A direct method for numerical analysis of relaxation of the statistical characteristics of brownian motion

We propose a numerical method for analyzing the relaxation of coordinate moments of the Brownian motion of a system described by a stochastic Liouville equation of the 1st or 2nd order with moderate-order polynomial nonlinearity. Using exact or approximate recurrence relations for the stationary values, at a certain step, we break the chain of equations for the moments of the Brownian motion. The evolution of the model probability distribution of coordinates is found from the numerical solution of the differential equations of relaxation of moments. This method is used for analyzing the nonstationary probability characteristics of a system with nonlinear rigidity described by a third-degree polynomial. The relaxation of moments and of the model probability distribution is plotted and tabulated. The results obtained allow us to draw certain conclusions on the statistical dynamics of the Brownian motion of the systems studied. Nizhny Novgorod Architecture and Civil Engineering University, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 42, No. 9, pp. 922–930, September 1999     

Spatial moments for the nonstationary one-velocity problem of transport theory with isotropic scattering. I. Point instantaneous source

Equations for the moments of spatial particle distribution in a homogeneous medium with isotropic scattering are derived for the nonstationary one-velocity problem of transport theory. Exact analytical and numerical solutions are found for five even moments (from the second to the tenth one) by the Laplace transform method, and an algorithm for calculating the moments of arbitrary order is described. Convergence to the corresponding moments is investigated in the diffusion approximation for t→∞, and its nonuniform character is established: higher moments differ significantly from the corresponding diffusion moments at any t. The physical causes for such behavior of the moments are discussed. Ul'yanovsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 88–94, February, 2000.     

The theory of nonstationary thermophoresis of a solid spherical particle

The theory of nonstationary thermophoresis of a solid spherical particle in a viscous gaseous medium is presented. The theory is constructed on the solutions of fluid-dynamics and thermal problems, each of which is split into stationary and strictly nonstationary parts. The solution of the stationary parts of the problems gives the final formula for determining the stationary component of the thermophoretic velocity of this particle. To determine the nonstationary component of the thermophoretic velocity of the particle, the corresponding formula in the space of Laplace transforms is derived. The limiting value theorems from operational calculus are used for obtaining the dependence of the nonstationary component of the thermophoretic velocity of the spherical particle on the strictly nonstationary temperature gradient for large and small values of time. The factors determining the thermophoretic velocity of the particle under investigation are determined.     

Inequalities generalizing the second law of thermodynamics for transitions between nonstationary states

We discuss the consequences of a variant of the Hatano-Sasa relation in which a nonstationary distribution is used in place of the usual stationary one. We first show that this nonstationary distribution is related to a difference of traffic between the direct and dual dynamics. With this formalism, we extend the definition of the adiabatic and nonadiabatic entropies introduced by M. Esposito and C. Van den Broeck in Phys. Rev. Lett. 104, 090601 (2010) for the stationary case. We also obtain interesting second-law-like inequalities for transitions between nonstationary states.     

Verallgemeinerung der Methode der singulären Normallösungen für isotherme irreversible Vlasov-Systeme

Irreversible Vlasov systems, i.e. systems governed by a Vlasov-type kinetic equation including entropy-producing collision terms, are treated by the techniques of singular normal modes and singular integral equations using a new indirect method which renders possible a straightforward generalization of the Case formalism as developed originally for collision-free Vlasov plasmas. This method is in contrast to a more complex method given by the present authors for the first application of the singular normal mode expansion to irreversible Vlasov systems (1970). The linearized Vlasov operator supplemented by complete Bhatnagar-Gross-Krook collision integrals as the most important model collision terms is analyzed in detail for a nonrelativistic, nondegenerate, stationary electron gas with neutralizing positive ions and neutral particles without a magnetic field at constant temperature, generalizations for more complex irreversible Vlasov systems being possible. The key of the indirect method given is the introduction of a transformed electron distribution function containing as an additive term an integral over the usual distribution function.     

Exact Green's function of the equations of quantum mechanics in an electromagnetic field. I

An exact Green's function of the Cauchy problem in arbitrary (nonparallel) stationary homogeneous electrical and magnetic fields is constructed for the Klein-Gordon-Fock and Dirac equations and for the Pauli equation in arbitrary nonstationary fields, on the basis of a single approach using the method of the canonical Maslov operator (extension of the WKB method to the multidimensional case) and a Fock idea about the proper time in relativitic mechanics.     

Spatial distribution of ions incident on solid target as a function of instantaneous energy

Integro-differential equations are derived which govern the spatial distribution of ions of energy E' moving in a solid target after the ions were incident on the target surface with energy E. These equations are specialized to integral equations which govern the first and second spatial moments of the distribution. Solutions to the moments equations are obtained to second order for an expansion of the integrands in powers of T, the energy transfer in a single collision between an ion and an atom (or electron) of the target. Similar equations are also derived for the average range along path of the ions at energy E'.     

圆柱形阳极层霍尔等离子体加速器的质点网格方法模拟

对圆柱形阳极层霍尔加速器内的放电等离子体运用二维质点网格方法（particle in cell）进行数值模拟,用蒙特卡罗碰撞方法处理带电粒子与中性粒子之间的碰撞. 得到了放电通道内离子与电子的分布以及离子流的运动,并且对出口外侧的能量分布进行了统计. 结果发现圆柱形阳极层等离子体加速器的磁场对电子有明显的约束作用,电子集中于阳极附近很小的区域内. 由于电磁场的特殊分布,离子流呈现出双峰式的分布. 离子能量范围从放电电压的20%到接近放电电压,平均能量在放电电压的40%—50%之间. 关键词： 质点网格方法 蒙特卡罗碰撞 数值模拟 阳极层霍尔等离子体加速器     

碰撞对两种正离子的电负性磁鞘影响的数值研究

建立包括两种正离子的电负性磁鞘的流体模型,利用四阶龙格库塔法数值求解描述一维稳态等离子体鞘层的方程组,考察离子与中性粒子碰撞对一维稳态等离子体鞘层的影响.结果表明:鞘边Ar⁺与He⁺的含量比值与碰撞参数对离子马赫数的取值范围都有影响.鞘边负离子含量越少,碰撞对鞘层中带电粒子密度的影响越明显.并且随碰撞参数的加大,鞘层中电子、负离子的密度下降越快,两种正离子的密度则呈现不同的波动变化.鞘边负离子含量越多,碰撞对鞘层中两种正离子的速度影响就越明显.此外,碰撞参数越大对鞘边δ越大的鞘层中的带电粒子密度影响越大.     

Analysis of oscillatory processes in lasers using the method of generalized eigenfunctions

The method of generalized eigenfunctions, which is used in the theory of diffraction, is applied to analyze stationary and narrow-band nonstationary processes in lasers. Using this method, one can avoid difficulties associated with integration of the eigenfunctions of an emitting system over the continuous spectrum, difficulties typical of the conventional frequency method. The method employs expansion in modes that are orthogonal inside the lasing medium. The problem of exponential growth of modes at infinity is eliminated. In addition, the field distribution inside the lasing medium is better described using the generalized eigenfunctions in a number of important cases.     

Solitons in collisionless cold plasma

The relativistic nonlinear self-consistent equations for a collisionless plasma with stationary ions are transformed into a form appropriate for finding exact analytic solutions. It is shown that for an axial system with planar geometry, the two-dimensional stationary equations for this system can be reduced to the sh-Gordon equation. The exact solution of this equation describing the charge-density equilibrium configuration is obtained, the solution having sharp transverse boundaries and a soliton form in longitudinal direction. The generalization to the nonstationary case is considered in an perturbative approach.     

Analytic-Numerical Construction of Nonstationary Probability Distributions for one Class of Nonlinear Stochastic Systems

We propose a method for constructing nonstationary model probability distributions for nonlinear dynamic systems related to the Verhulst stochastic equation. The proposed procedure is based on the numerical solution of relaxation differential equations for the mean and the variance. The set of the moment equations is closed and the probability density is constructed on the basis of rigorous analytical relations for the stationary probability characteristics. As a result, these distributions have correct stationary asymptotics. We show the possibility of numerical control of the accuracy of the proposed procedure. We consider the examples of relaxation of the probability characteristics of the amplitude of a self-oscillator and a parametric oscillator with a noise pump. The evolution of the amplitude probability distribution is found.     

中高能重离子碰撞中的阵发混沌

在量子分子动力学模型计算多重碎裂基础上,应用阶乘矩方法分析了多重碎裂.对¹⁹⁷An(200MeV/u)+¹⁹⁷An碰撞系统进行了计算.发现多重分布有阵发混沌存在.并对临界现象做了初步讨论.     

Kinetics of ions in a neutral gas upon abrupt application of an electric field. II: Various models of interaction

A new method for calculating the matrix elements of the collision integral in the Boltzmann equation is used for studying the behavior of an ion impurity upon an abrupt application of an electric field. For five models of interaction, the behavior of mobility is analyzed and the evolution of the distribution function for various values of the electric field is illustrated. The initial stage of runaway of ions is studied in the case of the Coulomb interaction. Two methods for improving the convergence of the polynomial expansions of the distribution function are considered.     

Nonstationary and Quasi-stationary Treatment of Distribution Anisotropy in the Temporal Relaxation of an Energetic Electron Group

A relaxation study of an electron group in collision dominated weakly ionized plasmas has been performed. The study is based on the two-term approximation of the Legendre polynomial expansion of the electron velocity distribution in the nonstationary Boltzmann equation. To overcome the limitation of the conventional quasi-stationary treatment of the distribution anisotropy, a very efficient solution approach of the nonstationary kinetic equation in two-term approximation has been developed which allows for a strict nonstationary treatment of the distribution anisotropy. By using this approach the temporal evolution of the isotropic and anisotropic distribution of the electrons has been investigated for a model plasma, which involves typical features of an inert gas plasma. A comparison of the results with corresponding ones obtained by applying the conventional approach under various parameter conditions clearly indicates a pronounced falsification of the real relaxation course by the latter approach.     

Bipolar harmonics method in the semiclassical theory of sub-doppler cooling

The bipolar harmonics method is extended to the case of complex elliptic polarization vectors. The method is used to study, on the basis of the semiclassical theory, the multipole moments of the ground state of atoms under conditions of sub-Doppler cooling with a monochromatic light field possessing spatial gradients of the polarization. It is shown that for stationary atoms with an initial isotropic distribution over sublevels the multipole moments of rank κ decompose, in accordance with the parity κ of the rank, according to one of two minimal sets of bipolar harmonics with different symmetry under inversion. An expansion of the corrections, which are linear in the velocity, to the multipole moments with respect to the indicated minimal sets of bipolar harmonics is studied for a stationary state, and the expansion coefficients are analyzed. The orientation vector J of the atomic ensemble is studied on the basis of the proposed method for the dipole transition 1/2 → 1/2, and the light-induced forces for a specific 2D configuration of the light field, including radiation friction forces and Lorentz-type forces, are analyzed.     

To the theory of the generalized collision integral. II

The method of nonstationary diagram summation, previously developed [1], is applied to constructing the collision integral for a classical plasma, in which ring diagrams with prior ladder renormalization are included, which is equivalent to including screening of a long-range potential and finite momentum transfer in binary collisions. For a spatially homogeneous plasma it is shown that in the collision integral obtained the collision cross section does not contain the characteristic Coulomb divergences.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, Vol. 16, No. 9, pp. 119–122, September, 1973.     

On the distribution of potential wells and its relation to collision-induced dipole moments in nonpolar solvents. - Part I.

The relaxation times of some rigid polar molecules in three non-polar solvents of different molecular geometries (and different collision induced dipole moments) are studied. There is some relationship to be observed between the arbritary distribution of surrounding potential wells and the dipole moments measured in the various solvents.     

Model-based classification of nonstationary vocal fold vibrations

Classification of vocal fold vibrations is an essential task of the objective assessment of voice disorders. For historical reasons, the conventional clinical examination of vocal fold vibrations is done during stationary, sustained phonation. However, the conclusions drawn from a stationary phonation are restricted to the observed steady-state vocal fold vibrations and cannot be generalized to voice mechanisms during running speech. This study addresses the approach of classifying real-time recordings of vocal fold oscillations during a nonstationary phonation paradigm in the form of a pitch raise. The classification is based on asymmetry measures derived from a time-dependent biomechanical two-mass model of the vocal folds which is adapted to observed vocal fold motion curves with an optimization procedure. After verification of the algorithm performance the method was applied to clinical problems. Recordings of ten subjects with normal voice and ten dysphonic subjects have been evaluated during stationary as well as nonstationary phonation. In the case of nonstationary phonation the model-based classification into "normal" and "dysphonic" succeeds in all cases, while it fails in the case of sustained phonation. The nonstationary vocal fold vibrations contain additional information about vocal fold irregularities, which are needed for an objective interpretation and classification of voice disorders.     

Electron Kinetics with Attachment and Ionization from Higher Order Solutions of Boltzmann's Equation

An appropriate approach is presented for solving the Boltzmann equation for electron swarms and nonstationary weakly ionized plasmas in the hydrodynamic stage, including ionization and attachment processes. Using a Legendre-polynomial expansion of the electron velocity distribution function the resulting eigenvalue problem has been solved at any even truncation-order. The technique has been used to study velocity distribution, mean collision frequencies, energy transfer rates, nonstationary behaviour and power balance in hydrodynamic stage, of electrons in a model plasma and a plasma of pure SF₆. The calculations have been performed for increasing approximation-orders, up to the converged solution of the problem. In particular, the transition from dominant attachment to prevailing ionization when increasing the field strength has been studied. Finally the establishment of the hydrodynamic stage for a selected case in the model plasma has been investigated by solving the nonstationary, spatially homogeneous Boltzmann equation in twoterm approximation.