Langevin Equation For Brownian Motion Research Articles

Single-mode nonlinear Langevin emulation of magnetohydrodynamic turbulence.

Based on the Langevin equation of Brownian motion, we present a simple model that emulates a typical mode in incompressible magnetohydrodynamic turbulence, providing a demonstration of several key properties. The model equation is consistent with von Kármán decay law and Kolmogorov's symmetries. We primarily focus on the behavior of inertial range modes, although we also attempt to include some properties of the large-scale modes. Dissipation scales are not considered. Results from the model are compared with results from published direct numerical simulations.

From the Underdamped Generalized Elastic Model to the Single Particle Langevin Description

The generalized elastic model encompasses several linear stochastic models describing the dynamics of polymers, membranes, rough surfaces, and fluctuating interfaces. While usually defined in the overdamped case, in this paper we formally include the inertial term to account for the initial diffusive stages of the stochastic dynamics. We derive the generalized Langevin equation for a probe particle and we show that this equation reduces to the usual Langevin equation for Brownian motion, and to the fractional Langevin equation on the long-time limit.

The Anomalous Distributions and Soret Coefficient in a Nonequilibrium Colloidal System

We study the density distribution and Soret coefficient in a nonequilibrium colloidal system by using the overdamped Langevin equation for Brownian motion in an inhomogeneous strong friction medium. Based on the relation between the temperature gradient, the interaction potential and the [Formula: see text]-parameter in nonextensive statistics, we show that the colloidal particle density can be a function of the temperature and anomalously follows the noted [Formula: see text]-distribution, or equivalently it can also be a function of the potential energy following Tsallis distribution. With the [Formula: see text]-parameter we can establish a new formula of Soret coefficient and thus, bridge the gap between the ideally theoretical Soret coefficient and available experiments.

Fluctuating neutron star magnetosphere: braking indices of eight pulsars, frequency second derivatives of 222 pulsars and 15 magnetars

Eight pulsars have low braking indices, which challenge the magnetic dipole braking of pulsars. 222 pulsars and 15 magnetars have abnormal distribution of frequency second derivatives, which also make contradiction with classical understanding. How neutron star magnetospheric activities affect these two phenomena are investigated by using the wind braking model of pulsars. It is based on the observational evidence that pulsar timing is correlated with emission and both aspects reflect the magnetospheric activities. Fluctuations are unavoidable for a physical neutron star magnetosphere. Young pulsars have meaningful braking indices, while old pulsars' and magnetars' fluctuation item dominates their frequency second derivatives. It can explain both the braking index and frequency second derivative of pulsars uniformly. The braking indices of eight pulsars are the combined effect of magnetic dipole radiation and particle wind. During the life time of a pulsar, its braking index will evolve from three to one. Pulsars with low braking index may put strong constraint on the particle acceleration process in the neutron star magnetosphere. The effect of pulsar death should be considered during the long term rotational evolution of pulsars. An equation like the Langevin equation for Brownian motion was derived for pulsar spin-down. The fluctuation in the neutron star magnetosphere can be either periodic or random, which result in anomalous frequency second derivative and they have similar results. The magnetospheric activities of magnetars are always stronger than those of normal pulsars.

Composite generalized Langevin equation for Brownian motion in different hydrodynamic and adhesion regimes.

We present a composite generalized Langevin equation as a unified framework for bridging the hydrodynamic, Brownian, and adhesive spring forces associated with a nanoparticle at different positions from a wall, namely, a bulklike regime, a near-wall regime, and a lubrication regime. The particle velocity autocorrelation function dictates the dynamical interplay between the aforementioned forces, and our proposed methodology successfully captures the well-known hydrodynamic long-time tail with context-dependent scaling exponents and oscillatory behavior due to the binding interaction. Employing the reactive flux formalism, we analyze the effect of hydrodynamic variables on the particle trajectory and characterize the transient kinetics of a particle crossing a predefined milestone. The results suggest that both wall-hydrodynamic interactions and adhesion strength impact the particle kinetics.

The precise time-dependent solution of the Fokker–Planck equation with anomalous diffusion

We study the time behavior of the Fokker–Planck equation in Zwanzig’s rule (the backward-Ito’s rule) based on the Langevin equation of Brownian motion with an anomalous diffusion in a complex medium. The diffusion coefficient is a function in momentum space and follows a generalized fluctuation–dissipation relation. We obtain the precise time-dependent analytical solution of the Fokker–Planck equation and at long time the solution approaches to a stationary power-law distribution in nonextensive statistics. As a test, numerically we have demonstrated the accuracy and validity of the time-dependent solution.

Aging scaled Brownian motion.

Scaled Brownian motion (SBM) is widely used to model anomalous diffusion of passive tracers in complex and biological systems. It is a highly nonstationary process governed by the Langevin equation for Brownian motion, however, with a power-law time dependence of the noise strength. Here we study the aging properties of SBM for both unconfined and confined motion. Specifically, we derive the ensemble and time averaged mean squared displacements and analyze their behavior in the regimes of weak, intermediate, and strong aging. A very rich behavior is revealed for confined aging SBM depending on different aging times and whether the process is sub- or superdiffusive. We demonstrate that the information on the aging factorizes with respect to the lag time and exhibits a functional form that is identical to the aging behavior of scale-free continuous time random walk processes. While SBM exhibits a disparity between ensemble and time averaged observables and is thus weakly nonergodic, strong aging is shown to effect a convergence of the ensemble and time averaged mean squared displacement. Finally, we derive the density of first passage times in the semi-infinite domain that features a crossover defined by the aging time.

A discrete impulsive model for random heating and Brownian motion

The energy of a mechanical system subjected to a random force with zero mean increases irreversibly and diverges with time in the absence of friction or dissipation. This random heating effect is usually encountered in phenomenological theories formulated in terms of stochastic differential equations, the epitome of which is the Langevin equation of Brownian motion. We discuss a simple discrete impulsive model that captures the essence of random heating and Brownian motion. The model may be regarded as a discrete analog of the Langevin equation, although it is developed ab initio. Its analysis requires only simple algebraic manipulations and elementary averaging concepts, but no stochastic differential equations (or even calculus). The irreversibility in the model is shown to be a consequence of a natural causal stochastic condition that is closely analogous to Boltzmann’s molecular chaos hypothesis in the kinetic theory of gases. The model provides a simple introduction to several ostensibly more advanced topics, including random heating, molecular chaos, irreversibility, Brownian motion, the Langevin equation, and fluctuation-dissipation theorems.

Rotational dynamics of CH3Cl and CH2CF2 in the fluid state

The far infrared absorptions of compressed gaseous and liquid CH3Cl and CH2CF2(1,1-difluoroethylene) have been measured in the frequency range 2-200 cm–1 and the results interpreted in terms of two statistical models of molecular rotational dynamics in fluids. A description based on an extended Langevin equation of Brownian motion reproduces the experimental features adequately using empirical intermolecular torque and width parameters. However, it does not reduce to free rotation in the limit of vanishing torque. The J-diffusion model of collision interrupted rotation in symmetric top molecules reduces to free rotation as the interval between elastic impacts becomes infinite, but fails to predict torsional oscillation at short times in denser fluids, so that it compares badly with the far infrared data for pure liquid CH3Cl and for a solution in ethane.

Fluctuating hydrodynamics and Brownian motion. II. Note on the slip boundary condition

Previous work by Hauge and Martin-Lof discussing the generalized Langevin equation for Brownian motion as a contraction from the more fundamental but still phenomenological description of a particle immersed in an incompressible fluid governed by fluctuating hydrodynamics with stick boundary condition is extended to the case of a Brownian particle with arbitrary shape and withslip boundary condition. The motivation for this extension is the fact that the latter condition naturally arises in a treatment of the problem from first principles.

On the Use of the Langevin Equation of Brownian Motion for the Analyses of Cold and Slow Neutron Scattering by Liquid Methane

The Langevin equation of Brownian motion is used to analyze the experimental data on the scattering of cold and slow neutrons by liquid methane. Scattering of neutrons by a molecule in the liquid is treated analytically as for a gas except the translational mode of the molecule is assumed to obey the Langevin equation. To obtain agreement with the experimental data using this model, it is necessary to adjust the viscous damping coefficient η appearing in the Langevin equation as the magnitude of the momentum transfer κ in the collision between the neutron and the molecule changes. Plotting the value of η needed to best fit the observed quasielastic peak against the momentum transfer κ resulted in a straight line on a semilogarithm plot. An extrapolation of the straight line to zero momentum transfer gave a value of 1.7×1013 sec−1 for the viscous damping coefficient. This value is in agreement with the experimental value of the diffusion coefficient which is related to the viscous damping coefficient by the Einstein relation. It is pointed out that the analysis is, at least qualitatively, in accord with results on the scattering of neutrons by other liquids. There is evidence, although not conclusive, that the rotational motion is partially hindered.