The scaling of higher cumulants in a diffusion problem

The scaling properties of higher cumulants for a diffusion problem are examined by means of numerical calculations. The exponent for the higher cumulants are found to be less than that of the first cumulant but larger than that of the second one. The calculations can be used for describing quantum particle diffusion in a random time-dependent potential, domain wall diffusion in a 2D magnet, etc.     

Mean first passage time of active Brownian particle in one dimension

We investigate the mean first passage time of an active Brownian particle in one dimension using numerical simulations. The activity in one dimension is modelled as a two state model; the particle moves with a constant propulsion strength but its orientation switches from one state to other as in a random telegraphic process. We study the influence of a finite resetting rate r on the mean first passage time to a fixed target of a single free active Brownian particle and map this result using an effective diffusion process. As in the case of a passive Brownian particle, we can find an optimal resetting rate r* for an active Brownian particle for which the target is found with the minimum average time. In the case of the presence of an external potential, we find good agreement between the theory and numerical simulations using an effective potential approach.     

Diffusion in a Weakly Random Hamiltonian Flow

We consider the motion of a particle governed by a weakly random Hamiltonian flow. We identify temporal and spatial scales on which the particle trajectory converges to a spatial Brownian motion. The main technical issue in the proof is to obtain error estimates for the convergence of the solution of the stochastic acceleration problem to a momentum diffusion. We also apply our results to the system of random geometric acoustics equations and show that the energy density of the acoustic waves undergoes a spatial diffusion.     

A note on confined diffusion

The random motion of a Brownian particle confined in some finite domain is considered. Quite generally, the relevant statistical properties involve infinite series, whose coefficients are related to the eigenvalues of the diffusion operator. Because the latter depend on space dimensionality and on the particular shape of the domain, an analytical expression is in most circumstances not available. In this article, it is shown that the series may in some circumstances sum up exactly. Explicit calculations are performed for 2D diffusion restricted to a circular domain and 3D diffusion inside a sphere. In both cases, the short-time behaviour of the mean square displacement is obtained.     

An invariance principle for reversible Markov processes. Applications to random motions in random environments

We present an invariance principle for antisymmetric functions of a reversible Markov process which immediately implies convergence to Brownian motion for a wide class of random motions in random environments. We apply it to establish convergence to Brownian motion (i) for a walker moving in the infinite cluster of the two-dimensional bond percolation model, (ii) for ad-dimensional walker moving in a symmetric random environment under very mild assumptions on the distribution of the environment, (iii) for a tagged particle in ad-dimensional symmetric lattice gas which allows interchanges, (iv) for a tagged particle in ad-dimensional system of interacting Brownian particles. Our formulation also leads naturally to bounds on the diffusion constant.     

A Brownian motion version of the directed polymer problem

Consider a Brownian particle in three dimensions in a random environment. The environment is determined by a potential random in space and time. It is shown that at small noise the large-time behavior of the particle is diffusive. The diffusion constant depends on the environment. This work generalizes previous results for random walk in a random environment. In these results the diffusion constant does not depend on the environment.     

Effect of the spatial distribution of probe beam on the results of measurements of the disperse composition of nanoparticles by dynamic light scattering method

The model of dynamic scattering on a finite ensemble of Brownian particles in liquid is considered. It is shown that an artifact characteristic relaxation time appears in the autocorrelation function of the scattered light intensity, which is much longer than the correlation time controlled by particle diffusion in the scattering volume.     

Brownian particles in random and quasicrystalline potentials: How they approach the equilibrium

We study the Brownian motion of an ensemble of single colloidal particles in a random square and a quasicrystalline potential when they start from non-equlibrium. For both potentials, Brownian dynamics simulations reveal a widespread subdiffusive regime before the diffusive long-time limit is reached in thermal equilibrium. We develop a random trap model based on a distribution for the depths of trapping sites that reproduces the results of the simulations in detail. Especially, it gives analytic formulas for the long-time diffusion constant and the relaxation time into the diffusive regime. Aside from detailed differences, our work demonstrates that quasicrystalline potentials can be used to mimic aspects of random potentials.     

Feedback control in a collective flashing ratchet

An ensemble of Brownian particles in a feedback controlled flashing ratchet is studied. The ratchet potential is switched on and off depending on the position of the particles, with the aim of maximizing the current. We study in detail a protocol which maximizes the instant velocity of the center of mass of the ensemble at any time. This protocol is optimal for one particle and performs better than any periodic flashing for ensembles of moderate size, but is defeated by a random or periodic switching for large ensembles.     

Relaxation of the Probability Characteristics of Brownian Motion under the Action of a Non-Delta-Correlated Random Force

We consider the relaxation of the moments of the coordinates of one-dimensional Brownian motion of particles in a symmetric potential profile under the action of a Gaussian, exponentially correlated random force. An analytical-numerical method of analysis based on obtaining and numerically solving a chain of differential equations for joint cumulants of some functions of particle coordinates and a random force is used. A priori constraints on the intensity and correlation time of noise are not imposed. Numerical procedure is checked by comparison with analytical results, which can be found in the limiting cases of delta-correlated and quasistatic random force. The dependence of the relaxation of the average value and variance on the intensity and spectrum of a random force and the character of the initial distribution of particles is elucidated. In particular, the presence of a variance minimum during distribution relaxation is established. The evolution of the model probability distribution of particle coordinates is constructed on the basis of the moment relaxation.     

A heavy ion in a fluid in presence of an electromagnetic field seen as an ordinary Brownian motion

An alternative method of how to characterize, at equilibrium, the diffusion process of a Brownian charged particle (heavy ion) in a fluid in presence of an electromagnetic field is presented. The theory is formulated via a Langevin equation associated with the ion's velocity vector, which is transformed to another velocity-space in which the diffusion process is quite similar to that of the ordinary Brownian motion. The diffusion process is characterized, in absence and in presence of the electric field, through the mean square displacement in the transformed configuration-space and then returned to the original variables, by means of the corresponding transformation. Under the action of the electric field, the diffusion process is studied for a general time-dependent electric field. Explicit results are obtained for a constant and oscillating electric field.     

Escape Through a Small Opening: Receptor Trafficking in a Synaptic Membrane

We model the motion of a receptor on the membrane surface of a synapse as free Brownian motion in a planar domain with intermittent trappings in and escapes out of corrals with narrow openings. We compute the mean confinement time of the Brownian particle in the asymptotic limit of a narrow opening and calculate the probability to exit through a given small opening, when the boundary contains more than one. Using this approach, it is possible to describe the Brownian motion of a random particle in an environment containing domains with small openings by a coarse grained diffusion process. We use the results to estimate the confinement time as a function of the parameters and also the time it takes for a diffusing receptor to be anchored at its final destination on the postsynaptic membrane, after it is inserted in the membrane. This approach provides a framework for the theoretical study of receptor trafficking on membranes. This process underlies synaptic plasticity, which relates to learning and memory. In particular, it is believed that the memory state in the brain is stored primarily in the pattern of synaptic weight values, which are controlled by neuronal activity. At a molecular level, the synaptic weight is determined by the number and properties of protein channels (receptors) on the synapse. The synaptic receptors are trafficked in and out of synapses by a diffusion process. Following their synthesis in the endoplasmic reticulum, receptors are trafficked to their postsynaptic sites on dendrites and axons. In this model the receptors are first inserted into the extrasynaptic plasma membrane and then random walk in and out of corrals through narrow openings on their way to their final destination.     

On the quantum-mechanical Fokker-Planck and Kramers-Chandrasekhar equation

In the classical theory of Brownian motion we can consider the Langevin equation as an infinitesimal transformation between the coordinates and momenta of a Brownian particle, given probabilistically, since the impulse appearing is characterized by a Gaussian random process. This probabilistic infinitesimal transformation generates a streaming on the distribution function, expressed by the classical Fokker-Planck and Kramers-Chandrasekhar equations. If the laws obeyed by the Brownian particle are quantum mechanical, we can reinterpret the Langevin equation as an operator relation expressing an infinitesimal transformation of these operators. Since the impulses are independent of the coordinates and momenta we can think of them as c numbers described by a Gaussian random process. The so resulting infinitesimal operator transformation induces a streaming on the density matrix. We may associate, according to Weyl functions with operators. The function associated with the density matrix is the Wigner function. Expressing, then, these operator relations in terms of these functions we can express the streaming as a continuity equation of the Wigner function. We find that in this parametrization the extra terms which appear are the same as in the classical theory, augmenting the usual Wigner equation.     

Diffusion coefficient for a Brownian particle in a periodic field of force: I. Large friction limit

The diffusion tensor for a Brownian particle in a periodic field of force is studied in the strong damping limit, in which the Smoluchowski equation is valid.A general relation between the diffusion tensor and the Smoluchowski “relaxation operator” is derived; the effect of the periodic force, at least in the simplest situation of diagonal and uniform friction, appears as a dressing of the bare particle mass to an effective tensor mass.From this the explicit form of the diffusion coefficient as a functional of the potential energy in the one-dimensional case is obtained, showing a temperature dependence which deviates at high temperatures from a simple Arrhenius behaviour.Finally, the expression for the mobility of the Brownian particle is derived, and by comparison with the expression for the diffusion coefficient the Einstein relation between diffusion and mobility is proved to be satisfied.     

Using bimodal probability distributions in the problems of Brownian diffusion

When analyzing nonlinear stochastic systems, we deal with the chains of differential equations for the moments or cumulants of dynamic variables. To disconnect such chains, the well-known cumulant approach, which is adequate to the quasi-Gaussian expansion of the higher-order moments is used. However, this method is inefficient in the problems of Brownian diffusion in bimodal potential profiles, and the disconnection problem should be solved on the basis of bimodal probability distributions. To this end, we propose to construct bimodal model distributions, in particular, the bi-Gaussian distribution. Cumulants and the expansions of the higher-order moments for symmetric and nonsymmetric bi-Gaussian models. On this basis, we consider relaxation of probability characteristics of one-dimensional Brownian motion in the bimodal potential profile. The dependences of relaxation of the mean value and variance of particle coordinate on the potential barrier “power,” the noise intensity, and the initial distribution of particles are analyzed numerically. In particular, it is shown that relaxation proceeds by stages with different temporal scales in the case of a powerful barrier. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 49, No. 8, pp. 718–729, August 2006.     

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.     

利用Matlab模拟布朗运动测量实验

利用matlab工具模拟了布朗运动测量的实验。通过一正态随机数产生函数模拟从而产生布朗运动步距。在假定粒子所受拖曳力满足斯托克斯关系的情况下,通过拟合多个粒子的均方位移随时间的变化曲线得到斜率,从而进一步可得出扩散系数和波尔兹曼常数。同时,根据模拟结果也对如何减小实验误差作了分析。     

In vivo anomalous diffusion and weak ergodicity breaking of lipid granules

Combining extensive single particle tracking microscopy data of endogenous lipid granules in living fission yeast cells with analytical results we show evidence for anomalous diffusion and weak ergodicity breaking. Namely we demonstrate that at short times the granules perform subdiffusion according to the laws of continuous time random walk theory. The associated violation of ergodicity leads to a characteristic turnover between two scaling regimes of the time averaged mean squared displacement. At longer times the granule motion is consistent with fractional Brownian motion.