Sonine Polynomial Expansion Research Articles

On the Einstein relation in a heated granular gas

Recent computer simulation results [Physica A 334 (2004) 513] for granular mixtures subject to stochastic driving have shown the validity of the Einstein relation ε≡ D/( T 0 λ)=1 between the diffusion D and mobility λ coefficients when the temperature of the gas T is replaced by the temperature of the impurity T 0 in the usual Einstein relation. This problem is analyzed in this paper by solving analytically the Boltzmann–Lorentz equation from the Chapman–Enskog method. The gas is heated by the action of an external driving force (thermostat) which does work to compensate for the collisional loss of energy. Two types of thermostats are considered: (a) a deterministic force proportional to the particle velocity (Gaussian thermostat), and (b) a white noise external force (stochastic thermostat). The diffusion and mobility coefficients are given in terms of the solutions of two linear integral equations, which are approximately solved up to the second order in a Sonine polynomial expansion. The results show that the violation of the Einstein relation ( ε≠1) is only due to the non-Maxwellian behavior of the impurity velocity distribution function (absence of the Gibbs state). At a quantitative level, the kinetic theory results also show that the deviation of ε from 1 is more significant in the case of the Gaussian thermostat than in the case of the stochastic one, in which case the deviation of the Einstein relation is in general smaller than 1%. This conclusion agrees quite well with the results found in computer simulations.

Diffusion of impurities in a granular gas.

Diffusion of impurities in a granular gas undergoing homogeneous cooling state is studied. The results are obtained by solving the Boltzmann-Lorentz equation by means of the Chapman-Enskog method. In the first order in the density gradient of impurities, the diffusion coefficient D is determined as the solution of a linear integral equation which is approximately solved by making an expansion in Sonine polynomials. In this paper, we evaluate D up to the second order in the Sonine expansion and get explicit expressions for D in terms of the coefficients of restitution for the impurity-gas and gas-gas collisions as well as the ratios of mass and particle sizes. To check the reliability of the Sonine polynomial solution, analytical results are compared with those obtained from numerical solutions of the Boltzmann equation by means of the direct simulation Monte Carlo method. In the simulations, the diffusion coefficient is measured via the mean-square displacement of impurities. The comparison between theory and simulation shows in general an excellent agreement, except for the cases in which the gas particles are much heavier and/or much larger than impurities. In these cases, the second Sonine approximation to D improves significantly the qualitative predictions made from the first Sonine approximation. A discussion on the convergence of the Sonine polynomial expansion is also carried out.

Probabilistic ballistic annihilation with continuous velocity distributions.

We investigate the problem of ballistically controlled reactions where particles either annihilate upon collision with probability p, or undergo an elastic shock with probability 1-p. Restricting to homogeneous systems, we provide in the scaling regime that emerges in the long time limit, analytical expressions for the exponents describing the time decay of the density and the root-mean-square velocity, as continuous functions of the probability p and of a parameter related to the dissipation of energy. We work at the level of molecular chaos (nonlinear Boltzmann equation), and using a systematic Sonine polynomials expansion of the velocity distribution, we obtain in arbitrary dimension the first non-Gaussian correction and the corresponding expressions for the decay exponents. We implement Monte Carlo simulations in two dimensions, which are in excellent agreement with our analytical predictions. For p<1, numerical simulations lead to the conjecture that unlike for pure annihilation (p=1), the velocity distribution becomes universal, i.e., does not depend on the initial conditions.

On the first Sonine correction for granular gases

We consider the velocity distribution for a granular gas of inelastic hard spheres described by the Boltzmann equation. We investigate both the free of forcing case and a system heated by a stochastic force. We propose a new method to compute the first correction to Gaussian behavior in a Sonine polynomial expansion quantified by the fourth cumulant a 2. Our expressions are compared to previous results and to those obtained through the numerical solution of the Boltzmann equation. It is numerically shown that our method yields very accurate results for small velocities of the rescaled distribution. We finally discuss the ambiguities inherent to a linear approximation method in a 2.

Energy Nonequipartition in a Sheared Granular Mixture

The kinetic granular temperatures of a binary granular mixture in simple shear flow are calculated by means of the Direct Simulation Monte Carlo method. The results show that the temperature ratio is clearly different from unity (as may be expected since the system is out of equilibrium) and strongly depends on the restitution coefficients as well as on the parameters of the mixture. The influence of the temperature differences on the rheological properties is also discussed. The results are compared with the theoretical predictions obtained from the Boltzmann kinetic theory by using a Sonine polynomial expansion. The comparison shows an excellent agreement over the range of parameters investigated.

Dynamics of ballistic annihilation.

The problem of ballistically controlled annihilation is revisited for general initial velocity distributions and an arbitrary dimension. An analytical derivation of the hierarchy equations obeyed by the reduced distributions is given, and a scaling analysis of the corresponding spatially homogeneous system is performed. This approach points to the relevance of the nonlinear Boltzmann equation for dimensions larger than 1 and provides expressions for the exponents describing the decay of the particle density n(t) proportional, variant t(-xi) and the root-mean-square velocity v proportional, variant t(-gamma) in terms of a parameter related to the dissipation of kinetic energy. The Boltzmann equation is then solved perturbatively within a systematic expansion in Sonine polynomials. Analytical expressions for the exponents xi and gamma are obtained in arbitrary dimension as a function of the parameter mu characterizing the small velocity behavior of the initial velocity distribution. Moreover, the leading non-Gaussian corrections to the scaled velocity distribution are computed. These expressions for the scaling exponents are in good agreement with the values reported in the literature for continuous velocity distributions in d=1. For the two-dimensional case, we implement Monte Carlo and molecular dynamics simulations that turn out to be in excellent agreement with the analytical predictions.

Transport coefficients of a heated granular gas

The Navier–Stokes transport coefficients of a granular gas are obtained from the Chapman–Enskog solution to the Boltzmann equation. The granular gas is heated by the action of an external driving force (thermostat) which does work to compensate for the collisional loss of energy. Two types of thermostats are considered: (a) a deterministic force proportional to the particle velocity (Gaussian thermostat), and (b) a random external force (stochastic thermostat). As happens in the free cooling case, the transport coefficients are determined from linear integral equations which can be approximately solved by means of a Sonine polynomial expansion. In the leading order, we get those coefficients as explicit functions of the restitution coefficient α. The results are compared with those obtained in the free cooling case, indicating that the above thermostat forces do not play a neutral role in the transport. The kinetic theory results are also compared with those obtained from Monte Carlo simulations of the Boltzmann equation for the shear viscosity. The comparison shows an excellent agreement between theory and simulation over a wide range of values of the restitution coefficient. Finally, the expressions of the transport coefficients for a gas of inelastic hard spheres are extended to the revised Enskog theory for a description at higher densities.

Steady-state properties of a mean-field model of driven inelastic mixtures.

We investigate a Maxwell model of inelastic granular mixture under the influence of a stochastic driving and obtain its steady-state properties in the context of classical kinetic theory. The model is studied analytically by computing the moments up to the eighth order and approximating the distributions by means of a Sonine polynomial expansion method. The main findings concern the existence of two different granular temperatures, one for each species, and the characterization of the distribution functions, whose tails are in general more populated than those of an elastic system. These analytical results are tested against Monte Carlo numerical simulations of the model and are in general in good agreement. The simulations, however, reveal the presence of pronounced non-Gaussian tails in the case of an infinite temperature bath, which are not well reproduced by the Sonine method.

Rheological properties in a low-density granular mixture

Steady simple shear flow of a low-density binary mixture of inelastic smooth hard spheres is studied in the context of the Boltzmann equation. This equation is solved by using two different and complementary approaches: a Sonine polynomial expansion and the Direct Simulation Monte Carlo method. The dependence of the shear and normal stresses as well as of the steady granular temperature on both the dissipation and the parameters of the mixture (ratios of masses, concentration, and sizes) is analyzed. In contrast to previous studies, the theory predicts and the simulation confirms that the partial temperatures of each species are different, even in the weak dissipation limit. In addition, the simulation shows that the theory reproduces fairly well the values of the shear stress and the phenomenon of normal stress differences. On the other hand, here we are mainly interested in analyzing transport in the homogeneous shear flow so that, the possible formation of particle clusters is ignored in our description.

Monte Carlo simulation of the homogeneous cooling state for a granular mixture

The Direct Simulation Monte Carlo (DSMC) method is used to numerically solve the Enskog equation for a granular binary mixture in the homogeneous cooling state (HCS). The fourth velocity moments, the temperature ratio, and also the velocity distribution functions are obtained and compared with approximate analytical results derived recently from a Sonine polynomial expansion [V. Garzo and J. W. Dufty, Phys. Rev. E 60, 5706 (1999)]. The comparison shows an excellent agreement between both approaches, even for strong dissipation or disparate values of the mechanical parameters of the mixture. In contrast to previous studies, the partial temperatures of each species are clearly different, so that the total energy is not equally distributed between both species. Finally, in the same way as in the one-component case, the simulation as well as the theory show a high energy tail of the distribution functions.

Multicomponent diffusion in gases and plasma mixtures

Molecular diffusion fluxes derived from kinetic theory are investigated. New expressions for molecular flux given by Stefan - Maxwell relationships derived for any level of Sonine's polynomial expansion are presented withcorresponding coefficients, and analysed. The accuracy of the coefficients is investigated. New simplified expressions for multicomponent diffusion coefficients are introduced, and compared with the predictions of kinetic theory for high orders of Sonine's polynomial expansions.

Deviation from Maxwell distribution in granular gases with constant restitution coefficient

We analyze the velocity distribution function of force-free granular gases in the regime of homogeneous cooling when deviations from the Maxwellian distribution may be accounted only by leading term in the Sonine polynomial expansion. These are quantified by the magnitude of the coefficient $a_2$ of the second term of the expansion. In our study we go beyond the linear approximation for $a_2$ and observe that there are three different values (three roots) for this coefficient which correspond to a scaling solution to the Boltzmann equation. The stability analysis performed showed, however, that among these three roots only one corresponds to a stable scaling solution. This is very close to $a_2$, obtained in previous studies in a linear with respect to $a_2$ approximation.

Homogeneous cooling state for a granular mixture.

The homogeneous cooling state for a binary mixture of inelastic hard spheres is studied using the Enskog kinetic theory. In the same way as for the one-component fluid, we propose a scaling solution in which the time dependence of the distribution functions occurs entirely through the temperature of the mixture. A surprising result is that the (partial) temperatures of each species are different, although their cooling rates are the same. Approximate forms for the distribution functions are constructed to leading order in a Sonine polynomial expansion showing a small deviation from Maxwellian, similar to that for the one-component case. The temperatures and overall cooling rate are calculated in terms of the restitution coefficients, the reduced density, and the ratios of mass, concentration, and sizes.

Charged-particle transport in gases in electric and magnetic fields crossed at arbitrary angles: Multiterm solution of Boltzmann's equation.

A multiterm solution of the Boltzmann equation has been developed and used to calculate transport coefficients of charged-particle swarms in gases under the influence of electric and magnetic fields crossed at arbitrary angles psi. The hierarchy resulting from a spherical harmonic decomposition of the Boltzmann equation in the hydrodynamic regime [Ness, Phys. Rev. A 47, 327 (1993)] is solved numerically by representing the speed dependence of the phase-space distribution function in terms of an expansion in Sonine polynomials about a weighted sum of Maxwellian distributions at different temperatures. Results are given for charged-particle swarms in certain model gases over a range of psi and field strengths. The variation of the transport coefficients with psi is addressed using physical arguments. The errors associated with the two-term approximation and inadequacies of Legendre polynomial expansions are highlighted.

Dense fluid transport for inelastic hard spheres.

The revised Enskog theory for inelastic hard spheres is considered as a model for rapid flow granular media at finite densities. A normal solution is obtained via the Chapman-Enskog method for states near the local homogeneous cooling state. The analysis is performed to first order in the spatial gradients, allowing identification of the Navier-Stokes order transport coefficients associated with the heat and momentum fluxes. In addition, the cooling rate is calculated to first order in the gradients and expressed in terms of the transport coefficients. The transport coefficients are determined from linear integral equations analogous to those for elastic collisions. The solubility conditions for these equations are confirmed and the transport coefficients are calculated as explicit functions of the density and restitution coefficient using a Sonine polynomial expansion. The results are not limited to small dissipation. Finally, the analysis is repeated using a simpler kinetic model. Excellent agreement is obtained with the results from the revised Enskog equation.

Calculation of entropy production in slightly rarefied gas flows by Sonine polynomial expansion

The expansion method in the kinetic theory of gases is often used for the calculation of entropy production in rarefied gas flows, usually in terms of Her-mite polynomials. In this paper, the method of Sonine polynomials expansion is used and the calculation of entropy production is carried out. First, the formal evaluation of the entropy production in rarefied gas flows is introduced. Then the calculation of the entropy production for arbitrary molecules with spherically symmetric intermolecular forces by Sonine polynomials expansion is presented and the formula for entropy production in the thirteen-moment approximation is obtained. It is shown that the entropy production can be expressed as a linear combination of square terms in the thirteen-moment approximation. This formula can be applied to slightly rarefied gas flows.

Kinetic theory of binary gas mixtures with large mass disparity

The Boltzmann equations for a binary mixture of gases are considered in the asymptotic limit when their molecular weight ratio and the light gas Knudsen number are small quantities. A first mass-ratio expansion reduces the cross-collision operator of the light gas Boltzmann equation to a Lorentz form, uncoupling its kinetic behavior from that of the heavy gas. The light gas distribution function is then determined to first order in the Knudsen number, independently of the degree of nonequilibrium characterizing the heavy gas, whose influence is felt only through its hydrodynamic quantities. All transport coefficients arising are determined variationally for arbitrary interaction potentials using Sonine polynomial expansions as trial functions. A remarkable feature of this analysis is that it yields binary transport information (i.e., diffusion and thermal diffusion coefficients) from considering only the Boltzmann equation for the light gas. A second mass expansion reduces the cross-collision operator of the heavy gas equation to a Fokker–Planck form. The corresponding coefficients involve integrals over the light gas distribution function determined previously and are evaluated explicitly in terms of the hydrodynamic quantities and transport coefficients of the light gas. The heavy gas distribution function can be determined by solving a Fokker–Planck equation at dilutions large enough to make heavy–heavy collisions negligible, or by a new Knudsen number expansion when the molar fraction of the heavy gas is of order 1. In this latter case, the heavy gas kinetic behavior is independent of the light gas, being characterized by the same transport coefficients of the pure heavy gas. The problem is then reduced to a set of two-fluid hydrodynamic equations.

Calculation of the electrical conductivity of a partially ionized gas

The theoretical calculation of electron transport properties in partially ionized gases using a Laguerre (or Sonine) polynomial expansion has met with convergence difficulties in certain cases, in particular, when there is a Ramsauer minimum in the electron-neutral momentum transfer cross section. In this work, which deals specifically with electrical conductivity, it is shown that the difficulties can be avoided by choosing a (nonorthogonal) expansion that includes a limiting-case analytical solution. With some simplifications, the approach leads to a practical formula for the conductivity involving generalized collision integrals. A formula based on a BGK-type approximation to the electron–electron collision operator is also given. Calculated conductivities are found to agree well with the more sophisticated expansion method.

The Enskog theory for multicomponent mixtures. II. Mutual diffusion

We present a detailed description of the mutual diffusion coefficients of binary and ternary dense fluid mixtures of hard spheres, as given by the revised Enskog theory (RET) of van Beijeren and Ernst and the standard Enskog theory (SET) of Tham and Gubbins. The formulas for the diffusion coefficients [see part I of this series, J. Chem. Phys. 78, 2746 (1983)] involve the contact values of the equilibrium pair distribution functions and the chemical potentials, for which the Carnahan–Starling approximation is used. The formulas, which were obtained by making an expansion in Sonine polynomials, are evaluated up to the third order and the convergence of the Sonine polynomial expansion is discussed. Except at low densities, the SET cannot be used to describe diffusion in hard-sphere mixtures. We find, using the CS approximation, that the SET gives values for the direct mutual diffusion coefficients that become negative at very high densities. A study is made of the dependence of the RET mutual diffusion coefficients on a variety of the parameters that determine the mixture (density, composition, molecular masses, and diameters) and a summary of the trends found is given. In particular, the influence of the addition of a third component on the mutual diffusion of two other components is discussed.

A classical dynamics study of rotational relaxation in nitrogen gas

Abstract Rotational relaxation, related to the experimentally accessible volume viscosity, is described to the first order in the Sonine polynomial expansion of nonequilibrium distribution functions by the well known Wang Chang, Uhlenbeck, and De Boer expression. By a classical trajectory evaluation of a classical limit of this expression, using a recently proposed anisotropic inter-molecular potential for N2-N2, results are obtained for the rotational relaxation cross section. A corresponding expression including second order Sonine terms is also investigated. The second order results are found to differ significantly from the first order treatment (by about 18% at room temperature) and to agree very well with ultrasonic measurements over a temperature range from 77 K to 293 K.