Fokker-Planck modeling for particle slowing down and thermalization in a Maxwellian plasma

In this paper the Fokker-Planck equation, which takes into account the medium absorption and the Maxwellian behavior of the field particles at low energy for Coulomb interactions is obtained. The analytical solution of the stationary distribution function is obtained in both angle and velocity variables. In particular the electron distribution for electron-ion collisions has been obtained using this diffusion approximation for beam and isotropic sources. If the absorption is neglected the solution recovers the classical stationary Maxwell distribution. For low absorption rates the solution shows a typical slowing down spectrum for high energy and a Maxwellian-like distribution at thermal energy. For moderate and high absorption rates the test particles do not reach the thermal equilibrium and the Maxwell distribution at low energies does not appear.Received: 1 December 2003, Published online: 16 March 2004PACS: 51.10. + y Kinetic and transport theory of gases - 41.75.-i Charged-particle beams - 52.65.Ff Fokker-Planck and Vlasov equation     

On the relaxation of anisotropic plasmas

The relaxation of anisotropic plasmas due to collisions between two kinds of particles is discussed on the basis of the Fokker-Planck equation, i.e. collective interactions are not taken into account. An initially elliptic distribution is assumed to remain elliptic during the relaxation process. This is the only approximation made, i.e. the usual restriction to small anisotropy is avoided.     

Dynamical properties of colloidal systems: II. Correlation- and response functions

For systems of interacting Brownian particles a Fokker-Planck equation is derived for the probability distribution function of the concentration fluctuations, using assumption of a Gaussian static distribution function. The drift- and the diffusion term are determined by static correlation functions. By this approach specific properties of different systems as e.g., suspensions of charged spherical particles or chain polymers are taken into account. Although the diffusion term is fluctuation dependent the properties of detailed balance and both fluctuation dissipation theorems are satisfied. Using the formalism of Martin, Siggia and Rose, Dyson- and vertex-equations for the two-particle correlation functions are derived. An explicit calculation of these functions, together with related quantities as the dynamic structure factor, and of the diffusion coefficients, is given in a mean-field approximation. The results are compared with several earlier theories, which were developed for specific systems.     

Tunneling migrational polarization in dielectrics

The effect of tunneling accompanying volume-charge relaxation is analyzed. The Fokker-Planck equation, in which tunneling transitions are taken into account in the diffusion coefficient and the mobility in the quasiclassical approximation for rectangular potential barriers, is derived from the condition of transitions of the relaxation oscillators between neighboring states. The distribution of the volume charge was found by solving simultaneously the Fokker-Planck and Poisson equations by the small-parameter method with auxiliary contacts on the electrodes. The region of non-Debye dispersion was determined by taking into account the tunneling of relaxation oscillators. Formulas for calculating the complex dielectric constant were derived.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 71–75, November, 1990.     

Calculation of pressure reflection ratio

For oscillators with non-linear damping, excited by white noise, an approximation to the stationary joint density function of the displacement and velocity response is derived. This involves reducing the basic two-dimensional Fokker-Planck equation for the transition density function to a one-dimensional equation relating to the energy envelope of the response. Results obtained from this approximation are compared with other theoretical predictions, and also with digital simulation results, in typical cases.     

Transport phenomena in a partially lonized Lorentzian magnetoplasma

Some transport coefficients are evaluated for a homogeneous, partially ionized Lorentzian plasma in the presence of a uniform external magnetic field. The electronion collisions are taken into account by means of the modified Fokker-Planck equation and the electron-neutral collision frequency, ν_en is taken as velocity (ω) dependent or velocity independent, depending on the energy range under consideration. The variation of the transport coefficients with magnetic field is determined for ν_en ∝ ω^s (s being a positive or negative integer) and one finds that qualitatively this behavior does not change by changing either the collision frequency or the velocity dependence of the collision frequency; however for weak magnetic fields the magnitudes of these transport coefficients increase with the decrease in ν_en or s, whereas for strong magnetic fields the transverse components of the transport coefficients decrease and the Hall components tend to saturate with the decrease of either the collision frequency or s.     

Broad band resonant light pressure

This paper treats the mechanical pressure of resonant light acting on a two-level system, where the degenerate magnetic sublevels are taken into account. The theory is developed with arbitrary relations between the quantization axis and the propagation and polarization of the light. Rate terms are obtained both for spontaneous and induced transitions; the requirements of incoherence put restrictions on the possible geometries of the experiment. The rate equations are restricted to motion along a light beam only; this one-dimensional case is simpler to handle. For small recoil velocities a Fokker-Planck equation is derived, and an adiabatic elimination procedure enables us to derive an equation for the velocity distribution of the total population. The assumptions and approximations are formulated and discussed.     

高能离子在稠密等离子体中的传输和能量沉积

用相对论福克-普朗克方程对高能离子在稠密氘氚等离子体中的碰撞动力学进行了研究，用球谐函数来展开方程的解：格林函数，然后简明地求出了不同能量质子和α粒子在等离子体中的停止时间、减速距离、纵向弥散距离和横向偏转距离.与以前研究离子在等离子体中运动的方法相比，没有假设高能离子在等离子体中损失能量远远小于入射离子能量，求解了纵向弥散距离；并且可以求解横向偏转距离.这些计算对实验上用高能离子加热冷的稠密等离子体，然后进行科学研究具有指导作用，并且可以用来研究快点火的可能性. 关键词： 高能离子 福克-普朗克方程     

A TWO PARTICLE CLUSTER THEORY FOR UNIAXIAL NEMATIC LIQUID CRYSTALS FORMED BY BIAXIAL MOLECULES

A two-particle cluster theory is presented to study uniaxial nematic liquid crystals formed by biaxial molecules, which interact with each other via dispersion forces. The equation for the molecular orientational distribution functions in two-particle cluster approximation is obtained and is solved by an terativemethod. The temperature dependences of the order parameter and of the internal energy per particle are calculated, and the values of the relevant quantities at the nematic isotropic phase transition are given. The present theory, by taking into account the short range correlations between molecules , yields improved results compared with the mean field theory.     

Master equation approach to the second harmonic and subharmonic generation

The statistical properties of the second harmonic and subharmonic are investigated on the basis of the generalized Fokker-Planck equation making use of the coherent state technique. An iterative procedure of solving the Fokker-Planck equation is adopted making it possible to obtain some recursion equations, particularly two iterations are explicitly performed providing approximate solutions for the normal and antinormal characteristic functions. Higher-order corrections to the superposition of coherent and chaotic fields are found in the subharmonic mode or if the coupling of modes is taken into account. Earlier results for the anticorrelation effect are rederived, the reservoir effect is obtained and the related problems of the existence of the Glauber-Sudarshan quasidistribution are discussed.     

Joule heating and viscous dissipation effects on MHD flow past a semi-infinite inclined plate with variable surface temperature

A numerical investigation is performed to study the MHD free convection flow past a semi-infinite inclined plate subjected to a variable surface temperature. The Joule heating and viscous dissipation effects are taken into account in the energy equation. The governing equations of the flow are transformed into a nondimensional form using suitable dimensionless quantities. A fully developed implicit finite-difference scheme of Crank-Nicolson type is engaged to solve the dimensionless governing equations, which is more accurate, fast convergent, and unconditionally stable. The effects of the MHD, inclination angle, power law, Grashof number, Prandtl number, Joule heating, and viscous dissipation effects are studied on the velocity, temperature, shear stress, and heat transfer coefficients during transient periods. It is observed that the MHD has retarding effects on velocity.     

Influence of hadronic hard core radius on detonations and deflagrations in quark matter

The parameters for deflagration and detonations transitions between baryon-rich quark matter and hadronic matter are calculated at high densities and low temperatures. The equation of state of hadronic matter takes into account the hard core repulsion character of nuclear forces and the bag pressure is included in the quark phase. The transition is considered first directly from the quark phase to the hadronic phase. It is found that the velocity of the front separating the quark and nuclear phases as well as the energy flux are substantially larger than in the case where no hard core repulsion is taken into account. Most of the calculated quantities are only weakly dependent on the temperature in the region considered. The transitions from the quark phase to the mixed phase and from the mixed phase to the hadronic phase are also considered. No realistic solutions are found for the case where the temperature remains zero throughout the transition. At small temperatures and high densities solutions are obtained where the transition involves only minimal superheating. The energy flux for such a transition is small.     

考虑二维效应和相对论效应时低杂波电流驱动部分模拟结果

本文介绍低杂波电流驱动模拟计算编码中福克－普朗克方程的数值求解。考虑了螺距角散射造成的高垂直温度效应（二维效应）和相对论效应。同时．福克－普朗克方程中包括了描述尾部电子约束的项。数值结果表明，二维效应和相对论效应的加入均能提高电流驱动效率。驱动电流随尾部电子约束改善而增加。     

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.     

Kinetics of crystalline configurations: I. General theory; Kinetics of homogeneous short range order

A simple and exact way of coarse graining the master equation for a Markov process in configuration space is shown to exist. The coarse grained master equation is applied to the Ising model of a homogeneous binary interstitial alloy and to a “magnetic” Ising model. Using an approximation analogous to the quasi-chemical approximation, for both models the macroscopic rate equations for the establishment of short range order and the Fokker-Planck equation governing the fluctuations are derived.     

The Fokker-Planck equation for arbitrary nonlinear noise

We present the Fokker-Planck equation for arbitrary nonlinear noise terms. The white noise limit is taken as the zero correlation time limit of the Ornstein-Uhlenbeck process. The drift and diffusion coefficients of the Fokker-Planck equation are given by triple integrals of the fluctuations. We apply the Fokker-Planck equation to the active rotator model with a fluctuating potential barrier which depends nonlinearly on an additive noise. We show that the nonlinearity may be transformed into the correlation of linear noise terms.     

Charged-particle distribution in velocity,angle and time by Fokker-Planck equation

Summary The present paper describes a direct method of obtaining are expression of the distribution function of charged particles which diffuse in a plasma. The distribution function in velocity, angle and time is presented starting from the Fokker-Planck equation. Temperature, average velocity and energy of charged particles (electron or ions) have also been calculated as functions of time. The authors of this paper have agreed to not receive the proofs for correction.     

Photon statistics of nonclassical radiation in second harmonic generation

The Fokker-Planck equation approach to the second harmonic generation is adopted to obtain a set of moment equations for this process. This set of equations is truncated in consistence with a model of generalized superposition of coherent fields and quantum noise, which provides equations of motion for unknown quantities in the model. Numerical solution of these equations provides at once the photon number distribution and its factorial moments including fully quantum features of radiation such as its sub-Poisson behaviour and squeezing of vacuum fluctuations. Limits of this approximation are discussed and numerical results are provided.     

Influence of Stress Dependence of Lubricant Shear Modulus on Self-Similar Behavior of Stress Time Series

Here we consider melting of an ultrathin lubricant layer between two atomically smooth solid surfaces taking into account the stress dependence of the lubricant shear modulus and its decrease with increasing stress (strain). In the adiabatic approximation with the stress relaxation time far longer than strain and temperature relaxation times, a Langevin equation is written and its respective Fokker-Planck equation is derived using the Stratonovich calculus. Phase diagrams for the steady case are presented illustrating the effect of the system parameters on the lubricant behavior. A joint numerical and analytical analysis demonstrates a very close match between probability distributions at different parameters. It is shown that in a limited stress range, a self-similar mode of dry friction is established showing up in self-similar behavior of stress time series.     

超短脉冲激光烧蚀绝缘体材料机理的耦合理论模型

基于电子能带理论,以动力论的Fokker_Planck方程为基础,从微观层次对超短脉冲激光烧蚀绝缘体材料的机理进行分析研究.源项中分别考虑了雪崩电离、多光子电离机制,并考虑了电子能量与散射机制对电子弛豫时间的影响.建立了绝缘体烧蚀机理的耦合数学模型,其计算的激光烧蚀临界能量密度阀值与实验结果很好的吻合.定量描述了超短脉冲激光对绝缘体材料烧蚀微观过程的影响.