Brownian motion in periodic potentials in the low-friction-limit; nonlinear response to an external force

The Brownian motion of particles in a periodic potential in response to a constant external force is investigated in the low-friction-limit. By introducing an energy and a space variable and by making a proper self consistent ansatz, the Fokker-Planck equation is solved in the stationary state for small friction constants. We found out that, for fixed force to friction ratios, the mobility times the friction constant is a linear function of the square root of the friction constant if the damping is small enough. Explicit results for this linear function are presented for a cosine potential and compared to previous results.     

Brownian motion in fluctuating periodic potentials

This work deals with the overdamped motion of a particle in a fluctuating one-dimensional periodic potential. If the potential has no inversion symmetry and its fluctuations are asymmetric and correlated in time, a net flow can be generated at finite temperatures. We present results for the stationary current for the case of a piecewise linear potential, especially for potentials being close to the case with inversion symmetry. The aim is to study the stationary current as a function of the potential. Depending on the form of the potential, the current changes sign once or even twice as a function of the correlation time of the potential fluctuations. To explain these current reversals, several mechanisms are proposed. Finally, we discuss to what extent the model is useful to understand the motion of biomolecular motors.     

Distribution functions for the Brownian motion of particles in a periodic potential driven by an external force

The stationary distribution functions for the Brownian motion of particles driven by an external force are calculated by expanding the velocity part into Hermite functions and the space part into a Fourier series. Insertion into the Fokker-Planck equation leads to a matrix continued fraction for the lowest two coefficients of the Hermite functions. Higher order terms are found by reverse iteration. Results are shown for a cosine potential. The good convergence allows the calculation in the full range of damping constants. For small friction the distribution function is in good agreement with previous results and the maxima are given by the solutions without noise.     

Brownian pump with an unbiased external force

We investigate Brownian pump transport in the presence of an unbiased external force. The pumping system is embedded in a finite region bounded by two particle reservoirs. In the adiabatic limit, we obtain the analytical expressions of the current and the concentration ratio. We find that Brownian particles can be pumped through an asymmetric potential from a particle reservoir at low concentration to one at the same or higher concentration in the presence of an unbiased external force.      

Interacting Brownian motion in a periodic potential

We study transport properties of a system composed of Brownian particles immersed in a periodic potential. Interaction among the Brownian particles is treated perturbationally in a framework of a generalized Fokker-Planck equation, which due to the interaction contains a renormalized periodic potential and an extra mean field term. We solve the kinetic equation numerically and discuss effects of the (repulsive) interaction on dynamic response functions and transport coefficients.     

Brownian transport in narrow channels subject to transverse periodic force

Brownian transport in two-dimensional narrow channels is extremely sensitive to the spatial modulation of the channel walls. The dependence of transport quantifiers, like the mobility and the diffusivity, on the longitudinal drive strongly varies with the sharpness of the channel corrugation. The two opposite regimes of smooth and septate channels are analyzed in the presence of a transverse periodic bias. Transverse periodic forces of appropriate frequencies enhance the particle mobility across sharp pores more effectively than across smooth bottlenecks.     

周期场中非各态历经布朗运动

将自由状态下呈弹道扩散的非各态历经系统置于周期场中,进而将非各态历经布朗运动分为两类.第一类是阻尼核的Laplace变换的低频为零的系统,当温度远大于势垒高度时,系统平均能量的动能部分依赖粒子的初始速度分布;随温度降低,系统的各态历经性得到恢复.然后将第一类系统的稳定速度变量作为一个内部噪声,再去驱动一个自由布朗粒子,则阻尼核的Laplace变换在零频时为无穷大.结果发现,粒子扩散系数随温度的增加而趋于零,显示一种经典局域化特征,系统的渐进分布依赖于初始坐标分布.这是第二类非各态历经性运动,不能通过外加势而恢复. 关键词： 非各态历经 非Markov布朗运动 扩散系数 噪声谱     

Eigenvalues for the extremely underdamped Brownian motion in an inclined periodic potential

The eigenvalues for the Brownian motion in a periodic potential with an additive constant force are investigated in the low friction limit. First the Fokker-Planck equation for the distribution function in velocity and position space is transformed to energy and position coordinates. By a proper averaging process over the position coordinate a differential equation for the distribution function depending on the energy only is obtained. Next the eigenvalues and eigenfunctions are calculated from this equation by a Runge-Kutta method. Finally the problem is formulated in terms of an integral equation from which the lowest non-zero eigenvalue is obtained analytically in the bistability region in the zero temperature limit.     

Brownian motion in a periodic potential: Application to dielectric relaxation

The relaxational dynamics of a planar rotator in anM-fold cosine potential subject to a random torque is investigated in detail. For the case of a periodic potential with large barrier height, the numerical results of the relaxation dynamics are in complete agreement with an approximate analytical solution. The latter, is derived on assuming a harmonic potential at the bottom of the potential minima and a large time-scale separation between the short-time libration inside each potential minima and a long-time hopping phenomenon over the potential barriers. ForM2, the hopping phenomenon is found to be the dominant feature of the orientational autocorrelation function. The average hopping time is explained satisfactorily in terms of the Kramers activation rate theory. In particular a complete agreement is found between the numerical results of the escape rate and those obtained from the modified Kramers' predictions valid for low friction coefficient. The cosine model is applied to the study of dielectric spectroscopy. The particle mobility and the complex permitivity, of a dielectric material are calculated by numerical solutions for rotational velocity and orientational auto-correlation functions, respectively. The main features of the experimental observables are determined analytically and compared to the corresponding numerical results. The applicability of the plane rotator model to dielectric spectroscopy is also discussed.From Jan. 1^st 1985: Institut für Theoretische Physik, RWTH, Aachen, FRG     

Brownian motion of a nonlinear oscillator

Starting with the Langevin equation for a nonlinear oscillator (the Duffing oscillator) undergoing ordinary Brownian motion, we derive linear transport laws for the motion of the average position and velocity of the oscillator. The resulting linear equations are valid for only small deviations of average values from thermal equilibrium. They contain a renormalized oscillator frequency and a renormalized and non-Markovian friction coefficient, both depending on the nonlinear part of the original equation of motion. Numerical computations of the position correlation function and its spectral density are presented. The spectral density compares favorably with experimental results obtained by Morton using an analog computer method.Technical Note BN-674. Research supported in part by NSF grant GP-12591, and in part by PHS Research Grant No. MG16426-02 from the National Institute of General Medical Sciences.     

Self-trapped nonlinear matter waves in periodic potentials

We demonstrate that the recent observation of nonlinear self-trapping of matter waves in one-dimensional optical lattices [Th. Anker, Phys. Rev. Lett. 94, 020403 (2005)10.1103/PhysRevLett.94.020403] can be associated with a novel type of broad nonlinear state existing in the gaps of the matter-wave band-gap spectrum. We find these self-trapped localized modes in one-, two-, and three-dimensional periodic potentials, and demonstrate that such novel gap states can be generated experimentally in any dimension.     

Kubo-Einstein relation for quantum Brownian motion in a periodic potential

Using a previously derived general formalism for a dissipative quantum particle in a boson bath, we prove that when the damping is Ohmic, the Kubo-Einstein relation between the diffusion constant and the linear mobilityD=kTM holds to all orders in V₀ for a periodic potentialV(x)=V ₀ cos(k)₀ x).     

Velocity spectrum for non-Markovian Brownian motion in a periodic potential

Non-Markovian Brownian motion in a periodic potential is studied by means of an electronic analogue simulator. Velocity spectra, the Fourier transforms of velocity autocorrelation functions, are obtained for three types of random force, that is, a white noise, an Ornstein—Uhlenbeck process, and a quasimonochromatic noise. The analogue results are in good agreement both with theoretical ones calculated with the use of a matrix-continued-fraction method, and with the results of digital simulations. An unexpected extra peak in the velocity spectrum is observed for Ornstein-Uhlenbeck noise with large correlation time. The peak is attributed to a slow oscillatory motion of the Brownian particle as it moves back and forth over several lattice spaces. Its relationship to an approximate Langevin equation is discussed.     

周期性外力驱动的混沌摆

为了简单直观地呈现混沌现象,研制了受周期驱动力驱动的混沌摆实验系统. 实验系统由力学机构、采集机构和数据处理分析仪组成. 力学机构是受周期外力驱动的复摆;采集机构采用无触点角度传感器,通过串口通信实时将数据传输给计算机;数据处理分析仪用虚拟仪器平台LabVIEW软件编程实现. 该系统可实时显示反映混沌基本特征的角度时序图、角速度时序图和相图.     

Hypersensitivity of a nonlinear system with multiplicative colored noise to an external periodic signal

A simple nonlinear stochastic system, an overdamped Kramers oscillator with multiplicative colored noise, is studied analytically and by numerical simulation. It is shown that in the region where on-off intermittency occurs, the system becomes hypersensitive to weak external periodic signals. Zh. éksp. Teor. Fiz. 116, 1484–1498 (October 1999)     

Bistability effects of the brownian motion in periodic potentials

In the stationary state the equation of motion for particles moving in a periodic potential has two solutions, a locked one and a running one, for low and intermediate damping constants and for suitable external forces. The effect of an additional Langevin force to this bistable behaviour is investigated. For finite noise strength, the mobility depends continuously on the external force, whereas in the limit of vanishing strength of the noise force one gets a sharp transition between the locked and the running solution at a critical external force. This critical force is calculated exactly in the low friction limit and approximately for intermediate friction constants. Furthermore the temperature dependence for various forces including the critical one is shown in the low friction limit.     

Brownian motion near an absorbing sphere

The spherically symmetric solution of the Fokker-Planck equation with absorbing boundary is given in terms of a solution of an equivalent integral equation whose explicit form is found.