The hydrodynamic equations for a gas of hard spheres with dissipative dynamics are derived from the Boltzmann equation. The heat and momentum fluxes are calculated to Navier-Stokes order and the transport coefficients are determined as explicit functions of the coefficient of restitution. The dispersion relations for the corresponding linearized equations are obtained and the stability of this linear description is discussed. This requires consideration of the linear Burnett contributions to the energy balance equation from the energy sink term. Finally, it is shown how these results can be imbedded in simpler kinetic model equations with the potential for analysis of more complex states.

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.

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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.

Hydrodynamic equations for a binary mixture of inelastic hard spheres are derived from the Boltzmann kinetic theory. A normal solution is obtained via the Chapman-Enskog method for states near the local homogeneous cooling state. The mass, heat, and momentum fluxes are determined to first order in the spatial gradients of the hydrodynamic fields, and the associated transport coefficients are identified. In the same way as for binary mixtures with elastic collisions, these coefficients are determined from a set of coupled linear integral equations. Practical evaluation is possible using a Sonine polynomial approximation, and is illustrated here by explicit calculation of the relevant transport coefficients: the mutual diffusion, the pressure diffusion, the thermal diffusion, the shear viscosity, the Dufour coefficient, the thermal conductivity, and the pressure energy coefficient. All these coefficients are given in terms of the restitution coefficients and the ratios of mass, concentration, and particle sizes. Interesting and new effects arise from the fact that the reference states for the two components have different partial temperatures, leading to additional dependencies of the transport coefficients on the concentration. The results hold for arbitrary degree of inelasticity and are not limited to specific values of the parameters of the mixture. Applications of this theory will be

An accurate analytical parametrization for the exchange-correlation free energy of the homogeneous electron gas, including interpolation for partial spin-polarization, is derived via thermodynamic analysis of recent restricted path integral Monte-Carlo (RPIMC) data. This parametrization constitutes the local spin density approximation (LSDA) for the exchange-correlation functional in density functional theory. The new finite-temperature LSDA reproduces the RPIMC data well, satisfies the correct high-density and low-and high-T asymptotic limits, and is well-behaved beyond the range of the RPIMC data, suggestive of broad utility. PACS numbers:The homogeneous electron gas (HEG) is a fundamentally important system for understanding many-fermion physics. In the absence of exact analytical solutions for its energetics, high-precision numerical results have been critical to insight. Recently published [1] restricted path integral Monte Carlo (RPIMC) data for the HEG over a wide range of temperatures and densities open the opportunity to obtain closed form expressions for HEG thermodynamics, in particular the exchange and correlation (XC) contributions. Such expressions extracted from Monte Carlo data are well-known for the zero-T HEG, where they have played a major role in understanding inhomogeneous electron-system behavior. We provide the corresponding thermodynamical expressions for wide temperature and density ranges.Density functional theory (DFT) is the motivating context. For ground-state DFT, the most basic exchangecorrelation (XC) density functional is the local density approximation (LDA). It approximates the local XC energy per particle, ε xc , as the value for the HEG at the local density, ε LDA xc (n(r)) ≈ ε HEG xc (n)| n=n(r) [also see Eq. (3) [5] for the spin-polarized T = 0 K HEG also validate the spininterpolation formulae used in that case, the local spin density approximation (LSDA). All more refined ε xc approximations reduce to the LSDA in the weak inhomogeneity limit.Finite-temperature DFT [6-8] increasingly is being used to study matter under diverse density and temperature conditions [9][10][11][12][13][14]. In it, the XC free-energy is defined by decomposition of the universal free-energy density functional (independent of the external potential). With the T -dependence suppressed for now, that functional isThe first two terms are the non-interacting kinetic energy and entropy (also known as the Kohn-Sham KE and entropy), F H [n] is the classical electron-electron Coulomb energy, and the XC free energy by definition iswith T [n] and S[n] the interacting system kinetic energy and entropy and U ee [n] the full quantum mechanical electron-electron interaction energy. Just as for T = 0 K, the existence theorems of finite-T DFT are not constructive for F xc , so approximations must be devised. Common practice [9] in simulations is to use a. This gives only the implicit T -dependence provided by n(r, T ). However, there is substantial evidence from both finite-T Hartree-Fock [12,15] and finite-T exac...

In contrast to normal fluids, a granular fluid under shear supports a steady state with uniform temperature and density since the collisional cooling can compensate locally for viscous heating. It is shown that the hydrodynamic description of this steady state is inherently non-Newtonian. As a consequence, the Newtonian shear viscosity cannot be determined from experiments or simulation of uniform shear flow. For a given degree of inelasticity, the complete nonlinear dependence of the shear viscosity on the shear rate requires the analysis of the unsteady hydrodynamic behavior. The relationship to the Chapman-Enskog method to derive hydrodynamics is clarified using an approximate Grad's solution of the Boltzmann kinetic equation.

The linear integral equations defining the Navier-Stokes (NS) transport coefficients for polydisperse granular mixtures of smooth inelastic hard disks or spheres are solved by using the leading terms in a Sonine polynomial expansion. Explicit expressions for all the NS transport coefficients are given in terms of the sizes, masses, compositions, density and restitution coefficients. In addition, the cooling rate is also evaluated to first order in the gradients. The results hold for arbitrary degree of inelasticity and are not limited to specific values of the parameters of the mixture. Finally, a detailed comparison between the derivation of the current theory and previous theories for mixtures is made, with attention paid to the implication of the various treatments employed to date.