From shape to randomness: A classification of Langevin stochasticity

The Langevin equation–perhaps the most elemental stochastic differential equation in the physical sciences–describes the dynamics of a random motion driven simultaneously by a deterministic potential field and by a stochastic white noise. The Langevin equation is, in effect, a mechanism that maps the stochastic white-noise input to a stochastic output: a stationary steady state distribution in the case of potential wells, and a transient extremum distribution in the case of potential gradients. In this paper we explore the degree of randomness of the Langevin equation’s stochastic output, and classify it à la Mandelbrot into five states of randomness ranging from “infra-mild” to “ultra-wild”. We establish closed-form and highly implementable analytic results that determine the randomness of the Langevin equation’s stochastic output–based on the shape of the Langevin equation’s potential field.     

Langevin dynamics of an interface near a wall

We study dynamical contact angles and precursor films using Langevin dynamics for SOS type models, near a wall which favors spreading. We first solve exactly the Gaussian model and discuss various asymptotic regimes. This is only appropriate to partial wetting. We then consider more general models. Using local equilibrium and scaling arguments, we derive the shape of the dynamical profile and the speed of the precursor film which exists when the spreading coefficient is strictly positive. Long-range potentials lead to a layered structure of the precursor film. We also consider the case of a meniscus in a capillary.     

Langevin simulation of scalar fields: Additive and multiplicative noises and lattice renormalization

We consider the Langevin lattice dynamics for a spontaneously broken λ?⁴$\lambda {?}^4$ scalar field theory where both additive and multiplicative noise terms are incorporated. The lattice renormalization for the corresponding stochastic Ginzburg–Landau–Langevin and the subtleties related to the multiplicative noise are investigated.     

Langevin dynamics simulations of phase separation in a one-component system

Langevin dynamics computer simulations have been performed for a two-dimensional Lennard-Jones fluid quenched into the coexistence region of its liquid-vapor phase diagram. For late stages of the phase-separation process, the average radius of the liquid clusters is found to grow proportional to (time)^1/4. This growth law is analyzed theoretically and compared to recent molecular dynamics and Monte Carlo results. Details of the different simulation methods are critically discussed.     

Return to thermal equilibrium by the solution of a quantum Langevin equation

A quantum-mechanical treatment of the evolution of an anharmonic oscillator coupled to a heat bath is given. It is shown that for a certain class of anharmonic potentials the heat bath drives the oscillator to an equilibrium state, close to the quantum Gibbs state associated to the potential. Thus a partial proof is provided for a conjecture of R. Benguria and M. Kac.This paper contains part of the author's Ph.D. work, done at the Institute for Theoretical Physics of Groningen State University, Groningen, the Netherlands.     

Langevin equation for a free particle driven by power law type of noises

We study a generalized Langevin equation for a free particle driven by N internal noises. The mean square displacement and velocity autocorrelation function are derived in case of a mixture of Dirac delta, power law and Mittag-Leffler noises. Additionally, a frictional memory kernel of distributed order is considered. The long time limit and short time limit are analyzed, and the dominant contributions of noises on particle dynamics is discussed. Various different diffusive behaviors (subdiffusion, superdiffusion, normal diffusion, ultraslow diffusion) are obtained. The considered problem may be used in the theory of anomalous diffusion in complex environment.     

Generalized Langevin equation driven by Lévy processes: A probabilistic, numerical and time series based approach

Lévy processes have been widely used to model a large variety of stochastic processes under anomalous diffusion. In this note we show that Lévy processes play an important role in the study of the Generalized Langevin Equation (GLE). The solution to the GLE is proposed using stochastic integration in the sense of convergence in probability. Properties of the solution processes are obtained and numerical methods for stochastic integration are developed and applied to examples. Time series methods are applied to obtain estimation formulas for parameters related to the solution process. A Monte Carlo simulation study shows the estimation of the memory function parameter. We also estimate the stability index parameter when the noise is a Lévy process.     

Construction of a Langevin model from time series with a periodical correlation function: Application to wind speed data

A Langevin-type equation for stochastic processes with a periodical correlation function is introduced. A procedure of reconstruction of the equation from time series is proposed and verified on simulated data. The method is applied to geophysical time series–hourly time series of wind speed measured in northern Italy–constructing the macroscopic model of the phenomenon.     

Numerical resolution of the modified Langevin equation using a differential expression: Application to the Jiles magnetostriction law of approach

The Langevin equation is classically used to model the anhysteretic magnetization curve. A modified version of this equation has been introduced by Jiles to take into account the effects of magnetostriction on the anhysteretic magnetization behavior when a ferromagnetic material undergoes mechanical stresses. The numerical resolution of the modified Langevin equation is usually performed with a root-finding algorithm. In this paper, a differential form of the modified Langevin equation is proposed, allowing a faster numerical resolution.     

Effect of Environment on Photo-detachment Dynamics of Halide Ions: A Model Approach

利用外部光场,采用二维模型的方法研究了闭壳阴离子系统在极性溶剂环境介质中光解动力学过程．同时用一个简单的模型研究了系统中溶剂极化和外界电荷之间强耦效应．卤离子的电子态用二维模型哈密尔顿函数表示,采用含时傅立叶格点哈密顿方法研究解离过程．通过上述两种不同的方法揭示溶剂环境对光解动力学的影响：引入影响阴离子系统能量的极性溶剂来改变有效势能面和利用井深的随机波动来模拟系统．另外对作为光场重要参数的光解平均速率常数进行计算,定性地解释了溶剂效应对于偶极束缚阴离子系统的影响．     

PbWO4闪烁晶体的发光动力学模型

在对PbWO₄闪烁晶体的光谱特性、发光衰减及其温度依赖以及热释光的研究基础上,并结合理论计算,提出了PbWO₄晶体发光的动力学模型,给出了PbWO₄晶体的基本能带结构及激子发光中心能态、陷阱能级在能隙中的位置。用此模型可以完整说明PbWO₄的发光过程,特别是导致室温下发光效率低的原因。最后还对其主发射成分蓝、绿发光中心的起源作了简要讨论。     

Stochastic motion of a charged particle in a magnetic field: II Quantum Brownian treatment

We study the quantum Brownian motion of a charged particle in the presence of a magnetic field. From the explicit solution of a quantum Langevin equation we calculate quantities such as the velocity correlation function and the mean-squared displacement. Our calculated expressions contain as special cases the motion of aclassical particle in a magnetic field and that of afree (but quantum) particle, in a dissipative environment.     

A Simple Generalized Excitability Model Mimicking Salient Features of Neuron Dynamics

A generalization of the FitzHugh—Nagumo model for excitability is provided to account for salient features of Inferior Olive neurons. The base state is a limit cycle and excitability appears as spiking over peaks of the oscillations. The response of the model to various types of external stimulus is also presented. In particular, we show the relevance of an appropriate balance between amplitude and duration of the stimulus.     

Chaotic Dynamics in an Electronic Model of a Genetic Network

We consider dynamics in a class of piecewise-linear ordinary differential equations and in an electronic circuit that model genetic networks. In these models, gene activity varies continuously in time. However, as in Boolean or discrete-time switching networks, gene activity is driven high or low based only on whether the activities of the regulating genes are high or low (i.e., above or below certain thresholds). Depending on the “regulatory logic”, these models can exhibit simple dynamics, like stable fixed points or oscillation, or chaotic dynamics. The observed qualitative and quantitative differences between the dynamics in the idealized equations and the dynamics in the electronic circuit lead us to focus attention on the analysis of the dynamics as a function of parameter values. We propose new techniques for solving the inverse problem – the problem of inferring the regulatory logic and parameters from time series data. We also give new symbolic and statistical methods for characterizing dynamics in these networks.     

Dissipative Dynamics and the Statistics of Energy States of a Hookean Model for Protein Folding

A generic model of a random polypeptide chain, with discrete torsional degrees of freedom and Hookean spring connecting pairs of hydrophobic residues, reproduces the energy probability distribution of real proteins over a very large range of energies. We show that this system with harmonic interactions, under dissipative dynamics driven by random noise, leads to a distribution of energy states obeying a modified one-dimensional Ornstein–Uhlenbeck process and giving rise to the so-called Wigner distribution. A tunably fine- or coarse-grained sampling of the energy landscape yields a family of distributions for the energies and energy spacings.     

Analysis and Experiments for a Computational Model of a Heat Bath

A question of some interest in computational statistical mechanics is whether macroscopic quantities can be accurately computed without detailed resolution of the fastest scales in the problem. To address this question a simple model for a distinguished particle immersed in a heat bath is studied (due to Ford and Kac). The model yields a Hamiltonian system of dimension 2N+2 for the distinguished particle and the degrees of freedom describing the bath. It is proven that, in the limit of an infinite number of particles in the heat bath (N), the motion of the distinguished particle is governed by a stochastic differential equation (SDE) of dimension 2. Numerical experiments are then conducted on the Hamiltonian system of dimension 2N+2 (N1) to investigate whether the motion of the distinguished particle is accurately computed (i.e., whether it is close to the solution of the SDE) when the time step is small relative to the natural time scale of the distinguished particle, but the product of the fastest frequency in the heat bath and the time step is not small—the underresolved regime in which many computations are performed. It is shown that certain methods accurately compute the limiting behavior of the distinguished particle, while others do not. Those that do not are shown to compute a different, incorrect, macroscopic limit.     

持久光谱烧孔动力学的理论模型和实验研究

建立了单光子光化烧孔的三能级模型，求解了反映质子转位变构机制的速率方程组，推导了成孔过程的动力学特性，获得了可与实验进行比较的解析解．用时域和频域两种方法测量了ＴＨＰ／ＰＭＭＡ和ＴＭＰ／ＰＭＭＡ薄膜样品光谱烧孔的动力学过程，得到与理论分析一致的规律，并测得了有效成孔速率和成孔时间等参数．这些参数对于评价频域光存贮材料性能有一定意义．     

基于Gay-Berne势能模型的粗粒化动力学模拟研究正丁醇的玻璃态相变

通过基于Gay-Berne势能模型的粗粒化动力学模拟,研究液态正丁醇体系的冷却过程. 采用密度泛函计算,拟合出适合正丁醇体系的GB势能参数. 体系的密度、平均势能等性质随温度的降低(由290 K降至50 K,间隔为10 K)发生特殊变化,即体系发生玻璃态相变,相变温度为T_g=120±10 K,与实验值110±1 K符合很好.     

液态氧化铝结构的分子动力学模拟

利用分子动力学模拟液态氧化铝从2663 K到过冷温度2223 K的结构变化,通过总双体分布函数得到的液态氧化铝的原子第一、第二近邻距离与Ansell研究小组的实验结果基本一致.通过配位数和偏角分布函数的分析,液态氧化铝的局域结构主要是由氧的三重铝原子配位和铝原子四重(AlO₄)和五重(AlO₅)氧原子配位组成.没有观察到可能因铝原子高、低配位数氧原子的团簇分离而出现的液-液相变.