On the Correlation Properties of Thermal Noise in Fluids

The properties of the thermal force driving micron particles in incompressible fluids are studied within the hydrodynamic theory of Brownian motion. It is shown that the assumption used for the hydrodynamic Langevin equation in its usual form with the initial time t = 0, according to which the random force at a time t > 0 and the velocity of the particle at t = 0 are uncorrelated, leads to an unexpected super-diffusion of the particle. To obtain the correct Einstein diffusion at long times, the mentioned hypothesis must be abandoned, which however does not contradict causality. The corresponding correlations are explicitly evaluated. The “color” of thermal noise, recently measured experimentally [Franosch et al. Nature 478, 85 2011)] is also considered, and the interpretation of these experiments is corrected. The time correlation functions for the thermal random force are obtained using the exact solution of the Langevin equation, and on the basis of the theorem that in the linear response theory connects the mobility of a particle and its velocity autocorrelation function.     

Trap of a Submicron Particle in a Quadrupole Acoustic Chamber

We investigate analytically and numerically trapping of submicron aerosol particles in a three-dimensional quadrupole acoustic chamber having hyperbolical configuration. The particle trajectories are described by the Langevin equation accounting for particle random Brownian motion. The particle trapping efficiency is investigated for a range of acoustic field parameters and particle properties. It is shown that submicron diffusive particles can be trapped in a small region near the chamber center. The effect of Brownian motion is to broaden the trapping region. The dimensions of the trapping region can be reduced by increasing the acoustic strength parameter.     

Dynamics of Fluctuating Bose–Einstein Condensates

We present a generalized Gross–Pitaevskii equation that describes the dissipative dynamics of a trapped partially Bose-condensed gas. It takes the form of a complex nonlinear Schrödinger equation with noise. We consider an approximation to this Langevin field equation that preserves the correct equilibrium for both the condensed and the noncondensed parts of the gas. We then use this formalism to describe the reversible formation of a one-dimensional Bose condensate, and compare with recent experiments. In addition, we determine the frequencies and the damping of collective modes in this case.     

Investigation of mode coupling in low and high NA step index plastic optical fibres using the Langevin equation

By employing the Langevin equation, we have examined a mode coupling in low and high NA step index plastic optical fibres. The numerical integration of the Langevin equation is based on the computer-simulated Langevin force. The solution matches the experimental data reported previously. We have shown that by solving the Langevin equation (stochastic differential equation) one can treat a mode coupling in multimode low and high NA step index plastic optical fibres, which is the result of fibre's intrinsic random perturbations.     

Multidimensional Kinetics of Nucleation in Liquid-Vapor Systems

A metastable system is described as an ensemble of locally isolated statistically independent centers, with a possibility of one and only one nucleus of a close-to- critical radius emerging on each one of these centers. It is assumed that this event occurs as a result of fluctuations in a heterophase subsystem and leads to the formation of a viable nucleus at a given point in space. The process of the emergence of this nucleus is treated as the first crossing of the potential barrier by a Brownian particle. Proceeding from the principles of nonequilibrium thermodynamics, dynamic equations of bubble (droplet) growth are derived, which correspond to the Onsager relations. These formulas are used as Langevin equations in multidimensional phase space and are related to the respective Fokker-Planck equation whose solution enables one to determine the local rate of emergence of a viable nucleus and, as a consequence, the rate of its emergence in the entire system. An alternative expression is given for the rate of homogeneous steady-state nucleation, which differs from the classical expression by the pre-exponential factor and, in the case where one parameter (radius) may be sufficient, gives close limits of attainable superheat (supersaturation). Given the expression for the nonequilibrium work of bubble (droplet) and the distribution of heterogeneous centers, the obtained result may be readily generalized to the case of heterogeneous nucleation.     

A buoyancy-assisted mechanism of scalable colloidal crystallization

An ingenious mechanism of scalable grain crystallization inside a 3D colloidal domain is proposed. The mechanism is materialized by random vibration either on a jammed particle assembly or on free-falling particle fronts traveling inside a colloidal domain under the influence of near-zero or limited buoyancy. Brownian dynamics is invoked by employing a generalized Langevin’s thermostat equation, which solves for a time-variant state of particles undergoing moderate random vibration under the effect of a viscose solvent until full occupancy of the box is reached at fixed temperature. The contacting particles admit a simple mass-spring-dashpot discrete-element model with negligible sliding friction. The problem seeks optimized states with the highest overall crystallinity and lowest grain-boundary effect by parametrization of the sedimentation domain geometry as well as the particles’ and solvent’s properties. The parametric study investigates the effects of geometry (box height, length and cross-sectional mismatch), particle density and solvent viscosity on evolving and ultimate-state crystallinity, chiefly quantified by an overall crystallinity ratio. Buoyancy-assisted sedimentation reflects the formation of FCC-dominant particle submanifolds, and is further suggestive of optimum ranges of box height, solvent viscosity and particle density as opposed to critical ranges of box length and in-plane aspect ratio. Depending on the desired level of crystallinity, the proposed mechanism can be regarded as supplant or supplement for other crystallization mechanisms including aging, magnetization, etc.     

Invalidity of the spectral Fokker–Planck equation for Cauchy noise driven Langevin equation

The standard Langevin equation is a first order stochastic differential equation where the driving noise term is a Brownian motion. The marginal probability density is a solution to a linear partial differential equation called the Fokker–Planck equation. If the Brownian motion is replaced by so-called -stable noise (or Lévy noise) the Fokker–Planck equation no longer exists as a partial differential equation for the probability density because the property of finite variance is lost. Instead it has been attempted to formulate an equation for the characteristic function (the Fourier transform) corresponding to the density function. This equation is frequently called the spectral Fokker–Planck equation.

This paper raises doubt about the validity of the spectral Fokker–Planck equation in its standard formulation. The equation can be solved with respect to stationary solutions in the particular case where the noise is Cauchy noise and the drift function is a polynomial that allows the existence of a stationary probability density solution. The solution shows paradoxic properties by not being unique and only in particular cases having one of its solutions closely approximating the solutions to a corresponding Langevin difference equation. Similar doubt can be traced Grigoriu's work [Stochastic Calculus (2002)].     

Brownian motion model of nanoparticle considering nonrigidity of matter-a systems modeling approach

The variance models for Brownian motion are developed either using the diffusion equation method or by using spectral analysis with a Langevin equation. The diffusion method approach does not consider properties of matter like inertia, elasticity, and dissipative capabilities whereas in the Langevin equation approach, although based on the property concept, the matter is considered rigid and there are no attempts of inclusion of elastic and other properties to study the Brownian motion. The concept of absoluteness (in rigidity) is debatable in nanodomains, and instead an analytical model for Brownian motion of nanosize particles has been obtained and explored in this work using the Langevin equation considering nonrigidity and dissipative capabilities of matter.     

分数阶双稳系统随机共振现象研究及FPGA实现

相比整数阶微积分,分数阶对一些具有记忆依赖性以及空间相关性的复杂系统的描述更简明与贴合.利用分数阶微积分的这一优点并结合广义郎之万方程阻尼具有幂律衰减特性,选择合理的核函数,将整数阶郎之万方程推广至分数阶.以此为理论基础,提出了一种分数阶双稳态随机共振系统的FPGA实现方法.对该系统进行了仿真实验验证,仿真结果表明:在...     

Asymptotic behavior of the Langevin equation for a nonhomogeneous medium

The Langevin equation has gained renewed interest through its role in modeling turbulent, reacting flow, and turbulent dispersion. Such applications often require an understanding of its long term, asymptotic behavior. The asymptotic, Smoluchowsky, equation is derived by constructing a uniformly valid bound on the velocity variance. Extra drift terms arise in the asymptotic equation when the coefficients are not constant.     

Solution of mode coupling in step-index optical fibers by the Fokker-Planck equation and the Langevin equation

The power-flow equation is approximated by the Fokker-Planck equation that is further transformed into a stochastic differential (Langevin) equation, resulting in an efficient method for the estimation of the state of mode coupling along step-index optical fibers caused by their intrinsic perturbation effects. The inherently stochastic nature of these effects is thus fully recognized mathematically. The numerical integration is based on the computer-simulated Langevin force. The solution matches the solution of the power-flow equation reported previously. Conceptually important steps of this work include (i) the expression of the power-flow equation in a form of the diffusion equation that is known to represent the solution of the stochastic differential equation describing processes with random perturbations and (ii) the recognition that mode coupling in multimode optical fibers is caused by random perturbations.     

A probability density closure model for turbulence

Summary Starting from the Navier-Stokes equation, the evolution equation of the first order probability density functions of turbulence is derived. Closure assumptions for the pressure terms are introduced and the closed pdf equation is obtained. The resulting evolution equation takes the form of the Fokker-Planck equation and contains the existing Langevin based models as special cases. It is shown that the model predicts the known behaviors of turbulence in the limit of extremely high Reynolds number as well as a decaying turbulence. The moment equations are also examined. It is shown that the model is consistent with the recent second and third order closure models of turbulence.     

Diffusion in a Medium with Nonlinear Friction

The Langevin equation describing the Brownian motion of particles is studied in the case when the friction depends nonlinearly on the particle velocity. The intensity \(D(\upsilon )\) of the thermal random force driving the particles then, in equilibrium systems, also depends on the velocity, and the equation of motion should contain an additional ‘spurious’ force proportional to the derivative of \(D(\upsilon )\) . This force is chosen in the kinetic representation, for which the stationary probability density for velocities is the Maxwell distribution and simultaneously the generalized Einstein relation is obeyed. A general formula for the diffusion coefficient of the particle is obtained and then specified for various models of dissipative forces studied in the literature, in particular, for the power-law friction. In this case the scaling of the diffusion coefficient with the model parameters is obtained also by a simple dimensional analysis, for systems both at equilibrium and far from it, when \(D\) is usually assumed to be independent of \(\upsilon \) .     

Thermal Noise and Hydrodynamic Memory Effects on Force Measurements at Microscales

The paper is devoted to the problem of the determination of regular forces acting on microscopic and smaller objects in fluids when the presence of thermal noise affects the results of measurements. One of the methods on how these forces are determined is the measurement of the drift velocity of Brownian particles. Traditionally such experiments are interpreted on the basis of the overdamped Langevin equation. An exact expression has been obtained for this velocity within the linear hydrodynamic theory of the Brownian motion in incompressible fluids, when the external force is constant and when the particle is in a harmonic potential well. It is shown that for sufficiently short experimental times the influence of the hydrodynamic memory effects in the drift velocity determination is significant. The obtained solutions contain algebraic long-time tails due to which the results expected from the standard theory are approached very slowly with the increase of time.     

Superparamagnetic properties of carbon-encapsulated Ni nanoparticle assemblies

Carbon-encapsulated Ni nanoparticles [Ni(C)] were synthesized using a modified arc-discharge reactor under methane atmosphere. The average particle size was revealed to be typically 10.5 nm with a spherical shape. The intimate and contiguous carbon fringe around these Ni nanoparticles is good evidence for complete encapsulation by carbon shell layers. Superparamagnetic property studies indicate that the blocking temperature (TB) is around 115 K at 1000 Oe applied field. Below TB, the temperature dependence of the coercivity is given by Hc = Hci[1 -(T/TB)1/2], with Hci approximately 500 Oe. Above TB, the magnetization M(H, T) can be described by the classical Langevin function L using the relationship M/Ms(T = 0) = coth(microH/kT)-kT/microH. The particle size can be inferred from the Langevin fit (particle moment mu) and the blocking temperature theory (TB), with values slightly larger than the high-resolution transmission electron microscopy observations. It is suggested that these assemblies of carbon-encapsulated Ni nanoparticles have typical single-domain, field-dependent superparamagnetic relaxation properties.     

Langevin acoustic radiation force of a high-order bessel beam on a rigid sphere

The acoustic radiation force of Langevin type resulting from the interaction of a high-order Bessel beam with a rigid immovable sphere in an ideal fluid is theoretically investigated. The analysis is based on applying the generalized Rayleigh series used in the near-field acoustic scattering problem to calculate the force. With appropriate selection of specific Bessel beam parameters, results for the rigid sphere unexpectedly reveal a negative radiation force caused by the Lagrangean energy density. Specifically, the negative force on the rigid sphere arises when the kinematic energy density is larger than the potential energy density. This condition provides an impetus for further designing acoustic tweezers operating with high-order Bessel beams of progressive waves for potential applications in particle entrapment and manipulation.     

Thermally activated magnetisation reversal

A model of thermally activated magnetisation reversal is presented. The approach is based on the numerical solution of the Langevin equation of the problem. It is shown that the random thermal perturbations give rise to correlated magnetisation fluctuations (spin waves) in coupled systems. Simulations of longitudinal thin films are presented which show an effective non-local damping arising from spin waves.     

Stochastic model of crack paths in composites based on the Langevin equation

A stochastic model based on the Langevin equation was used to describe crack paths in composites. A single crack path in a fiberglass reinforced epoxy matrix composite was used to predict Langevin parameters quantifying architectural drift and material variability. The predicted Langevin parameters were consistent with moderate architectural drift (Langevin drift parameter of 0.04 ± 0.02) and moderate material variability (Langevin variability parameter of 0.05 ± 0.03). These Langevin parameters were then used to predict a family of crack paths exhibiting the same stochastic characteristics as the original crack path. Architectural drift was qualitatively related to the orientation of the reinforcing phase. A strong correlation between the predicted and experimental crack path suggests the utility of the Langevin model to quantify crack paths.     

非线性分数阶双稳系统逻辑随机共振的研究

在整数阶逻辑随机共振的郎之万方程基础上构建了分数阶情况下的郎之万方程。对该方程描述的非线性分数阶双稳系统进行了仿真验证,分析分数阶阶次和系统参数的改变对逻辑随机共振现象的影响。结果表明当分数阶阶次小于临界值时,即使没有外加高斯白噪声或微弱周期信号也能观察到逻辑随机共振现象;当分数阶阶次大于临界值时,需要外加高斯白噪声或微弱周期信号才能实现逻辑随机共振,选择合适的噪声强度、微弱周期信号振幅、频率等可以提高逻辑输出的成功率。     

Cold Soliton of the General Nonlinear Discontinuity Equation

In this paper we consider generalized nonlinear discontinuity equation and try to obtain second-order approximation solution. In order to obtain these solutions we used modified Lindstedt-Poincare method. Then we extract solitonic solution from special case of generalized nonlinear discontinuity equation. These solution behave as particle like in the cold black holes and low temperature dark matter. Other application of these solution is in superfluid and superconductivity.