Sonine Polynomial Expansion Research Articles

Sonine expansion of range and damage distributions

Abstract To reconstract an approximate distribution function from its first few moments the Sonine polynomial expansion is proposed. Its zeroth approximant includes the mean value, the variance, and the skewness. The rapid convergence of the present expansion series is examined, comparing the reconstracted distribution function with the Monte Carlo results.

Quantum corrections to the momentum distribution of moderately dense Boltzmann gases. I. Smooth potentials

The momentum distribution of a quantum Boltzmann gas at equilibrium is shown to be the usual Maxwell–Boltzmann term plus a correction term of higher order in density and fully quantum -mechanical in origin. For smooth potentials, the correction may be described by a semiclassical expansion, and the leading quantum part depends on momentum in the simple manner postulated by Imam-Rahajoe and Curtiss, although higher-order terms have a more complicated momentum dependence. At lowest order in density, a Sonine polynomial expansion and expressions for the moments of the quantum distribution are derived. The pressure tensor and energy density of a quantum gas, as determined from kinetic theory, are shown to be consistent with thermodynamics in the equilibrium limit, but only when the quantum part of the momentum distribution is properly taken into account.

Sonine polynomial solution of the Fokker–Planck equation

Using the Sonine polynomial expansion method, the Fokker–Planck equation is accurately solved. The overall relaxation time is considerably shorter than the Spitzer self-collision time.

Collision integrals of ionized air components for a screened coulomb potential

Results of a numerical evaluation of the integrals Ωι,s (ι = 1, 2, 3, 4 and s = ι ... 8 -ι), in terms of which transport coefficients are expressed [1], are given in this paper. The results obtained are of practical use for calculating kinetic properties of ionized gases within the Chapman-Enskog theory, including third-order terms in the Sonine polynomial expansion. The integrals Ωι,s were calculated for singly-ionized colliding air particles at temperatures T = 10,000–40,000 °K and pressures p = 0.1, 1, and 100 atm. The screened Coulomb potential for attraction and repulsion was the model for electron-ion and ionic interactions, and the Debye screening length was chosen by taking into account screening both by electrons and by multiply-charged ions. Quantum effects are not important in the temperature and pressure ranges considered for air, and can, therefore, be neglected in calculating kinetic properties.

Polynomial Expansion Method for Problems Involving a Relativistic Maxwellian Distribution

With the aim of extending the Sonine polynomial expansion method to the region of relativistic Maxwellian distribution functions, a new set of polynomials has been introduced in terms of their orthogonality equation. Their coefficients have been derived and tabulated as functions of the temperature for several lowest orders.

Nonequilibrium Contributions to the Rate of Reactions. II. Isolated Multicomponent Systems

The nonequilibrium rates of reaction in multicomponent reactive systems are obtained by solving the Champman–Enskog equation. A Sonine polynomial expansion is used for the solution and its convergence is demonstrated for a wide range of system variables. It is found that the perturbation of the velocity distribution function for each reactive component depends on the deviation of the reactive collision frequency R(0)(cγ) from a special form R̄(0)(cγ). A model reactive system was examined with line-of-centers reactive cross sections and hard-sphere elastic cross sections chosen to fit the available data for the reaction H2+Cl=HCl+H. The deviation of the ratio of the forward and reverse rate coefficients (kf / kr) from the equilibrium value (kf(0) / kr(0)) was shown to be much smaller than the experimental deviation; this suggests that translational nonequilibrium is not significant, but does not exclude vibrational effects which could be more important.

Sonine Polynomial Solution of the Boltzmann Equation for Relaxation of Initially Nonequilibrium Distribution

Using the Sonine polynomial expansion method the nonlinear Boltzmann equation for time-dependent spatially uniform gases is solved to examine the relaxation of an initially nonequilibrium distribution toward the equilibrium. For an inverse-fifth power law force, a simple analytical expression is presented for equations for expansion coefficients. As a result the higher-order expansion coefficients as well as the lower-order ones are obtained, so that the decay of larger initial departure of the distribution function from the equilibrium is dealt with. The resulting distribution function is compared with the corresponding one obtained on the basis of the Bhatnagar-Gross-Krook model. The latter model is shown to be adequate only when the departure of distribution function from the equilibrium becomes small.

On the Validity of the Two-term Approximation of the Electron Distribution Function

The Boltzmann equation for electrons moving in a neutral gas under the influence of an externally applied field is solved by expanding the electron distribution function in terms of Legendre and Sonine polynomials. The solution is given in terms of infinite matrices which have elements ordered by the Sonine polynomial index, and which are dependent upon the field strength. From the structure of the formulae, it is possible to infer that truncation of the Legendre polynomial expansion after two terms is a good approximation at all field strengths. This is supported by calculations of the electron drift velocity at low field strengths, which show that the error introduced by making the two-term approximation is small, even when the deviation from equilibrium is significant. The convergence of the Sonine polynomial expansion is shown to be strongly depende:r;J.t upon field strength, and large matrices are required in the drift velocity formula at even small field strengths.

Time Correlation Functions in a Binary Gas Mixture

The low density forms of the correlation functions which yield the coefficients of mutual and thermal diffusion in a binary gas mixture are derived and examined from two points of view. One approach uses the binary collision expansion of the existing correlation function expressions, another uses the physical arguments given by Mori to extract these correlation functions from the Boltzmann equation by considering the evolution of the system from an arbitrary initial state. Both approaches indicate that the time correlation functions can be expressed in terms of single momentum averages of two dynamical functions which obey coupled integrodifferential equations of the Boltzmann type. These equations are solved by Sonine polynomial expansion and calculations are performed for a hard-sphere gas mixture in order to illustrate the results. The correlation functions which characterize the thermal diffusion coefficient are discussed in some detail and several interesting aspects of the dynamics are pointed out.

Nonequilibrium Contributions to the Rate of Reaction. I. Perturbation of the Velocity Distribution Function

The correction to the equilibrium rate of reaction in a one-component reactive system is calculated from the perturbation of the velocity distribution function obtained by solving the Chapman–Enskog and Burnett equations. A method of solution of the Chapman–Enskog equation with a Sonine polynomial expansion is described that is not limited by the number of terms retained and is applicable to realistic systems characterized by elastic and reactive cross sections which may be available only in tabulated form. The convergence of the Sonine polynomial expansion is demonstrated for a variety of model reactions with and without activation energy and for a set of cross sections obtained with a semiempirical potential for the reaction H2(i) + H2(j) → (products), where i and j denote the vibrational quantum numbers. The Sonine polynomial method is compared with the recent variational solutions of Present and Morris. It is shown that the extent of the departure from equilibrium is due to the deviation of the reactive collision number R(0)(c) from a special form R̄(0)(c). It was found that for the realistic systems considered here the decrease in the equilibrium rate of reaction due to a perturbation of the velocity distribution function by reaction is very small (≲1%) and that the Burnett correction to the distribution function can be neglected. For the set of (H2, H2) reactions, the greatest decrease in the equilibrium rate of reaction is 0.66% at 6400°K for (i, j) = (3, 2). A formal transport theory description of reaction rates analogous to the description of heat conduction and viscous flow by Chapman and Cowling is also presented.

Quantum-Mechanical Plasma Transport Theory

The quantum-mechanical convergent kinetic equation of Gould and DeWitt is used for the calculation of static transport coefficients through first order in the plasma parameter for a fully ionized, nondengenerate, two-component plasma near thermal equilibrium. The equation is solved with the standard Chapman-Enskog procedure using a three-term Sonine polynomial expansion. Since close collisions and dynamic screening effects are treated correctly, the results contain both logarithmic (dominant) and nonlogarithmic (subdominant) terms evaluated exactly to first order in the plasma parameter. Since close collisions are treated quantum mechanically the transport coefficients obtained are valid for a wide range of temperatures. For the case where only one Sonine polynomial is used in the evaluation of the electrical conductivity, the result obtained here reduces to that of Kivelson and DuBois in the high-temperature limit (kT ≫ Ry) and to that of Gould and DeWitt in the low-temperature, classical limit (kT ≪ Ry). Using more than one Sonine polynomial in the expansion of the distribution function, we obtain for the electrical conductivity σ(χ) = c(χ)σ(1), where χ is the number of terms in the Sonine expansion and σ(1) is the result for one term. In the limit where the Coulomb logarithm is large, c(3) = 1.95 and c(∞) = 1.973, so that our results are accurate to 1 or 2% for a wide range of temperatures (including kT ∼ Ry).

Variational Solution of the Chemical-Kinetic Boltzmann Equation

Variational principles are formulated for the linearized molecular Boltzmann equation and it is shown that if a perturbation of the Maxwell distribution maximizes the nonequilibrium correction to the reaction rate, subject to an equation of constraint, it is a solution of the integral equation describing a gas-phase reaction. A corollary is that the Chapman–Enskog–Burnett subapproximations generate lower bounds for the nonequilibrium correction to the reaction rate. Since it has been shown previously that the Chapman–Enskog method is useless in the treatment of slow activated reactions because the Sonine polynomial expansion diverges, the variational method is applied to the same problem using a trial function that is not in polynomial form. We use the trial perturbation function ψ = A [exp(γc2) + ac2 + b] where a and b are fixed by the auxiliary conditions and A by the equation of constraint and γ is an adjustable variation parameter. An order-of-magnitude improvement and increase in the nonequilibrium correction over the previous two-polynomial approximation is obtained for ε* / kT = 20 → 30 where ε* is the activation energy; however, the correction is still negligibly small. Small improvements are obtained for fast activated reactions and for free-radical recombinations.

KROOK模型について

Relaxation problem of an initially nonequilibrium distribution toward the equilibrium distribution is studied using both the BOLTZMANN equation and the KROOK model. By the use of SONINE polynomial expansion method, distribution is determined from the non-linear BOLTZMANN equation for spatially uniform gases. By comparing the solutions of the KROOK model with those of the BOLTZMANN equation, it is shown that the KROOK model can not reproduce the correct distribution obtained from the BOLTZMANN equation for the case where departures from the equilibrium distribution are large. The KROOK model, however, can reproduce the correct distribution when the distribution approaches the equilibrium distribution.

Chapman–Enskog Method in Chemical Kinetics

Nonequilibrium effects in gas-phase reactions associated with the perturbation of the Maxwell speed distribution by the reaction have been investigated by Prigogine and others using the Chapman–Enskog method in the lowest-order approximation. We have extended these calculations to the second order in the Burnett expansion of the perturbation in a series of Sonine polynomials. In the case of the reaction A + A→B + C in which the products B and C are removed and no inert gas is present, the appropriate elastic collision integrals have been evaluated for the model of rigid elastic spheres and the reactive collision integrals have been evaluated for two reaction cross sections, one corresponding to the simple collision theory of bimolecular reactions with activation energy ε* and the other to the “centrifugal barrier” theory of free-radical recombinations. For the activated reactions the detailed results show that the Chapman–Enskog method is useless when the reaction is fast (ε* / kT ≲ 5) because the perturbation method fails and is also useless when the reaction is slow because the Sonine polynomial expansion then diverges, as anticipated by Takayanagi. In free-radical reactions the Chapman–Enskog procedure is satisfactory and the nonequilibrium corrections to the reaction rate are negligibly small (<0.1%) in first and second orders.

Ionenbewegung unter dem Einflu� von Umladungsst��en

The motion of ions under the influence of an electric field and a density gradient is investigated by taking into account only charge-transfer collisions between the ions and a neutral gas background. The two model cases of constant mean free path and constant mean free time are considered. For weak electric fields the Boltzmann equation can be solved using a special expansion of the velocity distribution. The expansion in Sonine polynomials as given by Chapman and Cowling does not converge. For medium and high electric fields the velocity distribution is calculated by collecting the contributions of the different paths of the ions. The inhomogeneous drift in high electric fields is investigated for a small perturbation by gradients of the electric field, the ion density and of the mean free path.

Kinetic Theory of the Moderately Dense Rigid-Sphere Fluid. IV. Fluxes of Matter, Momentum, and Energy in a Mixture

The perturbation solution of the modified Boltzmann equation is used to obtain the fluxes of matter, momentum, and energy. The fluxes are evaluated to first order in the gradients of density, flow velocity, and temperature, and are left in terms of the Sonine polynomial expansion coefficients. As an example, the calculation of the isothermal diffusion coefficient is carried out for the general case of unequal masses and radii. In the limit of identical particles, D is proportional to (2kT/m)12(N/vσ2) {[1/g(σ)]+(const/v)+···} .