Binary correlation and convergent collision integral for a plasma

The BBGKY hierarchy equations for a homogeneous plasma are used to derive an explicit expression for the time-asymptotic binary correlation function. Closure of the hierarchy equations is shown possible, mainly due to a difference in scaling behavior, as functions of impact parameter(s), between the driving terms in the second and third hierarchy equations. The final result is shown to be a uniform approximation for interparticle distances much smaller than the minimum of the mean free path and the density inhomogeneity length. The resulting expressions for the free energy and the collision integral are convergent.     

Measurement of the electron-atom collision integral in low-temperature helium plasma

An expression is obtained for the electron-atom collision integral for axisymmetric low-temperature plasmas. The Legendre components of the collision integral are determined experimentally by the probe method. Comparison of the measured Legendre components with their theoretical values shows that, depending on the plasma conditions, the collisional regime or a regime of dominant collective interactions is established in the system. Zh. Tekh. Fiz. 67, 19–24 (April 1997)     

Infrared catastrophe in a two-quasiparticle collision integral

Relaxation of a nonequilibrium state in a disordered metal with a spin-dependent electron energy distribution is considered. The collision integral due to the electron-electron interaction is computed within the approximation of a two-quasiparticle scattering. It is shown that the spin-flip scattering processes with a small energy transfer may lead to the divergence of the collision integral for a quasi one-dimensional wire. This divergence is present only for a spin-dependent electron energy distribution that corresponds to the total electron spin magnetization M = 0 and only for nonzero interaction in the triplet channel. In this case, a nonperturbative treatment of the electron-electron interaction is needed to provide an effective infrared cutoff. The text was submitted by the authors in English. An erratum to this article is available at .     

Infrared catastrophe in a two-quasiparticle collision integral

Relaxation of a nonequilibrium state in a disordered metal with a spin-dependent electron energy distribution is considered. The collision integral due to the electron-electron interaction is computed within the approximation of a two-quasiparticle scattering. It is shown that the spin-flip scattering processes with a small energy transfer may lead to the divergence of the collision integral for a quasi one-dimensional wire. This divergence is present only for a spin-dependent electron energy distribution that corresponds to the total electron spin magnetization M = 0 and only for nonzero interaction in the triplet channel. In this case, a nonperturbative treatment of the electron-electron interaction is needed to provide an effective infrared cutoff.     

Construction of renormalized collision integral for a quantum gas

A quantum analog is obtained for the renormalized collision integral derived for a classical gas in [1]. As usual [2], considering lag effects in the renormalized collision integral corresponds to taking account of the nondissipative contribution of the interaction.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 107–109, August, 1978.It remains to thank L. Ya. Kobelev for his interest in the work, and for useful critical comments.     

Electron-ion collision integral in a strong magnetic field

Boltzmann’s collision integral is extended to the case of helical (Larmor) particle trajectories in a magnetic field of arbitrary strength. The main characteristics of collisions of electrons with positively charged ions in strong magnetic fields, where the Larmor radius of electrons becomes less than the characteristic impact parameter of close collisions in the absence of a magnetic field (Landau’s parameter), are investigated. The differential scattering cross section and the corresponding electron-ion collision integral in strong fields are found. The transport collision frequencies are calculated, and the results are compared with the similar quantities for weaker magnetic fields.     

Pair correlation function in a dense plasma and pycnonuclear reactions in stars

The short-range behavior of the pair correlation function in a dense onecomponent plasma (jellium) is investigated. As an intermediate step, the short-range behavior of the classical pair correlation function is obtained. Actually, although the temperature and the density are assumed to be such that the thermodynamic properties are almost classical, quantum mechanics (tunnel effect) always dominates the pair correlation function at short distances. The quantum pair correlation function is calculated by treating the many-body quantum effects by a perturbation theory, and by using a semiclassical approximation based on path integrals. The results are applied to the computation of the nuclear reaction rate in dense stellar matter (pycnonuclear reactions).Laboratoire associé au Centre National de la Recherche Scientifique.     

A bound for the pressure integral in a plasma equilibrium

An interpolation inequality for the total variation of the gradient of a composite function is derived by applying the coarea formula. A bound for the pressure integral is studied by establishing ana priori estimate for a solution of the Grad-Shafranov equation of plasma equilibrium. A weak formulation of the Grad-Shafranov equation is given to include singular current profiles.     

Isotropic matrix elements of the collision integral for the Boltzmann equation

We have proposed an algorithm for constructing matrix elements of the collision integral for the nonlinear Boltzmann equation isotropic in velocities. These matrix elements have been used to start the recurrent procedure for calculating matrix elements of the velocity-nonisotropic collision integral described in our previous publication. In addition, isotropic matrix elements are of independent interest for calculating isotropic relaxation in a number of physical kinetics problems. It has been shown that the coefficients of expansion of isotropic matrix elements in Ω integrals are connected by the recurrent relations that make it possible to construct the procedure of their sequential determination.     

Theory of the generalized collision integral. I

A diagram method previously suggested [1] for the construction of the integral collision in the single-particle kinetic equations is developed in this paper. The so-called diagram ladder summation, both with a variable external field and without it, is performed for systems with a strong short-range potential to any order in the density. It is shown that in a strong field the lowest order diagram in the density with preliminary ladder renormalization leads to a generalized Boltzmann collision integral.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 117–121, August, 1973.     

Relations between nonlinear kernels of the collision integral

Previously, to solve the Boltzmann equation by the moments method with expansion of the distribution function by spherical Hermit polynomials, a new computational method was suggested which allowed to construct nonlinear matrix elements of the collision integral with very large indices. This made it possible to substantially advance in construction of the distribution function. Limitations to convergence of the distribution function that appear in moment method are eliminated if we come to expansion by spherical harmonics from expansion by spherical Hermit polynomials. In this case, a complex five-fold collision integral is replaced by a set of comparatively simple integral operators, and kernels G _l1,l2 ^l (c, c ₁, c ₂) of these operators become the analog of matrix elements. We found the relations between expansions of the distribution function in the reference frames with various velocities of motion along marked axis. Starting from the invariance condition of the collision integral with respect to selection of such reference frames, we derived recurrent relations between the kernels with various indices. These relations allow us to construct any nonlinear kernel G _{l1, l2} ^l (c, c ₁, c ₂), if the kernel G _0,0⁰(c, c ₁, c ₂) is known.     

Theory of the generalized collision integral. IV

In the maximum time interval of kinetic evolution of the single-particle distribution function of a spatially homogeneous system summation of the step diagrams (without intersections of the lines of propagation and interaction) is carried out for an arbitrary nonequilibrium state in the Markov limit, as well as for a local equilibrium state with regard for memory effects. Procedures are indicated for the step renormalization of diagrams of arbitrary order with respect to the density in the Markov limit, with subsequent ladder renormalization. Ring diagrams with preliminary step and ladder renormalizations are summed. A more correct derivation is given for the generalized Boltzmann (Boltzmann-Landau) collision integrals for dense systems with a strong short-range (as well as strong long-range) potential. It is shown that the step, ring, and ladder renormalizations yield a nondivergent two-particle distribution function in a classical equilibrium plasma.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 86–91, August, 1978.     

液态~4He系统对关联函数的路径积分蒙特卡罗模拟

利用路径积分蒙特卡罗(PIMC)模拟方法研究了由256个4He原子组成的系统在不同条件下的对关联函数g(r).在标准大气压下(SVP),分别在1.38K,1.67K,2.50K和4.24K等四个温度时对系统不同势能的情况的对关联函数进行了计算;另外还讨论了系统的温度和密度对对关联函数的影响.结果发现,系统的势能变化及密度变化都或大或小影响对关联函数,而温度对其影响不是很大.虽然PIMC方法存在有限尺度效应,但是由128和256个4He原子组成系统的对关联函数存在极小的差异,因此可以得到:随着模拟系统尺度的增大,所模拟的有限尺度的系统已逐渐接近于真实系统.所以所得的256原子系统的性质可近似说明液态系统性质.     

Matrix elements and kernels of the collision integral in the Boltzmann equation

This article elaborates upon our previous work in which some general properties of the matrix elements and kernels of the gain and loss terms of the collision integral were found. The object of study is the loss term of the collision integral, since related analytical expressions are simple. Formulas to calculate the matrix elements are derived. The kernels of power-law interaction potentials are completely investigated and constructed using analytical and numerical approaches.     

General calculation of the collision integral for the linearized Boltzmann transport equation

We develop and describe a general method for evaluating collision integrals in the linearized Boltzmann transport equation which eliminates the necessity to repeat similar integration steps for each force law. Integrations not dependent on scattering cross-section variables have been carried out once and for all. The two mathematical innovations which facilitate these general integrations are (i) the development of an expansion of the Burnett functionX _NML(x+y) into products of Burnett functions of argument x with other functions; and (ii) the use of representations of the full rotation group to transform from space-fixed axes to axes aligned with the relative velocity vector of colliding atoms. The relations so derived allow rapid evaluation of the collision integral from a knowledge of the scattering cross section.