Microdynamics and nonlinear stochastic processes of gross variables

Starting from classical Hamiltonian mechanics, we derive for the dynamics of gross variables in nonequilibrium systems exact nonlinear generalized Fokker-Planck and Langevin equations in which the effect of the initial preparation is taken into account explicitly. This latter concept allows for the construction of a uniquely determined projection operator. The memory functions occurring in the Langevin equations are related to the random forces by a fluctuation-dissipation theorem of the second kind. We discuss the connection with the generalized Fokker-Planck equation. The known results for equilibrium fluctuations are recovered as a special case.Supported in part by the National Science Foundation, Grant CHE78-21460.     

Augmented Langevin approach to fluctuations in nonlinear irreversible processes

A Fokker-Planck equation derived from statistical mechanics by M. S. Green [J. Chem. Phys. 20:1281 (1952)] has been used by Grabertet al. [Phys. Rev. A 21:2136 (1980)] to study fluctuations in nonlinear irreversible processes. These authors remarked that a phenomenological Langevin approach would not have given the correct reversible part of the Fokker-Planck drift flux, from which they concluded that the Langevin approach is untrustworthy for systems with partly reversible fluxes. Here it is shown that a simple modification of the Langevin approach leads to precisely the same covariant Fokker-Planck equation as that of Grabertet al., including the reversible drift terms. The modification consists of augmenting the usual nonlinear Langevin equation by adding to the deterministic flow a correction term which vanishes in the limit of zero fluctuations, and which is self-consistently determined from the assumed form of the equilibrium distribution by imposing the usual potential conditions. This development provides a simple phenomenological route to the Fokker-Planck equation of Green, which has previously appeared to require a more microscopic treatment. It also extends the applicability of the Langevin approach to fluctuations in a wider class of nonlinear systems.     

A unified approach for deriving kinetic equations in nonequilibrium statistical mechanics. I. Exact results

A unified method for deriving exact kinetic equations for dynamical quantities of a many-body system is presented. The well-known results of Mori and Zwanzig are recovered as special cases. Furthermore, it is shown that they differ only by the way in which the system is prepared at the initial time. Connections between this method and others recently developed are also discussed.     

Dissipation and large thermodynamic fluctuations

The results of recent work of Kipnis, Olla, and Varadhan on the dynamic large deviations from a hydrodynamic limit for some interacting particle models are formally extended to a general hydrodynamic situation, including non-equilibrium steady states, as a fluctuation-dissipation hypothesis. The basic conjecture is that the exponent of decay in the probability of a large thermodynamic fluctuation is given by the dissipation of the force required to produce the fluctuation. It is shown that this hypothesis leads to a nonlinear version of Onsager-Machlup fluctuation theory that had previously been proposed by Graham. A direct consequence of the theory is a dynamic variational principle for the most probable thermodynamic history subject to imposed constraints (Onsager's principle of least dissipation). Following Graham, the theory leads also to a generalized potential, analogous to an equilibrium free energy, for the nonequilibrium steady state and an associated static variational principle. Finally, a formulation of nonlinear fluctuating hydrodynamics is proposed in which the noise enters multiplicatively so as to reproduce the conjectured large-deviations theory on a formal analogy with the results of Freidlin and Wentzell for finite-dimensional systems.     

Nonlinear Tripartite Entangled State Representation and Related Generalized Wigner Function

We construct the nonlinear tripartite entangled state representation and the related generalized Wigner operator. Then we discussed the Wigner functions of the nonlinear tripartite entangled state and the three-mode nonlinear squeezed vacuum state, and obtained the classical Weyl corresponding function of the three-mode nonlinear squeezed state.     

Nonlinear Tripartite Entangled State Representation and Related Generalized Wigner Function

We construct the nonlinear tripartite entangled state representation and the related generalized Wigner operator. Then we discussed the Wigner functions of the nonlinear tripartite entangled state and the three-mode nonlinear squeezed vacuum state, and obtained the classical Weyl corresponding function of the three-mode nonlinear squeezed state.     

时间非对称外力驱动分数阶布朗马达的定向输运

本文研究了周期对称势中时间非对称外力驱动的布朗粒子输运现象, 建立了分数阶布朗马达输运模型. 其中外力是零均值的, 而分数阶阶数则刻画了客观环境的非均匀性程度. 通过将模型离散化进行数值模拟, 讨论了分数阶阶数、系统参量和外部参量与定向流之间的依赖关系. 研究表明, 即使没有倾斜势场的作用, 时间非对称外力也可以诱导系统产生定向输运; 输运速度随分数阶阶数的增大而单调递增; 当阶数固定时, 系统的输运速度会随着势垒高度、噪声强度非单调变化, 表现出广义随机共振现象. 分析指出, 分数阶郎之万方程所刻画的输运现象是在整数阶模型基础上的一个推广, 进而为输运现象提供了一个可能更为真实的模型.     

A gas of Brownian particles in statistical dynamics

We consider a large number of particles on a one-dimensional latticel Z in interaction with a heat particle; the latter is located on the bond linking the position of the particle to the point to which it jumps. The energy of a single particle is given by a potentialV(x), x∈Z. In the continuum limit, the classical version leads to Brownian motion with drift. A quantum version leads to a local drift velocity which is independent of the applied force. Both these models obey Einstein's relation between drift, diffusion, and applied force. The system obeys the first and second laws of thermodynamics, with the time evolution given by a pair of coupled non linear heat equations, one for the density of the Brownian particles and one for the heat occupation number; the equation for a tagged Brownian particle can be written as a stochastic differential equation.     

Nonlinear transport in a dilute binary mixture of mechanically different particles

The hierarchy of moments of the Boltzmann equation for a binary mixture of mechanically different Maxwell molecules is exactly solved. The solution corresponds to a nonequilibrium homogeneous steady state generated by an external force that accelerates particles of each species (or color) along opposite directions. As a consequence, macroscopic fluxes are induced in spite of the absence of concentration gradients. Explicit expressions for the fluxes of mass and momentum as functions of the field strength, the mass ratio, the molar fractions, and the interaction constant ratio are obtained. In particular, the color conductivity coefficient reduces to the mutual diffusion coefficient in the zero-field limit. Some physically interesting limiting cases are discussed. The maximum-entropy method is used to construct an approximate velocity distribution function from the exact knowledge of the mass and momentum fluxes. This distribution is exact up to second order in the color field and also in the limit of large color field.     

On the derivation of the generalized Langevin equation for interacting Brownian particles

The main result of this paper is a derivation of a generalized nonlinear Langevin equation (GLE) forn interacting particles in a bath. A consequence of the derivation is that the exact form of the (generalized) fluctuation-dissipation theorem is obtained. We discuss also the relation between the memory kernel of the GLE and some corresponding correlation functions which can be easily obtained in a molecular dynamics computer experiment. In the same spirit it is shown that the approach applies to a Brownian particle subjected to a stationary external field. The technique presented in a previous paper to simulate generalized Brownian dynamics can be easily extended to the present case. Our derivation intends to clarify the uses and (possibly) abuses of the Langevin equation in Brownian dynamics studies.     

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 transport and classical dynamics in open billiards

We report numerical results of an investigation of quantum transport for a weakly opened integrable circle and chaotic stadium billiards with a pair of conducting leads. While the statistics of spacings of resonance energies commonly follow the Wigner (GOE)-like distribution, the electric conductance as a function of the Fermi wavenumber shows characteristic noisy fluctuations associated with a typical set of classical orbits unique for both billiards. The wavenumber autocorrelation for the conductance is stronger in the stadium than the circle billiard, which we show is related to the length spectrum of classical short orbits. We propose an explanation of these contrasts in terms of the effect of phase decoherence due to the underlying chaotic dynamics.     

Exact Solutions to Nonlinear Schroedinger Equation and Higher-Order Nonlinear Schroedinger Equation

We study solutions of the nonlinear Schroedinger equation （NLSE） and higher-order nonlinear Sehroedinger equation （HONLSE） with variable coefficients. By considering all the higher-order effect of HONLSE as a new dependent variable, the NLSE and HONLSE can be changed into one equation. Using the generalized Lie group reduction method （CLGRM）, the abundant solutions of NLSE and HONLSE are obtained.     

Nonlinear conductivity and entropy in the two-body Boltzmann gas

We find exact solutions of the two-particle Boltzmann equation for hard disks and hard spheres diffusing isothermally in an external field. The corresponding transport coefficient, one-particle current divided by field strength, decreases as the field increases. This nonlinear dependence of the current on the field and the corresponding nonlinear dependence of the distribution function on the current are compared to the predictions of single-time information theory. Our exact steady-state distribution function, from Boltzmann's equation, is quite different from the approximate information-theory analog. The approximate theory badly underestimates the nonlinear decrease of entropy with current. The exact two-particle solutions we find here should prove useful in testing and improving theories of steady-state and transient distribution functions far from equilibrium.     

On the effects of noise and drift on diffusion in fluids

We discuss some aspects of the intriguing problem of interplay between molecular diffusion and the geometry of the velocity field in the diffusion of test particles. By simple arguments one can understand how the diffusion coefficient can have a large enhancement from the combined effects of the noise and the drift terms in the Langevin equation ruling the motion of test particles. The same effects give rise to the superdiffusive transport observed in media with correlated random velocity fields.     

Active transport in complex media: Relationship between persistence and superdiffusion

We study the relationship between anomalous diffusion and persistent motion of micron-sized particles moving in a viscoelastic environment and subjected to an external noise. In the framework of a generalized Langevin equation, we compare the analytical expressions of the mean square displacement and the mean cosine of the turning angle. Both magnitudes can be easily computed from the particles trajectories, and allow us to investigate the different anomalous regimes typically obtained, for instance, in single particle tracking experiments within living cells. Finally, we analyze the directional changes occurring during the motion of pigment organelles driven by molecular motors in Xenopus laevis melanocytes, as an example of application of our model.     

Nonlinear transport of Bose--Einstein condensates in a double barrier potential

The stable nonlinear transport of the Bose-Einstein condensates through a double barrier potential in a waveguide is studied. By using the direct perturbation method we have obtained a perturbed solution of Cross-Pitaevskii equation. Theoretical analysis reveals that this perturbed solution is a stable periodic solution, which shows that the transport of Bose-Einstein condensed atoms in this system is a stable nonlinear transport. The corresponding numerical results are in good agreement with the theoretical analytical results.     

A New Algebraic Method and Its Application to Nonlinear Klein-Gordon Equation

In terms of the solutions of the generalized Riccati equation, a new algebraic method, which contains the terms of radical expression of functions f（ξ）, is constructed to explore the new exact solutions for nonlinear evolution equations. Being concise and straightforward, the method is applied to nonlinear Klein-Gordon equation, and some new exact solutions of the system are obtained. The method is of important significance in exploring exact solutions for other nonlinear evolution equations.