Stationary nonequilibrium solutions of model Boltzmann equation

We give an explicit solution of a model Boltzmann kinetic equation describing a gas between two walls maintained at different temperatures. In the model, which is essentially one-dimensional, there is a probability for collisions to reverse the velocities of particles traveling in opposite directions. Particle number and speeds (but not momentum) are collision invariants. The solution, which depends on the stochastic collision kernels at the walls, has a linear density profile and the energy flux satisfies Fourier's law.This paper is dedicated to Peter Gabriel Bergmann with affection and admiration on the occasion of his 70th birthday.     

库仑碰撞截面在等离子体粒子模拟中的应用

在等离子体粒子模拟中,TA模型和NanBu模型被广泛用于处理库仑碰撞,这两种模型要求每个时间步长内全部粒子参与计算。为了降低参与碰撞的粒子数,提高库仑碰撞的计算效率,提出了一种基于截面的库仑碰撞模拟方法,并给出了库仑碰撞概率的计算公式。采用该方法对不同温度不同密度电子气的弛豫过程进行模拟,分别对比了电子速度分布函数、电子温度以及电子x、y方向上的温度与电子温度之比的模拟值与理论值,验证了该方法的准确性。在相同的小时间步长上,该方法相比TA模型计算效率提升可达40%以上。对于较大的时间步长,该方法仍能得到与理论解近似的模拟结果,相比Nanbu模型,在相同的精度下可取更大的时间步长,计算效率也有所提升。研究表明,该方法同样适用于电子-离子碰撞。因此在提高库仑碰撞计算效率上,该方法具有碰撞粒子数少以及适用于大时间步长的优势。     

Cumulative author index of Volumes 351 to 360

The dynamic evolution of granular gases is fundamentally different from molecular gases due to the energy loss during collisions. Nevertheless techniques of kinetic theory are useful in a regime, when the granular particles are moving rapidly and the gas is sufficiently dilute. In these lecture notes we analyse in detail the collision of two rough particles which is inelastic due to incomplete normal and tangential restitution as well as Coulomb friction. Based on the Walton model a time evolution operator for the many particle system is introduced, a formalism which is well suited for simple approximations. We discuss free cooling of granular particles with particular emphasis on the exchange of energy between rotational and translational degrees of freedom.     

On Collisions Times of ‘Self-Sorting’ Interacting Particles in One-Dimension with Random Initial Positions and Velocities

We investigate a one-dimensional system of N particles, initially distributed with random positions and velocities, interacting through binary collisions. The collision rule is such that there is a time after which the N particles do not interact and become sorted according to their velocities. When the collisions are elastic, we derive asymptotic distributions for the final collision time of a single particle and the final collision time of the system as the number of particles approaches infinity, under different assumptions for the initial distributions of the particles’ positions and velocities. For comparison, a numerical investigation is carried out to determine how a non-elastic collision rule, which conserves neither momentum nor energy, affects the median collision time of a particle and the median final collision time of the system.     

一种补偿时间步长限制的粒子模拟-蒙特卡罗碰撞模型

 从碰撞次数的概率分布出发,推导出一种补偿蒙特卡罗碰撞模型,采用以正态分布计算得到的平均碰撞次数作为碰撞概率,来补偿传统方法中忽略的多次碰撞。通过模拟不同折合电场强度条件下He气放电产生电子的运动规律,验证了补偿蒙特卡罗碰撞模型的正确性。计算结果表明:补偿蒙特卡罗碰撞模型可以有效地提高计算效率,特别适用于高气压气体放电现象的粒子模拟。     

Zur Theorie der Rückstreuung

Electrons emitted from a cathode, are partially backscattered to the cathode surface by collisions with gas molecules. No equilibrium between electrons, gas and field is presumed for a theoretical treatment of this process, in contrary to the conception of Thomson [1]; instead of it the effect of single collisions is investigated assuming isotropy of scattering. The collision numbers, which may be calculated from successive integral transforms, are shown to form a monotone decreasing sequence. In the field-free case the collision numbers and the back diffusion rates for the first collisions may be explicitely stated; the convergence of this method is rather poor, however. A model for back diffusion in an electric field, whereby the effect of primary collisions only is considered, is completely treatable. The results depend on the angular and energy-distribution of the emitted electrons. A comparison with other formulas and with experimental results for back scattering shows a better agreement between the latter and our equations than in the case of the Thomson-Loeb relation. The consequences on the theory of cathode fall are briefly discussed.     

碰撞对尘埃等离子体鞘层中尘埃颗粒影响

在一维平板鞘层中应用流体模型研究了尘埃等离子体鞘层中碰撞对尘埃颗粒密度和带电量的影响。 研究所涉及的碰撞主要有电离碰撞,电子、离子分别与中性粒子的碰撞,以及电子、离子分别与尘埃颗粒的碰撞。通过采用四阶龙格库塔法,得到了数值解。结果表明,随着电离碰撞或者电子、离子分别与中性粒子碰撞的频率 增加,都将导致鞘层中尘埃颗粒的数密度增大,数密度的极值位置向鞘边位置靠近,尘埃颗粒带电量增多。电子、离子与尘埃颗粒的碰撞,使得尘埃带电量减小。此外,从研究的结果来看,由离子产生的碰撞要比电子产生的碰 撞对尘埃颗粒的影响明显得多。     

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The fluid dynamics model was used to study the influence of the existence of the collisions on the dust particles in a dusty plasma sheath. The main collisions in this study consist of the ionization collision, the collision between electron and neutral particle, the collision between ion and neutral particle, the collision between electron and dust particle, the collision between ion and dust particle. Numerical calculation results are obtained by the fourth-order-Rung-Kutta method. It is shown that both the dust particles’ density and corresponding electricity quantity increase as the following collisions’ frequency enhances, such as ionization collision, the collision between electron and neutral particle, the collision between ion and neutral particle. The charge reduces due to the collisions concluding electron-dust particle and ion-dust particle. In addition, the effect produced by ions’ collision is much more obviously compared with electrons’ collision.     

Effect of diffraction of molecules on collisional line narrowing

A new model of line profile is proposed, which simultaneously takes into account hard and soft (in velocity) collisions leading to collisional line narrowing. It is shown that parameters of collisional narrowing and broadening obtained by processing experimental profiles on the basis of the standard model of hard collisions contain almost no contribution of diffraction scattering of molecules by buffer gas particles. The model proposed for the line profile makes possible a correct comparison of frequencies of the collision integral measured by methods of linear and nonlinear spectroscopy.     

Central limit theorems for the one-dimensional Rayleigh gas with semipermeable barriers

A version of the one-dimensional Rayleigh gas is considered: a point particle of massM (molecule), confined to the unit interval [0,1], is surrounded by an infinite ideal gas of point particles of mass 1 (atoms). The molecule interacts with the atoms and with the walls via elastic collision. Central limit theorems are proved for a wide class of additive functionals of this system (e.g. the number of collisions with the walls and the total length of the molecular path).Research partially supported by the Hungarian National Foundation for Scientific Research, grant No. 819/1     

Die Methode der Normallösungen für die Vlasov-BGK-Gleichung

The linearized Vlasov equation with collision damping is solved by the method of normal modes of Van Kampen and Case. The system considered is an infinitely extended nonrelativistic nondegenerate electron gas with neutralizing ion background and neutral particles. There is no magnetic field. Collision damping is taken into account by complete Bhatnagar-Gross-Krook collision integrals for electron-electron collisions and electronion collisions; in the case of a partially ionized gas elastic collisions between electrons and neutrals can be included likewise. The Vlasov-BGK operator is transformed into an integral operator yielding complex singular normal modes even if the equilibrium distribution is a Maxwellian. The adjoint integral equation belongs to a more complicated type than that of a collision-free system. Its solutions must be orthogonalized. The set of all normal modes is complete rendering the exact solution of the initial value problem possible. The completeness is shown by transformations and regularization of the singular integral equation of the initial value problem, the techniques of Case not being applicable because of the complicated type of this equation.     

Formation time of hadrons and density of matter produced in relativistic heavy-ion collisions

Density of matter produced in relativistic heavy-ion collisions depends substantially on the spacetime evolution of the collision and on the formation time of hadrons produced. Interactions of hadrons younger than their formation time are attenuated with respect to their normal values (transparency of hadronic matter for newly formed hadrons). The system of secondary hadrons produced in a heavy-ion collision thus expands as a gas of almost non interacting particles before hadrons reach their formation time. Densities of interacting hadronic matter produced in oxygen-lead and sulphur-lead collisions at 200 GeV/nucleon are estimated as a function of the formation time of hadrons. Uncertainties in our knowledge of the critical temepratureT _c and of the formation time of hadrons τ₀ permit at present three scenarios: an optimistic one (QGP has already been produced in collisions of oxygen and sulphur with heavy ions and will be copiously produced in Lead collisions), a pessimistic one (QGP cannot be produced at 200 GeV/nucleon) and an intermediate one (QGP has not been produced in oxygen and sulphur interactions with heavy ions and will be at best produced only marginally in Pb-collisions). We find the last opinion as most probable.     

Probe-field spectroscopy in two-level systems with low Doppler broadening

The spectrum of absorption (amplification) of a weak probe field by two-level atoms colliding with atoms of a buffer gas in the field of a strong electromagnetic wave is studied theoretically. A universal, free of any collision model, solution is obtained for systems with small (compared to collision rates) Doppler broadening with no restrictions on the strong-field intensity and with the possible absence of dephasing of the light-induced dipole moment in elastic collisions of the gas particles. The relation of the collisional parameters of the problem with the characteristics of an elementary scattering event and with the macroscopic characteristics connected with the transport phenomena is established. It is found that the nontrivial features of the probe-field spectrum discovered by us previously [Zh. Éksp. Teor. Fiz. 127, 320 (2005)] in the framework of the strong-collision model are not connected with a particular model of collisions and have a universal nature.     

Ion-induced auger electron emission of Mg,Al and Si as a function of ion energy

Experiments have been performed in which ion-induced Auger electron yields of Mg, Al and Si were measured as a function of ion energy. A computer simulation based on a binary collision cascade model yielded results that were in close agreement with the experimental findings. From this a model is proposed, where the creation of inner shell vacancies occurs during symmetrical collisions in the collision cascade generated by the ion. From the minimum energy required for Auger emission a distance of closest approach between the particles can be derived, using the Molière approximation of the screened Coulomb potential to describe the collisions. This distance can be correlated with the radii of the interacting orbitals derived from Hartree-Fock-Slater calculations.     

System-size scan of dihadron azimuthal correlations in ultra-relativistic heavy ion collisions

System-size dependence of dihadron azimuthal correlations in ultra-relativistic heavy ion collision is simulated by a multi-phase transport model. The structure of correlation functions and yields of associated particles show clear participant path-length dependences in collision systems with a partonic phase. The splitting parameter and root-mean-square width of away-side correlation functions increase with collision system size from ¹⁴N + ¹⁴N to ¹⁹⁷Au + ¹⁹⁷Au collisions. The double-peak structure of away-side correlation functions can only be formed in sufficient “large” collision systems under partonic phase. The contrast between the results with partonic phase and with hadron gas could suggest some hints to study onset of confinement.     

Modeling process of the neutral beam re-ionization loss

The basic process of re-ionization loss was studied. In the drift duct there are three processes leading to re-ionization loss: the collision of neutral beam particles with the molecules of background gas, similar collisions with released molecules from the inner wall of the drift duct and the ferret-collisions among particles with different energy of the neutral beam. Mathematical models have been developed and taking EAST-NBI parameters as an example, the re-ionization loss was obtained within these models. The result indicated that in the early stage of the neutral beam injector operation the released gas was quite abundant. The amount of re-ionization loss owing to the released gas can be as high as 60%. In the case of a long-time operation of the neutral beam injector, the total re-ionization loss decreases from 13.7% to 5.7%. Then the re-ionization loss originating mainly from the collisions between particles of the neutral beam and the background molecules is dominant, covering about 92% of the total re-ionization loss. The drift duct pressure was the decisive factor for neutral beam re-ionization loss.     

On a density matrix equation for atomic systems subject to an external radiation field interaction and collision processes

Approximations are examined which are necessary for a density matrix equation of Rautian's form describing the interaction of a gas medium with an external radiation field by considering collision processes between active atoms as well as active and perturber atoms. The collision integrals of the kinetic density matrix equation calculated by the Bogolyubov method involve exitation processes, inelastic and elastic collisions. The features of phase- and velocity changing collisions are discussed by specializing the collision terms. Some phenomenological collision models for velocity changing collisions are examined and conditions for their validity given.     

Relativistic heavy-ion collisions: An approach based on non-equilibrium thermodynamics

A new theory of particle production in high energy collisions is proposed which is based on non-equilibrium thermodynamics. The non-equilibrium model is a major extension of the equilibrium thermodynamic model of relativistic heavy-ion collisions developed earlier. While the equilibrium thermodynamic theory is appropriate for the formation of light nuclei and for pions, the non-equilibrium theory applies to the creation of particles heavier than the pion, which include such particles as the strange mesons, strange baryons and the anti-nucleons. Using an approach based on the degree of the reaction of kinetic theory, the time evolution of the composition of hadronic systems in incomplete equilibrium is investigated. Densities of produced particles are related to space-time quantities and to the production cross sections of the underlying dynamic processes. An application of the non-equilibrium approach to the production of strange matter is given. The importance of secondary processes, following pion production, in the formation of strange matter is shown. In fact, the secondary production process for kaons is as important as the direct production process arising from initial nucleon-nucleon (NN) collision of a first collision picture. Thus, kaons can be produced in a late stage of the collision of two nuclei and they do not necessarily reflect the early stages of the collision as first thought. Using the experimental number of kaons, the time of reaction is also estimated. No evidence for a long-lived state of the nuclear system is found. Expressions for particle production ratios are developed. The results of an equilibrium theory and a non-equilibrium theory are found to be similar for such ratios. The chemical equilibrium constant is shown to be present in the non-equilibrium theory; the Boltzmann factor in the production threshold energy appears in the equilibrium theory. The K^?/K⁺ ratio is estimated. Surprisingly, reasonable agreement with experiment is found in the K^?/K⁺ ratio using the equilibrium theory, even though the production processes for K⁺'s and K^?'s treated individually, are not ones for which the equilibrium theory applies. It is shown that a fundamental difference between the equilibrium and non-equilibrium theory is lost when particle ratios for non-equilibrium particles are taken. Expressions for the production of complex composite structures made of strange particles are developed. The non-equilibrium model with some modifications may be useful for high energy NN and pion-nucleon collisions.     

The influence of surface forces on shear-induced tracer diffusion in mono and bidisperse suspensions

The shear-induced self-diffusivity of tracer particles of radius a _t = λa in a suspension of particles having a radius, a , is calculated by Stokesian dynamics for different values of the size ratio, λ , both in 2 and 3 dimensions in the binary-collision regime. The self-diffusion is found to decrease strongly when the size ratio becomes quite different from unity. On the other hand, for the same average distance of contact between two spheres, the presence of a soft force always increases greatly the diffusion compared to the effect of a hard shell which is used to model the roughness. This is particularly true for tracer particles smaller than the bath particles, where the shear-induced diffusion can be increased by many order of magnitudes in the presence of a soft force. For suspensions of monodisperse particles we show that, for low volume fraction, the diffusion coefficient is much smaller than the one predicted by the binary collision model, due to the existence of a layered structure. On the contrary at higher volume fraction, many-body collisions strongly enhance the diffusion and it appears that the value of the diffusion is quite sensitive to the presence of clusters of particles which are themselves determined by the range of interparticle forces.