One-dimensional displacement of active matter on curved substrates

We analytically find the diffusion of overdamped active Brownian particles (ABPs) constrained to move along curved one-dimensional channels. The autonomous motion of these particles is achieved by a projection of their internal propulsion force along the channels' long section. In particular, the diffusion of ABPs moving on one-dimensional channels with a form of a circle, an ellipse, and a limacon of second order is analysed. To characterise the effect of substrate's geometry and self-propulsion on their diffusion, analytical expressions for the ABPs short- and long-time variances, as well as their steady angular probability density functions are offered. Curvature effects are found to reduce the time an ABP reaches its steady state. Our theoretical results are validated using Brownian dynamics simulations. This model may be relevant for experiments dealing with catalytic driven systems, bacteria, and tumour cell dispersion in one-dimensional channels.     

Note on the fractal dimension of hard sphere trajectories

Using Monte Carlo molecular dynamics, a new, careful study is made of the approach of the trajectory of a typical particle in a hard sphere fluid to that of a Brownian particle, discussed before by Powles and Quirke and Rapaport. The apparent fractal dimension of the trajectory, as a function of reduced length scale,(), characterizes the transition from mechanical to Brownian motion and differs markedly from 2 in all present computer simulations.     

The fractal nature of molecular trajectories in fluids

The scaled lengths of molecular trajectories obtained by molecular dynamics simulation of a hard-sphere fluid are shown to have the same fractal dimensionD=2 as the random walk. Self-similarity first appears on length scales typically a factor of 25 greater than the mean free-path length, whereas for the simple random walk with constant step size the onset occurs after only six steps; the reason for the slow convergence is shown to be the near exponential distribution of intercollision path lengths of the fluid molecules. The influence of density on the scaled path lengths is also discussed.     

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.     

Brownian dynamics of colloidal hard spheres. 3. Extended investigations at the phase transition regime

The phase behavior of hard-sphere colloidal systems in the volume fraction regime 0.46<<0.64 has been studied in detail using a new and efficient algorithm to treat the nonanalytic interaction pair potential. In particular the influence of various initial configurations such as purely random and facecentered cubic (FCC) has been investigated, and former simulations have been extended toward much longer time scales. Thus, in the case of randomly initiated systems, crystallization could be suppressed for a comparably long time (500 _R , where _R is the structural relaxation time) where the system remained in a metastable glassy state. The concentration dependence of the long-time self-diffusion coefficients of these systems has been analyzed according to free volume theory (Doolittle equation). Numerical data fit excellently to the theoretical predictions, and the volume fraction of zero particle mobility was found close to the expected value of random close packing. In case of the FCC initiated systems, samples remained crystalline within the simulated evolution time of 500 _R if their volume fraction was above the predicted freezing transition _F = 0.494, whereas at lower concentrations rapid melting into a fluidlike disordered state is observed. It should be noted that this algorithm, which neglects higher-order correlations, considering only direct pair interactions, nevertheless yields the correct hard-sphere crystallization phase behavior as predicted in the literature.     

Brownian dynamics of polydisperse colloidal hard spheres: Equilibrium structures and random close packings

Recently we presented a new technique for numerical simulations of colloidal hard-sphere systems and showed its high efficiency. Here, we extend our calculations to the treatment of both 2- and 3-dimensional monodisperse and 3-dimensional polydisperse systems (with sampled finite Gaussian size distribution of particle radii), focusing on equilibrium pair distribution functions and structure factors as well as volume fractions of random close packing (RCP). The latter were determined using in principle the same technique as Woodcock or Stillinger had used. Results for the monodisperse 3-dimensional system show very good agreement compared to both pair distribution and structure factor predicted by the Percus-Yevick approximation for the fluid state (volume fractions up to 0.50). We were not able to find crystalline 3d systems at volume fractions 0.50–0.58 as shown by former simulations of Reeet al. or experiments of Pusey and van Megen, due to the fact that we used random start configurations and no constraints of particle positions as in the cell model of Hoover and Ree, and effects of the overall entropy of the system, responsible for the melting and freezing phase transitions, are neglected in our calculations. Nevertheless, we obtained reasonable results concerning concentration-dependent long-time selfdiffusion coefficients (as shown before) and equilibrium structure of samples in the fluid state, and the determination of the volume fraction of random close packing (RCP, glassy state). As expected, polydispersity increases the respective volume fraction of RCP due to the decrease in free volume by the fraction of the smaller spheres which fill gaps between the larger particles.     

少体硬球系统的动力学与统计研究

研究了少数几个封闭于箱子中的硬球组成的系统的动力学与统计行为.着重研究单粒子位形 空间的碰撞分布.计算表明，硬球的半径较小时，单粒子统计分布函数在空间主要是均匀分 布；随着半径的增大，均匀分布部分逐渐减小.当硬球半径与箱子尺寸比值超过临界值时， 单粒子分布函数呈现双峰形式.还利用少体硬球系统模拟布朗运动.研究表明，当硬球系统作 为介质时，系统不存在扩散过程；发现大粒子的平均平方位移与时间是平方关系，说明大粒 子在硬球介质中的输运是弹道输运过程. 关键词： 硬球 动力学 布朗运动 遍历     

一维磁性光子晶体的电磁场分析和数值计算

本文探讨一维磁性光子晶体的概念和结构.提出了一种分析一维磁性光子晶体的法拉第旋转效应的电磁场方法.分析计算表明:嵌于光子晶体中的一层很薄的磁性材料将可获得比单独一层同样厚度的磁性材料大得多的法拉第旋转效应,从而从理论上验证了文献上的相关实验.同时也分析了一维磁性光子晶体用作光子晶体结构下的光隔离器等器件中的法拉第旋转器的可能.     

Molecular dynamics study of nanodroplet diffusion on smooth solid surfaces

We perform molecular dynamics simulations of Lennard–Jones particles in a canonical ensemble to study the diffusion of nanodroplets on smooth solid surfaces. Using the droplet-surface interaction to realize a hydrophilic or hydrophobic surface and calculating the mean square displacement of the center-of-mass of the nanodroplets, the random motion of nanodroplets could be characterized by shorttime subdiffusion, intermediate-time superdiffusion, and long-time normal diffusion. The short-time subdiffusive exponent increases and almost reaches unity (normal diffusion) with decreasing droplet size or enhancing hydrophobicity. The diffusion coefficient of the droplet on hydrophobic surfaces is larger than that on hydrophilic surfaces.     

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。     

One-dimensional non-relativistic and relativistic Brownian motions: a microscopic collision model

We study a simple microscopic model for the one-dimensional stochastic motion of a (non-)relativistic Brownian particle, embedded into a heat bath consisting of (non-)relativistic particles. The stationary momentum distributions are identified self-consistently (for both Brownian and heat bath particles) by means of two coupled integral criteria. The latter follow directly from the kinematic conservation laws for the microscopic collision processes, provided one additionally assumes probabilistic independence of the initial momenta. It is shown that, in the non-relativistic case, the integral criteria do correctly identify the Maxwellian momentum distributions as stationary (invariant) solutions. Subsequently, we apply the same criteria to the relativistic case. Surprisingly, we find here that the stationary momentum distributions differ slightly from the standard Jüttner distribution by an additional prefactor proportional to the inverse relativistic kinetic energy.     

Brownian motion under a time-dependent periodic potential proportional to the square of the position of a particle and classical fluctuation squeezing

A general stationary case of a Brownian particle with a time-dependent periodic potential proportional to the square of the position of the particle is treated. Even though the vigorous change of the time-dependent proportionality coefficient is applied, there are cases where the fluctuation of the particle decreases in contrary to our intuition, which is called classical fluctuation squeezing. We obtain time-average variances analytically for general cases of an arbitrary change in the coefficient and find conditions favorable for classical fluctuation squeezing. We introduce an asymmetric function behaves like trigonometric cosine one and consider its behavior explicitly.     

长时相干效应与分形布朗运动

 我们研究了阻尼布朗粒子，在具有幂律长时相干C(t)~t^-β（0<β<1，1<β<2）的无规涨落力作用下的运动情况。我们发现它是作分形布朗运动，而不是作普通的布朗运动，而且，找出了分形布朗运动的有效Fokker-Planck方程，以及相应的精确解。于是第一次把长时相干效应和分形布朗运动建立了定量的联系。     

李群流形中的扩散方程

为了描述对称空间中的无规运动,建立了群流形中的扩散方程,给出了紧致黎曼空间中扩散方程的一种具体形式,并进一步讨论了紧致黎曼空间中量子扩散运动.     

The diffusion constant for two-state brownons

We calculate the diffusion constant for two-state brownons when the change of state is not, as usually assumed, Markovian. The correction to the non-interchanging species result is found to be exactly expressible in terms of the Laplace transforms of the sojourn time densities.     

Diffusion modeling of the Rayleigh piston

A Markov jump process in which a massive labeled particle undergoes random elastic collisions with a thermal bath is investigated. It is found that the behavior of the labeled particle can be divided into three distinct regimes depending on whether its velocity is (1) much less than, (2) on the order of, or (3) much greater than the mean speed of a bath particle. In each regime the jump process can be approximated by a particular continuous-path diffusion process. The first case corresponds to the Ornstein-Uhlenbeck process, while each of the latter can be modeled by a deterministic process with a nonlinear Langevin equation. In addition, in cases (2) and (3), the scaled deviation from the mean velocity can be modeled by a nonstationary diffusion. By scaling the time and letting the mass of the labeled particle become large, a continuous-path diffusion is constructed which approximates the jump process in each regime. Analytic solutions for the transition probability density are provided in each case, and numerical comparisons are made between the mean and variance of the diffusions and the original jump process.     

One-dimensional assemblies of silica-coated cobalt nanoparticles: Magnetic pearl necklaces

Silica-coated cobalt nanoparticles were found to organize into chains when driven by a weak external magnetic field. Strong dipole–dipole magnetic interactions are believed to be the driving force of the self-organization once the cobalt nanoparticles undergo the superparamagnetic to ferromagnetic (SP–FM) transition, as increasing their size during the synthesis process. The method, although simple, produces structures resembling pearl necklace-like structures, comparable to one-dimensional species obtained in more laborious processes. Molecular dynamic simulations taking magnetic dipolar forces into account reproduce the observed self-assembled structures. The nanoscale engineering of this type of colloids is expected to extend the spectrum of magnetic effects and functionalities.     

One-dimensional classical fluid with nearest-neighbor interaction in arbitrary external field

We consider the equilibrium statistical mechanics of a classical one-dimensional simple fluid, with nearest-neighbor interactions, and in an arbitrary external potential. The external potential is eliminated to yield relations between the truncated partition functions and the one-body density. These relations are solved for pure cores and for sticky cores, resulting in each case in both an explicit potential density relation and grand potential density functional. Both models maintain finite-range direct correlations and have grand potentials expressible in terms of simple linear density transforms.Rearch supported in part by NSF grant No. CHE-80011285, DOE contract No. DE-AC02-76ER03077, and CNPq 40.0438/79 (Brasil).     

First passage time statistics of Brownian motion with purely time dependent drift and diffusion

Systems where resource availability approaches a critical threshold are common to many engineering and scientific applications and often necessitate the estimation of first passage time statistics of a Brownian motion (Bm) driven by time-dependent drift and diffusion coefficients. Modeling such systems requires solving the associated Fokker-Planck equation subject to an absorbing barrier. Transitional probabilities are derived via the method of images, whose applicability to time dependent problems is shown to be limited to state-independent drift and diffusion coefficients that only depend on time and are proportional to each other. First passage time statistics, such as the survival probabilities and first passage time densities are obtained analytically. The analysis includes the study of different functional forms of the time dependent drift and diffusion, including power-law time dependence and different periodic drivers. As a case study of these theoretical results, a stochastic model of water resources availability in snowmelt dominated regions is presented, where both temperature effects and snow-precipitation input are incorporated.