Langevin representation of Coulomb collisions for bi-Maxwellian plasmas

Langevin model corresponding to the Fokker–Planck equation for bi-Maxwellian particle distribution functions is developed. Rosenbluth potentials and their derivatives are derived in the form of triple hypergeometric functions. The Langevin model is tested in the case of relaxation of the proton temperature anisotropy and implemented into the hybrid expanding box model. First results of this code are presented and discussed.     

A numerical solution of a nonlinear Langevin Equation

A one-dimensional Langevin Equation in which the friction term and the stochastic force term depend nonlinearly on the velocity is presented. Assuming that the Maxwell distribution is the stationary solution of the Fokker Planck Equation (which is equivalent to the nonlinear Langevin Equation) we derive a generalization of the Fluctuation Dissipation theorem. A numerical algorithm is developed which allows us to integrate the nonlinear Langevin Equation. From this numerical solution correlation functions are obtained.     

Nonlinear generalized Langevin equations

Exact generalized Langevin equations are derived for arbitrarily nonlinear systems interacting with specially chosen heat baths. An example is displayed in which the Langevin equation is nonlinear but approximately Markovian.Research supported by NSF grant GP-29534.     

Stochastic Resonance in a Linear Fractional Langevin Equation

The fractional Langevin equation is derived from the generalized Langevin equation driven by the additive fractional Gaussian noise. We investigate the stochastic resonance (SR) phenomenon in the underdamped linear fractional Langevin equation under the external periodic force and multiplicative symmetric dichotomous noise. Applying the Shapiro-Loginov formula and the Laplace transform technique, we obtain the exact expressions of the amplitude and signal-to-noise ratio (SNR) of the system. By studying the impacts of the driving frequency and the noise parameters, we find the non-monotonic behaviors of the output amplitude and SNR. The results indicate that the bona fide SR, conventional SR and the wide sense of SR phenomena occur in the proposed linear fractional system.     

On a Kinetic Equation for Coalescing Particles

Existence of global weak solutions to a spatially inhomogeneous kinetic model for coalescing particles is proved, each particle being identified by its mass, momentum and position. The large time convergence to zero is also shown. The cornestone of our analysis is that, for any nonnegative and convex function, the associated Orlicz norm is a Liapunov functional. Existence and asymptotic behaviour then rely on weak and strong compactness methods in L¹ in the spirit of the DiPerna-Lions theory for the Boltzmann equation.     

Non-Markovian Diffusion Over a Saddle with a Generalized Langevin Equation

The diffusion over a simple parabolic barrier is exactly solved with a non-Markovian Generalized Langevin Equation. For a short relaxation time, the problem is shown to be similar to a Markovian one, with a smaller effective friction. But for longer relaxation time, the average trajectory starts to oscillate and the system can have a very fast first passage over the barrier. For very long relaxation times, the solution tends to a zero-friction limit. PACS: 02.50.EY, 05.40.−a, 25.70.Jj     

On the Generalized Langevin Equation for a Rouse Bead in a Nonequilibrium Bath

We present the reduced dynamics of a bead in a Rouse chain which is submerged in a bath containing a driving agent that renders it out-of-equilibrium. We first review the generalized Langevin equation of the middle bead in an equilibrated bath. Thereafter, we introduce two driving forces. Firstly, we add a constant force that is applied to the first bead of the chain. We investigate how the generalized Langevin equation changes due to this perturbation for which the system evolves towards a steady state after some time. Secondly, we consider the case of stochastic active forces which will drive the system to a nonequilibrium state. Including these active forces results in an extra contribution to the second fluctuation–dissipation relation. The form of this active contribution is analysed for the specific case of Gaussian, exponentially correlated active forces. We also discuss the resulting rich dynamics of the middle bead in which various regimes of normal diffusion, subdiffusion and superdiffusion can be present.     

Langevin Equation Involving Three Fractional Orders

In this paper, the existence and uniqueness of initial value problems for nonlinear Langevin equation involving three fractional orders are discussed. We use a new norm that is convenient for the fractional and singular differential equations. This norm and the contraction mapping principles are the main tools for investigating the existence and uniqueness of the desired issue. The fractional derivatives are described in Caputo sense.

     

Injected Power Fluctuations in Langevin Equation

In this paper, we consider the Langevin equation from an unusual point of view, that is as an archetype for a dissipative system driven out of equilibrium by an external excitation. Using path integral method, we compute exactly the probability density function of the power (averaged over a time interval of length ) injected (and dissipated) by the random force into a Brownian particle driven by a Langevin equation. The resulting distribution, as well as the associated large deviation function, display strong asymmetry, whose origin is explained. Connections with the so-called Fluctuation Theorem are thereafter discussed. Finally, considering Langevin equations with a pinning potential, we show that the large deviation function associated with the injected power is completely insensitive to the presence of a potential.     

量子噪音与量子Langevin方程

介绍了量子噪音和量子Langevin方程 ,并与经典噪音和经典Langevin方程进行了比较 .量子噪音来源于两种途径 ,第一种与经典噪音相似 ,第二种则起源于Heisengberg测不准原理 .同时 ,也简略给出了量子Langevin方程的推导.The properties of quantum noise and Langevin equation are discussed. Comparisons between the quantum noise and Langevin eqution and the classic one are presented. A brief derivation for quantum Langevin equation is showed. The quantum noise comes from two ways, namely, the way as same as that of classic noise and the Heisenberg uncertainty.     

Moderate Deviations for the Langevin Equation with Strong Damping

In this paper, we establish a moderate deviation principle for the Langevin dynamics with strong damping. The weak convergence approach plays an important role in the proof.     

Ergodic Properties of the Non-Markovian Langevin Equation

We discuss the dissipative dynamics of a classical particle coupled to an infinitely extended heat reservoir. We announce a number of results concerning the ergodic properties of this model. The novelty of our approach is that it extends beyond Markovian dynamics to the case where the Langevin equation is driven by colored noise. Our method works in arbitrary space dimension, and for fully nonlinear systems.     

Semi-Integral Scheme for Simulation of Langevin Equation with Weak Inertia

A stable and accurate algorithm for simulating massive damped Brownian motion is proposed and discussed. The algorithm, being fully integral for the friction and noise terms and predictor-corrector for the potential force in the Langevin equations, is stable upon changing time step and for various masses of the particle. In particular, the limit of zero inertia can be safely taken, and the algorithm yields naturally the corresponding overdamped case. The steady velocity of a particle moving in a titled periodic potential is calculated and three algorithms are compared.     

The Nonlinear Schrodinger Equation for a Many-Dimensional System in an Oscillator Field

The method of semi-classical asymptotics is applied to a many-dimensional nonlinear Schrodinger equation with an external anisotropic oscillator field. Solutions asymptotic in a small parameter ħ, ħ → 0, are constructed in the class of functions localized in the neighborhood of an unclosed parabolic surface associated with the phase curve that describes the evolution of the vertex of this surface. In the normal direction to the surface, the functions have the form of the one-soliton solution of a one-dimensional nonlinear Schrodinger equation. The asymptotic evolution operator is considered accurate to O(ħ^3/2), ħ → 0, in the specified class of solutions. The phenomenon of collapse is discussed. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 70–75, July, 2005.     

蛋白质构型变化的二维广义郎之万方程描述

采用二维广义郎之万方程描述蛋白质构型的随机变化,并与电子输运过程的原子密度填充模型相协调. 假设通过键和通过空间的两类电子输运路径分别受到高斯分数噪声和高斯白噪声的影响. 推导了电子转移给体-受体距离涨落和荧光寿命涨落自相关函数的一般表达式. 采用数值拉普拉斯反变换计算了蛋白质构型涨落动力学,并详尽讨论了长时间和短时间行为的渐近解析. 最后,明确了基于二维广义郎之万方程的蛋白质构型描述与一维描述之间的关系.     

Numerical Integration of a Non-Markovian Langevin Equation with a Thermal Band-Passing Noise

An accurate and fast approach for numerically solving a non-Markovian Langevin equation with a thermal band-passing noise is proposed. The algorithm combines the closed integration for both damping and noise terms with the Runge–Kutta method for nonlinear force in the Markovian Langevin equation transferred from the original equation. The present algorithm is tested through simulating diffusion of a free particle by using different initial distributions, and then a strong superdiffusion is shown. The mean velocity of a particle in a flashing ratchet driven by the band-passing colored noise is calculated numerically. The dependence of the resulting mean velocity on temperature, asymmetry of the ratchet potential, and inertia of the particle is discussed, and some novel behaviors in comparison with the usual model are observed.     

Similarity Reductions for a Nonlinear Diffusion Equation

Abstract

Similarity reductions and new exact solutions are obtained for a nonlinear diffusion equation. These are obtained by using the classical symmetry group and reducing the partial differential equation to various ordinary differential equations. For the equations so obtained, first integrals are deduced which consequently give rise to explicit solutions. Potential symmetries, which are realized as local symmetries of a related auxiliary system, are obtained. For some special nonlinearities new symmetry reductions and exact solutions are derived by using the nonclassical method.     

The Functional Callan-Symanzik Equation for the Coulomb Gas

A non-perturbative scheme, based on the functional generalization of the Callan-Symanzik equation is developed to treat the Coulomb interaction in an electron gas. The one-particle irreducible vertex functions are shown to satisfy an evolution equation whose initial condition is given by means of the classical action and the final point corresponds to the physical system. This equation is truncated by expanding it in momenta and excitation energies, leaving the electric charge as an arbitrary, not necessarily small parameter. Exact coupled partial differential equations up to first order in the frequencies and excitation energies are derived. The numerical integration of these equations is left to a later stage. Nevertheless, in order to demonstrate the relation with the perturbation expansion the one-loop Lindhard function and screening are reproduced in the independent mode approximation of the evolution equation.     

Composite Lane-Emden Equation as a Nonlinear Poisson Equation

After a quick review of the Lane-Emden equation and its properties, a composite of two different polytropes is introduced and some of the consequences are explored. The results are used to build a nonlinear electromagnetism with non-singular, solitonic solutions as charged particles.