been calculated for two main classes of Brownian ratchets operating in the adiabatic regime of fluctuations of the potential energy: first, the stationary periodic potential and dichotomic fluctuations of an external force with zero average value (rocking ratchet) and, second, dichotomic fluctuations of the periodic potential itself. It has been shown that, in contrast to passive transport at which the inertial correction always reduces the effective mobility and diffusion coefficients, inertial corrections for Brownian ratchets can play a constructive role, increasing the drift velocity at least at high temperatures.     

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.     

温库边界对布朗热机性能的影响

研究了单势垒锯齿势中,布朗粒子在外力和空间周期温度场作用下构成的布朗热机的热力学性能.考虑布朗粒子动能变化以及高、低温库之间热漏引起的热流.用Smoluchowski方程描述粒子在黏性介质中的动力学特性,推导出高、低温库的热流以及热机功率和效率的解析表达式.通过数值计算分析势垒高度、外力和温库边界对热机性能的影响.研究表明:由于动能变化和热漏引起的不可逆热流的存在,布朗热机为不可逆热机,热机的功率效率特性为一闭合的关系曲线;势垒边界与温库边界重合时,热机的功率达到最大值;通过改变温库边界的位置,可以在一定范围内提高热机的效率,但同时减小了热机的输出功率.     

Dynamical properties of colloidal systems: 1. Derivation of stochastic transport equations

To describe dynamical properties of a system of interacting Brownian particles stochastic transport equations are derived for the positions of the particles and their concentration fluctuations. This is achieved by an expansion of the Langevin equation for the momenta in terms of the reciprocal of the friction coefficient. As a by-product this procedure gives a new derivation of the generalized Smoluchowski equation. Using a local equilibrium approximation for the configurational distribution function a mode-mode coupling equation is derived for the local concentration, which still depends on the random forces of the solvent. For the interaction free case the relation to the ordinary diffusion approach is established.     

对利用动态光散射法测量颗粒粒径和液体黏度的改进

光散射技术通过测量悬浮液中布朗运动颗粒的平移扩散系数,得到颗粒流体力学直径或液体黏度.本文由单参数模型入手,建立了低颗粒浓度下,单颗粒平移扩散系数与颗粒集体平移扩散系数和颗粒浓度之间的线性依存关系并将其引入光散射法中,从而对现有的测量方法进行了改进.改进后的测量方法可实现纳米尺度球型颗粒标称直径的测量和液体黏度的绝对法测量.以聚苯乙烯颗粒+水和二氧化硅颗粒+乙醇两个分散系为参考样本,通过实验,验证了改进后方法的可行性.此外,还针对上述两个分散系,实验探讨了温度和颗粒浓度对颗粒集体平移扩散系数的影响规律,发现聚苯乙烯颗粒+水分散系中,颗粒间相互作用表现为引力;二氧化硅颗粒+乙醇分散系中,颗粒间相互作用表现为斥力.讨论了颗粒集体平移扩散系数随颗粒浓度变化规律与第二渗透维里系数的关系.     

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.     

Hydrodynamic interactions in Brownian dynamics: I. Periodic boundary conditions for computer simulations

Because of the long range nature of hydrodynamic interactions, the problem of boundary conditions on a finite simulation cell of a hydrodynamically dense suspension of particles in Brownian motion is quite as complicated as the analogous problem in simulation of the statistical mechanics of charged and dipolar systems. One resolution of this problem is to use periodic boundary conditions and to view them as a way of describing a physical system composed of a large spherical array of periodic replicas of the simulation cell. The hydrodynamic interactions are calculated using the quasi-static linearized Navier-Stokes equation. This requires that the suspending fluid velocity remains small throughout the array. That the sum of the particle velocities in the simulation cell be zero is insufficient to force boundedness of the fluid velocity as the array becomes large. Boundedness in the array of the suspending fluid velocity is achieved if a rigid wall boundary condition is applied at the outer edge of the array as the array becomes large. With this condition the net particle velocity equals zero condition is not needed. The condition allows lattice sum representations for the suspending fluid velocity to be derived. These lattice sums are absolutely and rapidly convergent and periodic. Representations of the velocity in the array with boundary condition allow calculation of mobility tensors which are also periodic and can be evaluated numerically in tolerable amounts of computer time. A major effect of these calculations is to identify the physical model system corresponding to a truly periodic fluid velocity and mobility tensor as a large array with rigid wall boundary condition.     

Particle transport in space-periodic potentials in underdamped systems

We consider the motion of an underdamped Brownian particle in a tilted periodic potentialin a wide temperature range. Based on the previous data and the new simulation results weshow that the underdamped motion of particles in space-periodic potentials can beconsidered as overdamped motion in the velocity space in the effective double-wellpotential. Simple analytic expressions for the particle mobility and diffusion coefficientare derived with the use of the presented model. These accurately match numericalsimulation results.     

On the derivation of Smoluchowski equations with corrections in the classical theory of Brownian motion

Differential equations governing the time evolution of distribution functions for Brownian motion in the full phase space were first derived independently by Klein and Kramers. From these so-called Fokker-Planck equations one may derive the reduced differential equations in coordinate space known as Smoluchowski equations. Many such derivations have previously been reported, but these either involved unnecessary assumptions or approximations, or were performed incompletely. We employ an iterative reduction scheme, free of assumptions, and calculate formally exact corrections to the Smoluchowski equations for many-particle systems with and without hydrodynamic interaction, and for a single particle in an external field. In the absence of hydrodynamic interaction, the lowest order corrections have been expressed explicitly in terms of the coordinate space distribution function. An additional application of the method is made to the reduction of the stress tensor used in evaluating the intrinsic viscosity of particles in solution. Most of the present work is based on classical Brownian motion theory, but brief consideration is given in an appendix to some recent developments regarding non-Markovian equations for Brownian motion.Supported by the National Science Foundation.     

Mobility in a periodic potential: A simple derivation

A simple derivation is given for the mobility of Brownian particles in a periodic potential in the overdamped regime. The method makes use of the fact that the steady state current density, in response to a uniform external force, is uniform in space and can be expressed as a product of the particle density and mean velocity field. To lowest order in the external force, the particle density is given by the equilibrium density in the absence of the external force.     

Expansion of the kinetic hierarchy for a massive particle: The repeated ring and Fokker-Planck equations

The tagged particle BBGKY hierarchy is systematically expanded in inverse powers of the square root of the particle mass. In the Brownian limit, for fixed Knudsen number, the hierarchy reduces to the Brownian limit of the repeated ring equation which itself reduces to the Fokker-Planck equation. The friction coefficient of the Fokker-Planck equation is found to be a functional of the solution of Dorfman, van Beijeren, and McClure's extended Boltzmann equation for a fixed object in a flowing gas. As a consequence, the tagged particle diffusion coefficient calculated in the Brownian limit of the repeated ring equation is valid for all particle sizes.     

Many-body effects in diffusion-limited kinetics

We review a novel approach to treating many-body effects in diffusion-limited kinetics. The derivation of the general expression for the survival probability of a Brownian particle in the presence of randomly distributed traps is given. The reduction of this expression to both the Smoluchowski solultion and the wellknown asymptotic behavior is demonstrated. It is shown that the Smoluchowski solution gives a lower bound for the particle survival probability. The correction to the Smoluchowski solution which takes into account the particle death slowdown in the initial process stage is described. The steady-state rate-constant concentration dependence and the reflection of many-body effects in it are discussed in detail.     

Microscopic derivation of non-Markovian thermalization of a Brownian particle

In this paper, the first microscopic approach to Brownian motion is developed in the case where the mass density of the suspending bath is of the same order of magnitude as that of the Brownian (B) particle. Starting from an extended Boltzmann equation, which describes correctly the interaction with the fluid, we derive systematically via multiple-time-scale analysis a reduced equation controlling the thermalization of the B particle, i.e., the relaxation toward the Maxwell distribution in velocity space. In contradistinction to the Fokker-Planck equation, the derived new evolution equation is nonlocal both in time and in velocity space, owing to correlated recollision events between the fluid and particle B. In the long-time limit, it describes a non-Markovian generalized Ornstein-Uhlenbeck process. However, in spite of this complex dynamical behavior, the Stokes-Einstein law relating the friction and diffusion coefficients is shown to remain valid. A microscopic expression for the friction coefficient is derived, which acquires the form of the Stokes law in the limit where the meanfree path in the gas is small compared to the radius of particle B.Knowing the interest of Matthieu Ernst in the subtle and fundamental problems of kinetic theory, we have the pleasure to dedicate this study to him.