Application of the Gillespie algorithm to a granular intruder particle

We show how the Gillespie algorithm, originally developed to describe coupled chemical reactions, can be used to perform numerical simulations of a granular intruder particle colliding with thermalized bath particles. The algorithm generates a sequence of collision ``events'' separated by variable time intervals. As input, it requires the position-dependent flux of bath particles at each point on the surface of the intruder particle. We validate the method by applying it to a one-dimensional system for which the exact solution of the homogeneous Boltzmann equation is known and investigate the case where the bath particle velocity distribution has algebraic tails. We also present an application to a granular needle in bath of point particles where we demonstrate the presence of correlations between the translational and rotational degrees of freedom of the intruder particle. The relationship between the Gillespie algorithm and the commonly used Direct Simulation Monte Carlo (DSMC) method is also discussed.Comment: 13 pages, 8 figures, to be published in J. Phys. A Math. Ge

Fluctuating lattice Boltzmann

The lattice Boltzmann algorithm efficiently simulates the Navier Stokes equation of isothermal fluid flow, but ignores thermal fluctuations of the fluid, important in mesoscopic flows. We show how to adapt the algorithm to include noise, satisfying a fluctuation-dissipation theorem (FDT) directly at lattice level: this gives correct fluctuations for mass and momentum densities, and for stresses, at all wavevectors $k$. Unlike previous work, which recovers FDT only as $k\to 0$, our algorithm offers full statistical mechanical consistency in mesoscale simulations of, e.g., fluctuating colloidal hydrodynamics.Comment: 7 pages, 3 figures, to appear in Europhysics Letter

Gaussian density fluctuations and Mode Coupling Theory for supercooled liquids

The equations of motion for the density modes of a fluid, derived from Newton's equations, are written as a linear generalized Langevin equation. The constraint imposed by the fluctuation-dissipation theorem is used to derive an exact form for the memory function. The resulting equations, solved under the assumption that the noise, and consequently density fluctuations, of the liquid are gaussian distributed, are equivalent to the random-phase-approximation for the static structure factor and to the well known ideal mode coupling theory (MCT) equations for the dynamics. This finding suggests that MCT is the canonical mean-field theory of the fluid dynamics.Comment: 4 pages, REVTE

On the velocity distributions of the one-dimensional inelastic gas

We consider the single-particle velocity distribution of a one-dimensional fluid of inelastic particles. Both the freely evolving (cooling) system and the non-equilibrium stationary state obtained in the presence of random forcing are investigated, and special emphasis is paid to the small inelasticity limit. The results are obtained from analytical arguments applied to the Boltzmann equation along with three complementary numerical techniques (Molecular Dynamics, Direct Monte Carlo Simulation Methods and iterative solutions of integro-differential kinetic equations). For the freely cooling fluid, we investigate in detail the scaling properties of the bimodal velocity distribution emerging close to elasticity and calculate the scaling function associated with the distribution function. In the heated steady state, we find that, depending on the inelasticity, the distribution function may display two different stretched exponential tails at large velocities. The inelasticity dependence of the crossover velocity is determined and it is found that the extremely high velocity tail may not be observable at ``experimentally relevant'' inelasticities.Comment: Latex, 14 pages, 12 eps figure

Enskog Theory for Polydisperse Granular Mixtures. I. Navier-Stokes order Transport

A hydrodynamic description for an $s$-component mixture of inelastic, smooth hard disks (two dimensions) or spheres (three dimensions) is derived based on the revised Enskog theory for the single-particle velocity distribution functions. In this first portion of the two-part series, the macroscopic balance equations for mass, momentum, and energy are derived. Constitutive equations are calculated from exact expressions for the fluxes by a Chapman-Enskog expansion carried out to first order in spatial gradients, thereby resulting in a Navier-Stokes order theory. Within this context of small gradients, the theory is applicable to a wide range of restitution coefficients and densities. The resulting integral-differential equations for the zeroth- and first-order approximations of the distribution functions are given in exact form. An approximate solution to these equations is required for practical purposes in order to cast the constitutive quantities as algebraic functions of the macroscopic variables; this task is described in the companion paper.Comment: 36 pages, to be published in Phys. Rev.

Diffusion in a Granular Fluid - Theory

Many important properties of granular fluids can be represented by a system of hard spheres with inelastic collisions. Traditional methods of nonequilibrium statistical mechanics are effective for analysis and description of the inelastic case as well. This is illustrated here for diffusion of an impurity particle in a fluid undergoing homogeneous cooling. An appropriate scaling of the Liouville equation is described such that the homogeneous cooling ensemble and associated time correlation functions map to those of a stationary state. In this form the familiar methods of linear response can be applied, leading to Green - Kubo and Einstein representations of diffusion in terms of the velocity and mean square displacement correlation functions. These correlation functions are evaluated approximately using a cumulant expansion and from kinetic theory, providing the diffusion coefficient as a function of the density and the restitution coefficients. Comparisons with results from molecular dynamics simulation are given in the following companion paper

Electroviscous effects of simple electrolytes under shear

On the basis of a hydrodynamical model analogous to that in critical fluids, we investigate the influences of shear flow upon the electrostatic contribution to the viscosity of binary electrolyte solutions in the Debye-H\"{u}ckel approximation. Within the linear-response theory, we reproduce the classical limiting law that the excess viscosity is proportional to the square root of the concentration of the electrolyte. We also extend this result for finite shear. An analytic expression of the anisotropic structure factor of the charge density under shear is obtained, and its deformation at large shear rates is discussed. A non-Newtonian effect caused by deformations of the ionic atmosphere is also elucidated for $\tau_D\dot{\gamma}>1$. This finding concludes that the maximum shear stress that the ionic atmosphere can support is proportional to $\lambda_D^{-3}$, where $\dot{\gamma}$, $\lambda_D$ and $\tau_D=\lambda_D^2/D$ are, respectively, the shear rate, the Debye screening length and the Debye relaxation time with $D$ being the relative diffusivity at the infinite dilution limit of the electrolyte.Comment: 13pages, 2figure

Hydrodynamic modes, Green-Kubo relations, and velocity correlations in dilute granular gases

It is shown that the hydrodynamic modes of a dilute granular gas of inelastic hard spheres can be identified, and calculated in the long wavelength limit. Assuming they dominate at long times, formal expressions for the Navier-Stokes transport coefficients are derived. They can be expressed in a form that generalizes the Green-Kubo relations for molecular systems, and it is shown that they can also be evaluated by means of $N$-particle simulation methods. The form of the hydrodynamic modes to zeroth order in the gradients is used to detect the presence of inherent velocity correlations in the homogeneous cooling state, even in the low density limit. They manifest themselves in the fluctuations of the total energy of the system. The theoretical predictions are shown to be in agreement with molecular dynamics simulations. Relevant related questions deserving further attention are pointed out

Temperature-dependent quantum pair potentials and their application to dense partially ionized hydrogen plasmas

Extending our previous work \cite{filinov-etal.jpa03ik} we present a detailed discussion of accuracy and practical applications of finite-temperature pseudopotentials for two-component Coulomb systems. Different pseudopotentials are discussed: i) the diagonal Kelbg potential, ii) the off-diagonal Kelbg potential iii) the {\em improved} diagonal Kelbg potential, iv) an effective potential obtained with the Feynman-Kleinert variational principle v) the ``exact'' quantum pair potential derived from the two-particle density matrix. For the {\em improved} diagonal Kelbg potential a simple temperature dependent fit is derived which accurately reproduces the ``exact'' pair potential in the whole temperature range. The derived pseudopotentials are then used in path integral Monte Carlo (PIMC) and molecular dynamics (MD) simulations to obtain thermodynamical properties of strongly coupled hydrogen. It is demonstrated that classical MD simulations with spin-dependent interaction potentials for the electrons allow for an accurate description of the internal energy of hydrogen in the difficult regime of partial ionization down to the temperatures of about $60 000$ K. Finally, we point out an interesting relation between the quantum potentials and effective potentials used in density functional theory.Comment: 18 pages, 11 figure

Interaction of photons with plasmas and liquid metals: photoabsorption and scattering

Formulas to describe the photoabsorption and the photon scattering by a plasma or a liquid metal are derived in a unified manner with each other. It is shown how the nuclear motion, the free-electron motion and the core-electron behaviour in each ion in the system determine the structure of photoabsorption and scattering in an electron-ion mixture. The absorption cross section in the dipole approximation consists of three terms which represent the absorption caused by the nuclear motion, the absorption owing to the free-electron motion producing optical conductivity or inverse Bremsstrahlung, and the absorption ascribed to the core-electron behaviour in each ion with the Doppler correction. Also, the photon scattering formula provides an analysis method for experiments observing the ion-ion dynamical structure factor (DSF), the electron-electron DSF giving plasma oscillations, and the core-electron DSF yielding the X-ray Raman (Compton) scattering with a clear definition of the background scattering for each experiment, in a unified manner. A formula for anomalous X-ray scattering is also derived for a liquid metal. At the same time, Thomson scattering in plasma physics is discussed from this general point of view.Comment: LaTeX file: 18 pages without figur