A Flashing Model for Transport of Brownian Motors

A flashing coloured noise model is proposed to describe the motion of a molecular motor.In this model,the overdamped Brownian particle moves in an asymmetric periodic potential with a flashing Ornstein-Ulenbeck coloured noise.The relationship between the current and the parameters-such as the intensity,the correlation time of coloured noise and the flip rate of the noise-is discussed using the Monte Carlo simulation method.Current reversal occurs with the change of the correlation time and the flip rate of coloured noise,which may be related to the directed motion and the current reversal of molecular motors.     

Asymmetrical Diffusion-Induced Directional Motion

Competition between anomalous diffusion and normal diffusion along two different directions of the track for a Brownian motor, combined with a periodic potential flashing, can lead to a macroscopic motion. The current is calculated analytically by using the Astumian-Bier‘s approach of the step number per cycle. It is shown that the direction of current occurs reversal for different waiting times of the potential off and the magnitude of current is prominently enhanced. Moreover, a thermal “green“ noise is proposed to produce the ballistic diffusion, numerical simulations for the average velocity of the particle in the presence of ballistic and normal diffusions support the present theoretical findings.     

Asymmetrical Diffusion-Induced Directional Motion

Competition between anomalous diffusion and normal diffusion along two different directions of the track for a Brownian motor, combined with a periodic potential flashing, can lead to a macroscopic motion. The current is calculated analytically by using the Astumian-Bier‘s approach of the step number per cycle. It is shown that the direction of current occurs reversal for different waiting times of the potential off and the magnitude of current is prominently enhanced. Moreover, a thermal “green“ noise is proposed to produce the ballistic diffusion, numerical simulations for the average velocity of the particle in the presence of ballistic and normal diffusions support the present theoretical findings.     

Harmonic Noise-Induced Resonant Passing in an Inverse Harmonic Potential

The dynanfics of a particle passing over the sad- dle point plays a crucial role in many branches of physics and chemistry.Previous studies on the saddle-point passing problem mostly considered par- ticles that are subjected to a random force. Noise- induced transport has been studied in many fields including mathematics, physics, chemistry, biology, and economics. A classic example is the fusion reaction of massive nuclei, in which the fusion is in- duced by diffusion. In previous works, the fusion probability was obtained by determining the passing probability of a Brownian particle over the top of an inverse harmonic potential. Great progress has been achieved by regarding the study of passing prob- lems since tile pioneering work published by Kramers. The simplest situation assumes a Brownian particle driven by white noise, which however is not satisfied due to the multiple spectral density of the realistic ran- dora force. One of the realistic models of random force is the Gaussian colored noise, which requires the study of the motion of a particle driven by structured noise, such as harmonic noise which can be generated by the coordinate of a harmonic oscillator driven by white noise. This type of colored noise allows the consideration of resonance phenomena due to the fact that a peak in its spectrum is presented here.     

The Onsager reciprocity relation and generalized efficiency of a thermal Brownian motor

Based on a general model of Brownian motors,the Onsager coefficients and generalized efficiency of a thermal Brownian motor are calculated analytically.It is found that the Onsager reciprocity relation holds and the Onsager coefficients are not affected by the kinetic energy change due to the particle’s motion.Only when the heat leak in the system is negligible can the determinant of the Onsager matrix vanish.Moreover,the influence of the main parameters characterizing the model on the generalized efficiency of the Brownian motor is discussed in detail.The characteristic curves of the generalized efficiency varying with these parameters are presented,and the maximum generalized efficiency and the corresponding optimum parameters are determined.The results obtained here are of general significance.They are used to analyze the performance characteristics of the Brownian motors operating in the three interesting cases with zero heat leak,zero average drift velocity or a linear response relation,so that some important conclusions in current references are directly included in some limit cases of the present paper.     

Particle Transport with Asymmetric Unbiased Forces and Entropic Barrier

Transport of a Brownian particle moving along the axis of a three-dimensional （313） symmetric and periodic tube is investigated in the presence of asymmetric unbiased forces. It is found that in the presence of entropic barrier, the asymmetry of the unbiased forces is a way of inducing a net particle current. The particle current is a peaked function of temperature, which indicates that the thermal noise may facilitate the transport even in the presence of entropic barrier. There exists an optimized radius at the bottleneck at which the particle current takes its maximum value. The current can be influenced by the slope of tube walls and there exists an optimized slope for which the particle current takes its maximum value.     

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.     

Stochastic Resonance in a Spatially Symmetric and Flashing Periodic Potential Subjected to Correlated Noises

A Brownian particle in a spatially symmetric and flashing periodic potential subjected to correlated noises is investigated. The exact expression of its current is analytically derived. The numerical results indicate that its current as a function of noise intensity exhibits two peaks in the case of positive correlations, and two vales in the case of negative correlations, i.e., a novel stochastic resonance （SR） phenomenon. The SR is attributed to the harmonic cooperation between the noises and the flashing periodic potential. The conditions under which the SR occurs are also presented.     

Towards an Understanding of the Influence of Sedimentation on Colloidal Aggregation by Peclet Number

The Peclet number is a useful index to estimate the importance of sedimentation as compared to the Brownian motion. However, how to choose the characteristic length scale for the Peclet number evaluation is rather critical because the diffusion length increases as the square root of the time whereas the drifting length is linearly related to time. Our Brownian dynamics simulation shows that the degree of sedimentation influence on the coagulation decreases when the dispersion volume fraction increases. Therefore using a fixed length, such as the diameter of particle, as the characteristic length scale for Peclet number evaluation is not a good choice when dealing with the influence of sedimentation on coagulation. The simulations demonstrated that environmental factors in the coagulation process, such as dispersion volume fraction and size distribution, should be taken into account for more reasonable evaluation of the sedimentation influence.     

Analytical Solutions of a Model for Brownian Motion in the Double Well Potential

In this paper, the analytical solutions of Schr¨odinger equation for Brownian motion in a double well potential are acquired by the homotopy analysis method and the Adomian decomposition method. Double well potential for Brownian motion is always used to obtain the solutions of Fokker–Planck equation known as the Klein–Kramers equation, which is suitable for separation and additive Hamiltonians. In essence, we could study the random motion of Brownian particles by solving Schr¨odinger equation. The analytical results obtained from the two different methods agree with each other well. The double well potential is affected by two parameters, which are analyzed and discussed in details with the aid of graphical illustrations. According to the final results, the shapes of the double well potential have significant influence on the probability density function.     

Current reversals in an inhomogeneous system with asymmetric unbiased fluctuations

We present a study of the transport of a Brownian particle moving in a periodic symmetric potential in the presence of asymmetric unbiased fluctuations. The particle is considered to move in a medium with periodic space dependent friction. By tuning the parameters of the system, the direction of the current exhibits reversals, both as a function of temperature as well as the amplitude of rocking force. We found that the mutual interplay between the opposite driving factors is the necessary term for current reversals.Received: 11 October 2003, Published online: 9 April 2004PACS: 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion - 02.50.Ey Stochastic processes - 87.10. + e General theory and mathematical aspects     

活性布朗粒子运动的稳态解

拟可积Hamilton系统随机平均法可以用来研究活性布朗粒子运动.介绍了该随机平均法，利用它详细求解了布朗粒子运动的动力学方程，该方程描述了活性布朗粒子在平面上的运动，粒子受到的激励是Gaussian白噪声，受到的阻尼是Schienbein-Gruler速度依赖的磨擦模型.通过与数字模拟和与实验数据的比较，证明所得稳态解正确.对于Rayleigh和Erdamnn速度依赖的磨擦模型，也给出了稳态解. 关键词： 活性布朗粒子 拟可积Hamilton系统随机平均法 可积性 稳态解     

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.     

Effect of the Noise Correlation Time on the Stationary Current of Brownian Particle in a Spatially Symmetric Periodic Potential

We investigate the noise-induced transport of Brownian particle in a deterministic spatial symmetrical periodic potential driven by colored cross correlation between a multiplicative white noise and an additive white noise. We derive the general formula of the stationary current. Based on numerical computation, we found that directed motion of the Brownian particles can be induced by the correlation time τ of cross correlation between the multiplicative noise and the additive noise and the current reversal and the direction of the current is controlled by the τ.     

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.     

Semi-Integral Scheme for Simulation of Langevin Equation with Weak Inertia

A stable and accurate algorithm for simulating massive damped Brownian motion is proposed and discussed. The algorithm, being fully integral for the friction and noise terms and predictor-corrector for the potential force in the Langevin equations, is stable upon changing time step and for various masses of the particle. In particular, the limit of zero inertia can be safely taken, and the algorithm yields naturally the corresponding overdamped case. The steady velocity of a particle moving in a titled periodic potential is calculated and three algorithms are compared.     

On the theory of Brownian motion. I. Interaction between Brownian particles

The molecular theory of the Brownian motion of heavy particles in a homogeneous solvent of light particles is extended to cover the case of interactions between the Brownian particles. This will have physical effects in the concentration dependence of the Brownian particle self-diffusion coefficient. A density expansion for the Brownian particle friction coefficient is derived, and an approximation permitting the first density correction to be calculated is suggested.This work, part of research supported by NSF Grant GP-8497, was done under the tenure of a National Science Foundation Senior Postdoctoral Fellowship, and of a sabbatical leave granted by the University of Oregon.     

Mean-square displacement of a stochastic oscillator: Linear vs quadratic noise

Using the Langevin equations, we calculated the stationary second-order moment (mean-square displacement) of a stochastic harmonic oscillator subject to an additive random force (Brownian motion in a parabolic potential) and to different types of multiplicative noise (random frequency or random damping or random mass). The latter case describes Brownian motion with adhesion, where the particles of the surrounding medium may adhere to the oscillator for some random time after the collision. Since the mass of the Brownian particle is positive, one has to use quadratic (positive) noise. For all types of multiplicative noise considered, replacing linear noise by quadratic noise leads to an increase in stability.     

Stochastic rolling of a rigid sphere in weak adhesive contact with a soft substrate

We study the rolling motion of a small solid sphere on a fibrillated rubber substrate in an external field in the presence of a Gaussian noise. From the nature of the drift and the evolution of the displacement fluctuation of the ball, it is evident that the rolling is controlled by a complex non-linear friction at a low velocity and a low noise strength (K), but by a linear kinematic friction at a high velocity and a high noise strength. This transition from a non-linear to a linear friction control of motion can be discerned from another experiment in which the ball is subjected to a periodic asymmetric vibration in conjunction with a random noise. Here, as opposed to that of a fixed external force, the rolling velocity decreases with the strength of the noise suggesting a progressive fluidization of the interface. A state (K) and rate (V) dependent friction model is able to explain both the evolution of the displacement fluctuation as well as the sigmoidal variation of the drift velocity with K. This research sets the stage for studying friction in a new way, in which it is submitted to a noise and then its dynamic response is studied using the tools of statistical mechanics. Although more works would be needed for a fuller realization of the above-stated goal, this approach has the potential to complement direct measurements of friction over several decades of velocities and other state variables. It is striking that the non-Gaussian displacement statistics as observed with the stochastic rolling is similar to that of a colloidal particle undergoing Brownian motion in contact with a soft microtubule.