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.     

Transport properties under the influence of finite friction

Using the Langevin Monte Carlo method,the influence of friction on the directed motion of a Brownian particle driven by an external noise source is investigated.The results show that the exitence and change of the environment friction influence the establishment and development of the steady motion of a Brownian particle derived by nonequilibrium fluctuation.The most probable correlation time,which corresponds to the maximum current,is inversely proportional to the friction coefficient.The abnormal transition of the current with different friction appears because of the coupling between the effective ratchet potential and coloured noise intensity.     

A composite micromotor driven by self-thermophoresis and Brownian rectification

Brownian motors and self-phoretic microswimmers are two typical micromotors, for which thermal fluctuations play different roles. Brownian motors utilize thermal noise to acquire unidirectional motion, while thermal fluctuations randomize the self-propulsion of self-phoretic microswimmers. Here we perform mesoscale simulations to study a composite micromotor composed of a self-thermophoretic Janus particle under a time-modulated external ratchet potential. The composite motor exhibits a unidirectional transport, whose direction can be reversed by tuning the modulation frequency of the external potential. The maximum transport capability is close to the superposition of the drift speed of the pure Brownian motor and the self-propelling speed of the pure self-thermophoretic particle. Moreover, the hydrodynamic effect influences the orientation of the Janus particle in the ratched potential, hence also the performance of the composite motor. Our work thus provides an enlightening attempt to actively exploit inevitable thermal fluctuations in the implementation of the self-phoretic microswimmers.     

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.     

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.     

Systematic adiabatic elimination for stochastic processes

I describe a method to adiabatically eliminate fast variables from stochastic processes. Exact and perturbative results are discussed for Brownian motion of a particle in an external potential.     

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.     

Asymmetry of the shape of surroundings as a mechanism for generating directed motion

A mechanism for generating a directed motion of a Brownian particle in an asymmetric channel under the action of a varying force field is considered. The setup implementing such a mechanism resembles typical Brownian motors using asymmetry of the energy potential (ratchet effect). It is shown that under certain conditions, the asymmetry of the shape may ensure the maximal level of rectification for a large intensity of the external field drawing the system from equilibrium. The main question formulated here is the dependence of the rectifying ability of such a mechanism on the external action parameters and, above all, on the form of the time dependence. The results obtained here for a sine signal and an aperiodic train of pulses are compared with the previous results obtained by the authors for bipolar rectangular pulses. General estimates obtained for pulses of an arbitrary shape determine its influence on the velocity of directed motion. Analysis of the proposed mechanism is based of the combination of analytic calculations for a strong external field with simulation by the method of Brownian dynamics for arbitrary parameters.     

Relativistic dynamics of stochastic particles

Particle motion in stochastic space, i.e., space whose coordinates consist of small, regular stochastic parts, is considered. A free particle in this space resembles a Brownian particle the motion of which is characterized by a dispersionD dependent on the universal length l. It is shown that in the first approximation in the parameter l the particle motion in an external force field is described by equations coincident in form with equations of stochastic mechanics due to Nelson, Kershow, and de la Pena-Auerbach. A method is proposed for the relativization of the scheme used to describe the processes in the stochastic space; by using this method, the equations of particle motion can be written in a covariant form.     

Diffusion of testing particles in turbulent nematic liquid crystals

Summary We report on the diffusive motion of dielectric testing particles floating in a turbulent nematic fluid under the effect of an external oscillating electric field. The particle is viewed as a tracer of the underlying turbulence. We calculate the diffusion coefficients and the probability distribution of the particle displacements, thus showing that the motion of the particle is a classical Brownian diffusion. Paper presented at the I International Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994.     

Quantum Brownian motion on potential surfaces coupled via tunneling in an external electric field[-2mm]

The present paper deals with the motion of a Brownian particle on two identical but shifted potential surfaces, coupled via a tunneling matrix element in an external electric field. Dissipation is induced by a heat bath represented by an infinite set of harmonic oscillators with a continuum range of frequencies. We derive a perturbative solution for the quantum coherence term of the particle system after performing a small-polaron-like transformation. This is subsequently necessary for the extraction of an equation that describes the reduced dynamics and the minimal action path of the Brownian particle. Finally we extract expressions for the population relaxation rate and the pure quantum-dephasing rate of the two-level system. Received 4 January 2001 and Received in final form 12 March 2001     

Application of integral transforms to a description of the Brownian motion by a non-Markovian random process

The one-dimensional Brownian motion and the Brownian motion of a spherical particle in an infinite medium are 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. A harmonic oscillator in a viscous medium is also considered within the framework of the examined model. It is demonstrated that for rheological models, random dynamic processes are also non-Markovian in character. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 66–74, February, 2009.     

Temperature Dependence of Thermal Conductivity of Nanofluids

Mechanism of thermal conductivity of nanofluids is analysed and calculated, including Brownian motion effects, particle agglomeration and viscosity, together influenced by temperature. The results show that only Brown- Jan motion as reported is not enough to describe the temperature dependence of the thermal conductivity of nanofluids. The change of particle agglomeration and viscosity with temperature are also important factors. As temperature increases, the reduction of the particle surface energy would decrease the agglomeration of nanopartieles, and the reduction of viscosity would improve the Brownish motion. The results egree well with the experimental data reported.     

Near-surface Brownian motor with synchronously fluctuating symmetric potential and applied force

A model is suggested which accounts for the unidirectional surface-parallel motion of a Brownian particle under the action of fluctuating surface-inclined unbiased external force. The surface-normal force component induces amplitude fluctuations of the symmetric periodic near-surface potential, whereas the surface-parallel component makes the particle move along the surface. The combined effect of synchronous fluctuations of the symmetric potential and the applied force leads to the longitudinal drift of the particle. It is shown that the temperature dependence of the motor velocity is nonmonotonic, with the maximum governed by the range of the near-surface potential.     

Chaotic Behavior of a Brownian Particle in a Periodic Potential

The classical deterministic dynamics of a Brownian particle with a time-dependent periodic perturbation in a spatially periodic potential is investigated. We have constructed a perturbed chaotic solution near the heteroclinic orbit of the nonlinear dynamics system by using the Constant-Variation method. Theoretical analysis and numerical result show that the motion of the Brownian particle is a kind of chaotic motion. The corresponding chaotic region in parameter space is obtained analytically and numerically.     

Chaotic Behavior of a Brownian Particle in a Periodic Potential

The classical deterministic dynamics of a Brownian particle with a time-dependent periodic perturbation in a spatially periodic potential is investigated. We have constructed a perturbed chaotic solution near the heteroclinic orbit of the nonlinear dynamics system by using the Constant-Variation method. Theoretical analysis and numerical result show that the motion of the Brownian particle is a kind of chaotic motion. The corresponding chaotic region in parameter space is obtained analytically and numerically.     

Fluctuations and dissipations in stochastic energetic resonance

By analyzing the fluctuations and dissipations of a Brownian particle colliding with the molecules in a fluid, the work exchanged between the Brownian particle constrained in a bistable potential well and an external periodic force is investigated. Characters of the stochastic energetic resonance are found and studied at different intensities of fluctuations and dissipations. The microscopic mechanism of energy exchange between the Brownian particle and the external force is revealed. The method used in this study provides a novel way of controlling the stochastic energetic resonance.     

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.