Stochastic model for the dynamics of a spin-oscillator coupled system

For a model system consisting of bilinearly coupled harmonic oscillators and pseudospins, a generalized Langevin equation for the oscillator subsystem, valid for general spatial dimensionality and interaction range, is derived exactly, using Mori's projection operator technique. The retarded friction kernel is derived to second order in the spinoscillator interaction constant. It is shown that for a one-dimensional model system treated in an earlier paper, the oscillator susceptibility predicted by this approximation, is identical to the susceptibility obtained then by a phenomenological Langevin equation approach.     

用一种分数阶算法研究非马尔可夫过程中阻尼与涨落的竞争机制

基于分数阶朗之万方程和随机行走理论, 建立了一种用于研究非马尔可夫系统中随机变量随时间演化的数值模拟算法, 称之为分数阶随机行走模拟法. 进一步运用此算法分别数值研究了无阻尼有涨落、 有阻尼无涨落和阻尼与涨落兼备三种情况下, 受欠扩散分数阶朗之万方程约束的随机变量随时间的演化行为. 结果显示阻尼和涨落存在竞争关系: 高斯型涨落的影响会随着时间的增长被"抹平", 从而凸显阻尼使系统趋于平衡的作用; 而长尾型涨落则由于包含"小概率大贡献"事件, 使得长时间演化之后系统变量仍以一定概率出现突然变化. 关键词： 非马尔可夫 欠扩散 阻尼与涨落 分数阶朗之万方程     

Quantum Dynamics of a Harmonic Oscillator in a Defomed Bath in the Presence of Lamb Shift

In this paper, we investigate the dissipative quantum dynamics of a harmonic oscillator in the presence a deformed bath by considering the Lamb shift term. The deformed bath is modelled by a collection of deformed quantum harmonic oscillators as a generalization of Hopfield model. The Langevin equation for both the photon number and the fluctuation spectrum under the Weisskopf–Winger approximation are obtained and discussed.     

Quantum fluctuations in a distributed Josephson junction and the spectral properties of Josephson oscillators

Within the framework of a tunneling Hamiltonian, we obtain an equation for the reduced density matrix to describe quasiclassical dynamics and fluctuation effects in distributed Josephson junctions for voltages comparable with the superconducting gap. For quasiclassical dynamics, we derive the Langevin equation describing in a self-consistent way the resistive state and fluctuations due to both the tunneling current and the electromagnetic field. Current-voltage characteristics of a Josephson oscillator are calculated in the high magnetic field approximation. The intensity and shape of the spectral line of radiation due to vortices moving in a distributed Josephson junction are found.     

Langevin Dynamics,Large Deviations and Instantons for the Quasi-Geostrophic Model and Two-Dimensional Euler Equations

We investigate a class of simple models for Langevin dynamics of turbulent flows, including the one-layer quasi-geostrophic equation and the two-dimensional Euler equations. Starting from a path integral representation of the transition probability, we compute the most probable fluctuation paths from one attractor to any state within its basin of attraction. We prove that such fluctuation paths are the time reversed trajectories of the relaxation paths for a corresponding dual dynamics, which are also within the framework of quasi-geostrophic Langevin dynamics. Cases with or without detailed balance are studied. We discuss a specific example for which the stationary measure displays either a second order (continuous) or a first order (discontinuous) phase transition and a tricritical point. In situations where a first order phase transition is observed, the dynamics are bistable. Then, the transition paths between two coexisting attractors are instantons (fluctuation paths from an attractor to a saddle), which are related to the relaxation paths of the corresponding dual dynamics. For this example, we show how one can analytically determine the instantons and compute the transition probabilities for rare transitions between two attractors.     

Path integral representation of fractional harmonic oscillator

Fractional oscillator process can be obtained as the solution to the fractional Langevin equation. There exist two types of fractional oscillator processes, based on the choice of fractional integro-differential operators (namely Weyl and Riemann-Liouville). An operator identity for the fractional differential operators associated with the fractional oscillators is derived; it allows the solution of fractional Langevin equations to be obtained by simple inversion. The relationship between these two fractional oscillator processes is studied. The operator identity also plays an important role in the derivation of the path integral representation of the fractional oscillator processes. Relevant quantities such as two-point and n-point functions can be calculated from the generating functions.     

Self-synchronization of nonlinear oscillations in the presence of fluctuations

A statistical mechanical theory is presented for the self-organization of a macroscopic oscillation with the presence of external fluctuations in a system of Van der Pol oscillators coupled through dissipative interactions. Starting from Langevin equations for the Van der Pol oscillators, the static and dynamic characteristics are studied. The threshold condition is given by the relative size between the fluctuation and the interaction. The transitions between synchronous and asynchronous phases are well discussed by a Landau-type equation. The steady state value of the order parameter and the onset time are compared between the theory and the computer experiments and a good agreement is obtained.     

Precise fabrication of nanomaterials: a nonlinear dynamics approach

Modeling of the precise fabrication in the self-assembling of particles is studied using the nonlinear Langevin equation system. The numerical simulation showed a marked ordering of the particles as a function of time after some induction period. The abnormally enlarged fluctuation was found around the start of the evident ordering. After the fluctuation, a sudden increase of the cluster size was observed. The results corresponded well to the dynamics due to the formation of the critical cluster. The shape of the critical cluster around the enlarged fluctuation was not compact and showed fractal-like structures. The fluctuation of the cluster size around the formation of the critical cluster was explained by the anomalous fluctuation theorem for the generalized Langevin equation. The characterization of the stochastic dynamics of the critical clusters rationalized the concept of dynamic templating for the fabrication technique of the self-assembling of nanoparticles, that is, the structural constraint on the particle assembly by externally adding the resonance frequencies that match with the localized nonlinear vibrational modes of the target structures originating from thermal (Brownian) activation.     

耦合系统的朗之万动力学产生法

提出一种朗之万动力学方法获取处于热平衡态耦合系统内部振子坐标,数值模拟了单端固定简谐振子链的时间演化行为,并将其平衡性质与解析解进行了比较.结果表明了朗之万动力学方法的有效性.推广应用于非简谐四次方型耦合系统,模拟得到振子的四次方均坐标,与理论值验证;以模拟结果作为样本点计算哈密顿量,其能量分布与Boltzmann分布相符.     

Exchange fluctuation theorems for a chain of interacting particles in presence of two heat baths

The exchange fluctuation theorem for heat exchanged between two systems at different temperatures, when kept in direct contact, has been investigated by Jarzynski and Wójcik [Phys. Rev. Lett. 92, 230602 (2004)]. We extend this result to the case where two Langevin reservoirs at different temperatures are connected via a conductor made of interacting particles, and are subjected to an external drive or work source. The Langevin reservoirs are characterized by Gaussian white noise fluctuations and concomitant friction coefficients. We first derive the Crooks theorem for the ratio between forward and reverse paths, and discuss the first law in this model. Then we derive the modified detailed fluctuation theorems (MDFT) for the heat exchanged at each end. These theorems differ from the usual form of the detailed fluctuation theorems (DFT) in literature, due the presence of an extra multiplicative factor. This factor quantifies the deviation of our MFDT from the DFT. Finally, we numerically study our model, with only two interacting particles for simplicity.     

Ward Takahashi identities and fluctuation-dissipation theorem in a superspace formulation of the Langevin equation

A class of Langevin equations is formulated as a field theory in superspace. The Ward Takahashi identities associated with the hidden supersymmetry are derived which in turn are shown to lead to the fluctuation dissipation theorems.     

Foundations of quantum mechanics: The Langevin equations for QM

Stochastic derivations of the Schrödinger equation are always developed on very general and abstract grounds. Thus, one is never enlightened which specific stochastic process corresponds to some particular quantum mechanical system, that is, given the physical system—expressed by the potential function, which fluctuation structure one should impose on a Langevin equation in order to arrive at results identical to those comming from the solutions of the Schrödinger equation. We show, from first principles, how to write the Langevin stochastic equations for any particular quantum system. We also show the relation between these Langevin equations and those proposed by Bohm in 1952. We present numerical simulations of the Langevin equations for some quantum mechanical problems and compare them with the usual analytic solutions to show the adequacy of our approach. The model also allows us to address important topics on the interpretation of quantum mechanics.     

Fokker-Planck description of multi-degree-of-freedom systems with correlated fluctuations

A Fokker-Planck equation is derived for a many-degree-of-freedom nonlinear Langevin equation driven by parametric gaussian fluctuations with finite correlation times. An oscillator with a fluctuating frequency is presented as an example.     

Dynamics of driven Brownian inverted oscillator

In this paper, we derive the quantum Langevin equation for a driven Brownian inverted oscillator in the framework of the Heisenberg picture for the Caldeira-Leggett model. We describe the influence of an arbitrary time-dependent force on an open inverted oscillator dynamics. We take into account environment through the integral operator of relaxation and the force correlation function. The resulting behavior of the system is represented as a combination the time evolution of the position expectation and the variance, being induced simultaneously by spreading the wave packet and the chaotic Brownian motion. We discuss the possibility of stabilization of an open inverted oscillator, when applying external alternating force.     

Entropy production along a stochastic trajectory and an integral fluctuation theorem

For stochastic nonequilibrium dynamics like a Langevin equation for a colloidal particle or a master equation for discrete states, entropy production along a single trajectory is studied. It involves both genuine particle entropy and entropy production in the surrounding medium. The integrated sum of both Delatas(tot) is shown to obey a fluctuation theorem (exp([-Deltas(tot) = 1 for arbitrary initial conditions and arbitrary time-dependent driving over a finite time interval.     

Linear relaxation processes governed by fractional symmetric kinetic equations

The fractional symmetric Fokker-Planck and Einstein-Smoluchowski kinetic equations that describe the evolution of systems influenced by stochastic forces distributed with stable probability laws are derived. These equations generalize the known kinetic equations of the Brownian motion theory and involve symmetric fractional derivatives with respect to velocity and space variables. With the help of these equations, the linear relaxation processes in the force-free case and for the linear oscillator is analytically studied. For a weakly damped oscillator, a kinetic equation for the distribution in slow variables is obtained. Linear relaxation processes are also studied numerically by solving the corresponding Langevin equations with the source given by a discrete-time approximation to white Levy noise. Numerical and analytical results agree quantitatively.     

Stochastic thermodynamics: principles and perspectives

Stochastic thermodynamics provides a framework for describing small systems like colloids or biomolecules driven out of equilibrium but still in contact with a heat bath. Both, a first-law like energy balance involving exchanged heat and entropy production entering refinements of the second law can consistently be defined along single stochastic trajectories. Various exact relations involving the distribution of such quantities like integral and detailed fluctuation theorems for total entropy production and the Jarzynski relation follow from such an approach based on Langevin dynamics. Analogues of these relations can be proven for any system obeying a stochastic master equation like, in particular, (bio)chemically driven enzyms or whole reaction networks. The perspective of investigating such relations for stochastic field equations like the Kardar-Parisi-Zhang equation is sketched as well.     

Analysis of a few numerical integration methods for the Langevin equation

We study the position recurrence relation of several existing numerical integrators for the Langevin equation and use the modified equation approach to analyse their accuracy. We show that for the harmonic oscillator, the BBK integrator converges weakly with order 1 while the vGB82 and Langevin impulse (LÎ)? integrator converge weakly with order 2. We also study a restricted class of velocity definitions—those that lead to explicit starting procedures. We show that some recurrence relations exact for constant force, can achieve the exact virial relation by a proper definition of velocity, extending the result of Pastor et al. on the analysis of BBK integrators in 1988.