Nonequilibrium dynamics of scalar fields in a thermal bath

We study the approach to equilibrium for a scalar field which is coupled to a large thermal bath. Our analysis of the initial value problem is based on Kadanoff-Baym equations which are shown to be equivalent to a stochastic Langevin equation. The interaction with the thermal bath generates a temperature-dependent spectral density, either through decay and inverse decay processes or via Landau damping. In equilibrium, energy density and pressure are determined by the Bose-Einstein distribution function evaluated at a complex quasi-particle pole. The time evolution of the statistical propagator is compared with solutions of the Boltzmann equations for particles as well as quasi-particles. The dependence on initial conditions and the range of validity of the Boltzmann approximation are determined.     

升力系数不变时跨音速机翼阻力优化设计

基于共轭方程的优化设计理论,应用三维欧拉方程进行了升力系数不变时跨音速机翼阻力优化设计研究,根据给定的目标函数推导了在物理空间上表述的共轭方程及边界条件,研究了共轭方程的数值求解方法及目标函数对设计变量的敏感性导数求解问题,发展了一种跨音速机翼阻力优化设计方法,应用该设计方法进行了跨音速机翼阻力优化设计研究,优化后机翼表面的激波强度减弱很多,有效减少了波阻.     

Numerical solution of inverse scattering for near-field optics

A novel regularized recursive linearization method is developed for a two-dimensional inverse medium scattering problem that arises in near-field optics, which reconstructs the scatterer of an inhomogeneous medium located on a substrate from data accessible through photon scanning tunneling microscopy experiments. Based on multiple frequency scattering data, the method starts from the Born approximation corresponding to weak scattering at a low frequency, and each update is obtained by continuation on the wavenumber from solutions of one forward problem and one adjoint problem of the Helmholtz equation.     

Exact c-number representation of non-Markovian quantum dissipation

The reduced dynamics of a quantum system interacting with a linear heat bath finds an exact representation in terms of a stochastic Schr?dinger equation. All memory effects of the reservoir are transformed into noise correlations and mean-field friction. The classical limit of the resulting stochastic dynamics is shown to be a generalized Langevin equation, and conventional quantum state diffusion is recovered in the Born-Markov approximation. The non-Markovian exact dynamics, valid at arbitrary temperature and damping strength, is exemplified by an application to the dissipative two-state system.     

Theory of open systems I

A new method of treating open systems is presented. The normal treatment using the generalized master equation with the projection of Argyres and Kelley is meaningful only if the state of the reservoir never deviates appreciably from the reference state which appears in the projection. Otherwise, one must make at least a partial resummation of the perturbative expansion of the kernel of the generalized master equation. The present method avoids the introduction of a projection operator and allows us to overcome such resummation difficulties. It is based on an integrodifferential equation for the statistical operator of the composite system, which naturally provides a hierarchy of equations involving the statistical operator ?(t) of the open system and suitable quantities describing higher and higher order bath-system correlations. Treating the deviations of the bath from its initial equilibrium or stationary state as expansion parameters, one gets an approximation scheme, each step of which gives a closed system of equations for ?(t) and a suitable set of correlation quantities.Eliminating such quantities one obtains a closed linear integrodifferential equation for ?(t). The zeroth approximation in the deviations coincides with the Born approximation of the generalized master equation which uses the projection of Argyres and Kelley.On the other hand, even the first approximation is equivalent to the resummation of infinite contribution of the Born series of such a generalised master equation. When it is suitable, the concentration of the bath can also be used as an expansion parameter to handle the hierarchy.     

Some basic remarks on eigenmode expansions of time-delay dynamics

We describe in elementary terms how eigenmode expansions can be used to deal with differential-difference equations. As particular applications we present the full analytical solution of linear stochastic time-delay systems and the weakly nonlinear analysis of nonlinear differential-difference equations in the limit of large time delay. Our exposition is essentially based on an explicit analytical expression for the linear spectrum in terms of the Lambert W-function and on the explicit formula for the eigenfunctions of the adjoint equation.     

Deformed quantum harmonic oscillator with diffusion and dissipation

A master equation for the deformed quantum harmonic oscillator interacting with a dissipative environment, in particular with a thermal bath, is derived in the microscopic model by using perturbation theory. The coefficients of the master equation and of equations of motion for observables depend on the deformation function. The steady-state solution of the equation for the density matrix in the number representation is obtained and the equilibrium energy of the deformed harmonic oscillator is calculated in the approximation of small deformation.     

Stochastic diffusion in a periodic potential

This paper deals with two equations for classical stochastic diffusion in a potential. First, the full Fokker-Planck equation in phase-space for a Brownian particle in a periodic potential and linearly coupled to an external field is considered. The solution discussed previously by the author and co-worker is improved upon. An alternative approximation is introduced. Then, the simpler Smoluchowski equation, which is derivable from the Fokker-Planck equation under suitable conditions, is solved using Hill's determinant method. Finally a WKB-type method is proposed to solve the Smoluchowski equation for a general class of potentials.     

双光子光学双稳态的研究

用量子的方法,退关联近似,获得了三能级简并双光子双稳态方程。所得结果与实验相符合。指出用双光子矢量模型方法处理的缺点以及它与实验不符的原因。 关键词：     

Non-linear models of geophysical hydrodynamics and the problem of forecasting stream patterns I

The dynamic stochastic approach to the study of mathematical models of thermohydrodynamic large-scale fields is developed in which the mathematical image of stochasticity is the strange attractor of the real atmosphere. This approach is based on methods of analysing non-linear equations of atmosphere. The approximation of these equations is mostly effected using Galerkin's procedures. This reduction is based on the theorem of invariant manifold, especially on the theorem of central manifold for semi-flow of the Navier-Stokes equation. According to the theorem of the central manifold, all significant phenomena related to dynamic systems of thermohydrodynamic equations of atmosphere occur in a particular finite-dimensional formulation. General circulation could be modelled with the aid of relatively single dynamic systems. We shall continue deal with the finite-dimensional approximations of the dissipative, non-divergent, barotropic flow and the bifurcation analysis of the spectral models with a small number of spectral modes and an external force acting only on fundamental modes.     

Low-pressure RF discharge in the free-flight regime

The self-consistent equations system for low-pressure RF discharge in the free-flight regime is formulated. The expressions for the electron energy diffusion coefficient due to electron-neutral collisions and to the electron collisions with the plasma-space charge moving boundary (stochastic heating) are derived. If the electron-neutral elastic collisions frequency exceeds the inelastic one, the conventional two-term approximation for the electron distribution function (EDF) can be generalized, and the space-time-averaged electron kinetic equation can be reduced to the one-dimensional energy diffusion one. The fast electrons attached to the electrode surface can also be accounted for in this equation. It is shown that in the cases of (a) spatially uniform ion profile, (b) for frequencies that are small compared with the electron bounce frequency, and (c) for frequencies exceeding the electron plasma one in the sheath, the stochastic heating vanishes     

On a transformed stochastic equation

A linear stochastic equation is considered. As a result of the transformation used in the theory of integral equations for improving the convergence of successive approximations, transformed stochastic equations are obtained. The latter are exact and are equivalent to the original equation. By solving the transformed stochastic equations by the method of small perturbations the conditions are derived for the applicability of the approximate Keller equations for a value of the field averaged over the ensemble, which satisfies the original stochastic equation. As an application, the applicability boundaries of the Dyson equations are estimated in the Foldy and Burre approximation. In the first case it is assumed that the medium consists of Rayleigh scatters, while in the second case it is assumed that the fluctuations of the permeability of the medium are small-scale ones. If the medium is bounded and has the form of a sphere, the applicability condition of the Dyson equations impose an upper constraint on the radius of the sphere which nevertheless may take values that exceed the extinction length.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 15, No. 1, pp. 66–72, January, 1972.     

非单色激光场与同位素原子体系相互作用

本文研究了多模激光场与同位素原子体系相互作用动力学问题。采用混沌模型描述多模激光场，用Ｆｏｋｋｅｒ－Ｐｌａｎｃｋ方程方法，导出了有限带宽混沌场与同位素原子体系相互作用动力学方程。分析了激光线宽对同位素原子激发电离效率和选择性因子的影响。     

The Systematic Approximation and the Systematic Adiabatic Approximation for the Stochastic Systems Driven by O-U Noises

In this paper, we generalize the systematic approximation and the systematic adiabatic approximation developed by G. Schöner and H.aken^[1,2] to the stochastic systems driven by O-U noises. After eliminating the fast variables st, we introduce a new family of stochastic processes K_t^(υ) (υ≥2) in the equations for the order parameters vt. A typical model serves to illustrate how to derive the equation for the order parameter and how to calculate the statistical properties of the new stochastic processes K_t^(υ) (υ≥2). Some significant discussions and conclusions are given in Sec. V.     

Problems for the elimination of atomic variables and the single-mode dye laser equations

The stochastic generalization of the adiabatic approximation has been developed earlier by Wunderlin and Haken [6] and recently by Schöner and Haken [7] and the authors [8]. Using these theories, a complex field amplitude equation for the single-mode dye laser is derived based on a set of semiclassical single-mode laser equations. An effective Fokker-Planck equation in which the laser intensity is decoupled from the phase variable is obtained. The relations between our theory and the phenomenological single-mode dye laser equation are discussed.     

Modulation Equations: Stochastic Bifurcation in Large Domains

We consider the stochastic Swift-Hohenberg equation on a large domain near its change of stability. We show that, under the appropriate scaling, its solutions can be approximated by a periodic wave, which is modulated by the solutions to a stochastic Ginzburg-Landau equation. We then proceed to show that this approximation also extends to the invariant measures of these equations.     

The Stochastic Adiabatic Approximation of Multidimensional Non-Markovian Systems Driven by the Ornstein-Uhlenbeck Noises

A set of nonlinear stochastic differential equations (NSDE'S) that describes a large class of nonlinear multidimensional non-Markovian dynamical systems driven by the Ornstein- Uhlenbeck(O-U) noises is studied. By virtue of the stochastic generalization of usual adiabatic approximation, we obtain the equation for the order parameter. The statistical properties of the new stochastic variables occurred are studied. The effective Fokker-Planck equation (EFPE) corresponding to the equation for the order parameter is derived and the stationary solution of EFPE is calculated.     

Convergence from Boltzmann to Landau Processes with Soft Potential and Particle Approximations

Our aim in this paper is to show how a probabilistic interpretation of the Boltzmann and Landau equations gives a microscopic understanding of these equations. We firstly associate stochastic jump processes with the Boltzmann equations we consider. Then we renormalize these equations following asymptotics which make prevail the grazing collisions, and prove the convergence of the associated Boltzmann jump processes to a diffusion process related to the Landau equation. The convergence is pathwise and also implies a convergence at the level of the partial differential equations. The best feature of this approach is the microscopic understanding of the transition between the Boltzmann and the Landau equations, by an accumulation of very small jumps. We deduce from this interpretation an approximation result for a solution of the Landau equation via colliding stochastic particle systems. This result leads to a Monte-Carlo algorithm for the simulation of solutions by a conservative particle method which enables to observe the transition from Boltzmann to Landau equations. Numerical results are given.