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.     

Energetics of stochastic resonance

In this paper, we discuss the motion of a Brownian particle in a double-well potential driven by a periodic force in terms of energies delivered by the periodic and the noise forces and energy dissipated into the viscous environment. It is shown that, while the power delivered by the periodic force to the Brownian particle is controlled by the strength of the noise, the power delivered by the noise itself is independent of the amplitude and frequency of the periodic force. The implications of this result for the mechanism of stochastic resonance in an equilibrium system is that it is not energy from the noise force which enhances a small periodic force, but rather an increase of energy delivered by the periodic force, regulated by the strength of the noise. We further re-evaluate the frequency dependence of stochastic resonance in terms of energetic terms including efficiency.     

Can colored noise improve stochastic resonance?

The phenomenon of stochastic resonance is studied in the presence of colored noise. Several sources of colored noise are introduced and the consequences for the asymptotic time-periodic probability and the (phase-averaged) power spectrum are discussed. Based on space-time symmetry considerations, selection rules for the occurrence of-spikes in the power spectrum are derived. The effect of colored noise on the amplification of small periodic signals is studied in terms of effective, time-periodic Fokker-Planck equations: In overdamped systems driven by colored noise, we find that SR is suppressed with increasing noise color. In contrast, for colored noise induced by inertia (as well as for asymmetric dichotomic noise), one obtains an enhancement of SR. This latter result is obtained by studying the Kramers equation perturbed by a small periodic force.     

Entropic Stochastic Resonance Driven by Colored Noise

The phenomenon of entropic stochastic resonance （ESR） in a two-dimensional confined system driven by a transverse periodic force is investigated when the colored fluctuation is included in the system. Applying the method of unified colored noise approximation, the approximate Fokker-Planck equation can be derived in the absence of the periodic force. Through the escaping rate of the Brownian particle from one well to the other, the power spectral amplification can be obtained. It is found that increasing the values of the noise correlation time and the signal frequency can suppress the ESR of the system.     

Statistical Fluctuation of Linear-System Driven by Colored Noise

The Brownian motion driven by colored noise in the harmonic oscillator potential is studied, and the "best Fokker-Planck equationn" is proposed. The analytical formulse of the second moments of coordinate and velocity are obtained. The fluctuation behavior of a system under influence of the correlation time of colored noise is discussed.     

Numerical Integration of a Non-Markovian Langevin Equation with a Thermal Band-Passing Noise

An accurate and fast approach for numerically solving a non-Markovian Langevin equation with a thermal band-passing noise is proposed. The algorithm combines the closed integration for both damping and noise terms with the Runge–Kutta method for nonlinear force in the Markovian Langevin equation transferred from the original equation. The present algorithm is tested through simulating diffusion of a free particle by using different initial distributions, and then a strong superdiffusion is shown. The mean velocity of a particle in a flashing ratchet driven by the band-passing colored noise is calculated numerically. The dependence of the resulting mean velocity on temperature, asymmetry of the ratchet potential, and inertia of the particle is discussed, and some novel behaviors in comparison with the usual model are observed.     

Brownian motors: Joint effect of non-Gaussian noise and time asymmetric forcing

Previous works have shown that time asymmetric forcing on the one hand, as well as non-Gaussian noises on the other, can separately enhance the efficiency and current of a Brownian motor. Here, we study the result of subjecting a Brownian motor to both effects simultaneously. Our results have been compared with those obtained for the Gaussian white noise regime in the adiabatic limit. We find that, although the inclusion of the time asymmetry parameter increases the efficiency value up to a certain extent, for the present case this increase is much less appreciable than in the white noise case. We also present a comparative study of the transport coherence in the context of colored noise. Though the efficiency in some cases becomes higher for the non-Gaussian case, the Péclet number is always higher in the Gaussian colored noise case than in the white noise as well as non-Gaussian colored noise cases.     

外力作用下反馈耦合布朗棘轮的定向输运

本文研究了处于外力作用下双阱棘轮势中两个反馈耦合布朗粒子的定向输运性能. 通过对过阻尼朗之万方程的数值求解, 详细讨论了外力、热噪声与势阱的不对称参数等对耦合布朗粒子的平均速度、 有效扩散系数及Pe数的影响. 研究发现, 平均速度随外力呈周期性的变化规律. 同时耦合系统存在最优噪声强度会使定向输运达到最强. 值得指出的是棘轮系统可通过改变双阱势的结构来获得较强的定向流. 关键词： 耦合布朗棘轮 外力 双阱棘轮势 平均速度     

Fluctuation-magnification of non-equilibrium membranes near a wall

Membranes in thermal equilibrium are well known to exhibit Brownian motion type shape fluctuations. Membranes containing active force centers -- such as chemically active membrane proteins -- suffer additional non-equilibrium shape fluctuations due to the activity of these force centers. We demonstrate, using scaling arguments, that non-equilibrium shape fluctuations are in general greatly amplified by the presence of a nearby wall or membrane due to the absence of a fluctuation-dissipation theorem. For adhesive membranes, this fluctuation magnification effect may facilitate the establishment of bonding. For non-adhesive membranes, fluctuation magnification produces a long-range repulsive pressure which can exceed the well known Helfrich repulsion due to purely thermal fluctuations. Received: 1 September 1997 / Accepted: 3 December 1997     

Microscopic basis for Fick's law for self-diffusion

We investigate self-diffusion in a classical fluid composed of two species which are distinguished through the color of their particles, either black or white, but are identical as regards their mechanical properties. Disregarding color the fluid is in thermal equilibrium. We show that if a single test particle in the one-component fluid moves asymptotically as Brownian motion, then the color density and current in certain classes of nonequilibrium states are related, on the appropriate macroscopic scale, through Fick's law, and the former is governed by the diffusion equation. If in addition several test particles move asymptotically as independent Brownian motions, then the colored fluid is, on a macroscopic scale, in local equilibrium with parameters governed by the solution of the diffusion equation.Part of this work was done while both authors were at IHES, Bures-sur-Yvette, France.Supported in part by NSF Grant No. PHY 78-15920-02.Supported by a Heisenberg fellowship of the Deutsche Forschungsgemeinschaft.     

Transient Properties of One-Dimensional Stochastic Processes: Unified-Colored-Noise Approximation

By virtue of the unified-colored-noise approximation the formulas of mean, variance and skewness of first-passage-time distributions (FPTD) for the systems driven by colored noise are derived. The mean and variance of FPTD of bistable system driven by colored noise are calculated. The approximation expressions of the mean, variance and skewness of FPTD for the colored cubic model of a dye laser are obtained. The role of the "color" of pump noise in the dye laser transient is analyzed.     

Fluctuation on Brownian motor in the presence of entropic barrier

Transporting velocity of a loaded Brownian motor with entropic barrier is investigated in the presence of an asymmetric unbiased force. It is found that in the presence of the entropic barrier, the stall force of the Brownian motor does not change monotonously with temperature. The average velocity of the Brownian motor is a peaked function of thermal noise and amplitude of the asymmetric unbiased external force, which indicates that a definite fluctuation can facilitate the loaded Brownian motor moving. With the increase of the load, the range of temperature and amplitude of the asymmetric unbiased external force for Brownian motor working become smaller. The limited area for Brownian motor working is given on the load-temperature plane. The threshold of fluctuation for Brownian motor working is found, and the minimum of asymmetric parameter of unbiased external force for Brownian motor working is given.     

Brownian Ratchet Driven by a Rocking Forcing with Broken Temporal Symmetry

The ratchet motion of a Brownian particle in a symmetric periodic potential under a rocking force thatbreaks the temporal symmetry is studied. As long as the relaxation time in the thermal background is much shorter thanthe forcing period, the unidirectional transport can be analytically treated. By solving the Fokker-Planck equations, weget an analytical expression of the current. This result indicates that with an appropriate match between the potentialfield, the asymmetric ac force and the thermal noise, a net current can be achieved. The current versus thermal noiseexhibits a stochastic-resonance-like behavior.     

色关联乘性和加性色噪声驱动的多稳态系统的稳态特性

研究了色关联乘性和加性色噪声作用下的三稳态系统的稳态问题.首先利用一致有色噪声近似方法,推导出稳态概率密度函数的表达式,然后分析了乘性噪声和加性噪声的强度以及关联性对稳态概率密度函数的影响,研究结果表明:加性噪声强度、加性噪声和乘性噪声的关联强度和关联时间可以诱导系统的非平衡相变.     

具有质量及频率涨落的欠阻尼线性谐振子的随机共振

以往的研究大多考虑线性谐振子模型受频率涨落噪声的影响, 而当布朗粒子处于具有吸附能力的复杂环境时, 粒子质量也存在随机涨落. 因此, 本文研究具有质量及频率涨落两项噪声的二阶欠阻尼线性谐振子模型的随机共振现象. 利用Shapiro-Loginov公式和Laplace变换, 推导了系统响应一阶稳态矩及稳态响应振幅的解析表达式. 并根据稳态响应振幅的解析表达式, 建立了稳态响应振幅关于质量涨落噪声及频率涨落噪声各自的噪声强度能够诱导随机共振现象产生的充分必要条件. 仿真实验表明, 当系统参数满足本文所给出的充分必要条件要求时, 系统稳态响应振幅关于噪声强度的变化曲线具有明显的共振峰, 即此选定参数组合能够诱导系统产生随机共振现象.     

反馈控制棘轮的定向输运效率研究

研究了反馈耦合布朗棘轮中粒子处于负载力、时变外力及噪声作用下的定向输运问题.详细讨论了外力作用时间的不对称性、外势空间的不对称性及外力周期等对反馈耦合棘轮中粒子输运效率的影响.研究发现,外力的时间不对称度能促进反馈棘轮中粒子的定向输运,随时间不对称度的增大,反馈棘轮中粒子能获得较大的效率.然而,外势空间的不对称度能有效抑制耦合棘轮中粒子的扩散,达到增强耦合粒子定向输运的效果.同时还发现,存在最优的噪声强度能使耦合粒子的输运效率达到最大.     

Thermally activated state transition technique for femto-Newton-level force measurement

We develop and test a thermally activated state transition technique for ultraweak force measurement. As a force sensor, the technique was demonstrated on a classical Brownian bead immersed in water and restrained by a bistable optical trap. A femto-Newton-level flow force imposed on this sensor was measured by monitoring changes in the transition rates of the bead hopping between two energy states. The treatment of thermal disturbances as a requirement instead of a limiting factor is the major feature of the technique, and provides a new strategy by which to measure other ultraweak forces beyond the thermal noise limit.     

Effect of thermal inhomogeneity on the performance of a Brownian heat engine

We explore the effect of thermal inhomogeneity on the performance of a Brownian heat engine by considering exactly solvable models. We first consider a Brownian heat engine which is modeled as a Brownian particle in a ratchet potential moving through a highly viscous medium driven by the thermal kick it receives from a linearly decreasing background temperature. We show that even though the energy transfer due to kinetic energy is neglected, Carnot efficiency cannot be achieved at quasistatic limit. At quasistatic limit, the efficiency for such a Brownian heat engine approaches the efficiency of endoreversible engine η = 1 ? √T _c /T _h [F.L. Curzon, B. Ahlborn, Am. J. Phys. 43, 22 (1975)]. Moreover, the dependence of the current, the efficiency and the coefficient of performance of the refrigerator on the model parameters is also explored via Brownian dynamic simulations and analytically. We show that such a Brownian heat engine has a higher performance when acting as a refrigerator than when operating as a device subjected to a piecewise constant temperature [M. Asfaw, M. Bekele, Eur. Phys. J. B 38, 457 (2004), M. Asfaw, M. Bekele, Physica A 384, 346 (2007)]. Furthermore, for a Brownian heat engine driven by a piecewise constant temperature, we show that systematic removal of the inhomogeneous medium leads to a homogeneous medium with a uniform temperature where the effect of temperature inhomogeneity is replaced by an effective load.