Critical Behavior of a Heavy Particle in a Granular Fluid

Behavior analogous to a second order phase transition is observed for the homogeneous cooling state of a heavy impurity particle in a granular fluid. The order parameter $\phi$ is the ratio of impurity mean square velocity to that of the fluid, with a conjugate field $h$ proportional to the mass ratio. A parameter $\beta$, measuring the fluid cooling rate relative to the impurity--fluid collision rate, is the analogue of the inverse temperature. For $\beta <1$ the fluid is ``normal'' with $\phi =0$ at $h=0$, as in the case of a system with elastic collisions. For $\beta >1$ an ``ordered'' state with $\phi \neq 0$ occurs at $h=0$, representing an extreme breakdown of equipartition. Critical slowing and qualitative changes in the velocity distribution function for the impurity particle near the transition are notedComment: 4 pages (4 figures included

Fluctuation-Dissipation relations in Driven Granular Gases

We study the dynamics of a 2d driven inelastic gas, by means of Direct Simulation Monte Carlo (DSMC) techniques, i.e. under the assumption of Molecular Chaos. Under the effect of a uniform stochastic driving in the form of a white noise plus a friction term, the gas is kept in a non-equilibrium Steady State characterized by fractal density correlations and non-Gaussian distributions of velocities; the mean squared velocity, that is the so-called {\em granular temperature}, is lower than the bath temperature. We observe that a modified form of the Kubo relation, which relates the autocorrelation and the linear response for the dynamics of a system {\em at equilibrium}, still holds for the off-equilibrium, though stationary, dynamics of the systems under investigation. Interestingly, the only needed modification to the equilibrium Kubo relation is the replacement of the equilibrium temperature with an effective temperature, which results equal to the global granular temperature. We present two independent numerical experiment, i.e. two different observables are studied: (a) the staggered density current, whose response to an impulsive shear is proportional to its autocorrelation in the unperturbed system and (b) the response of a tracer to a small constant force, switched on at time $t_w$, which is proportional to the mean-square displacement in the unperturbed system. Both measures confirm the validity of Kubo's formula, provided that the granular temperature is used as the proportionality factor between response and autocorrelation, at least for not too large inelasticities.Comment: 11 pages, 7 figures, submitted for publicatio

Self-diffusion in granular gases

The coefficient of self-diffusion for a homogeneously cooling granular gas changes significantly if the impact-velocity dependence of the restitution coefficient $\epsilon$ is taken into account. For the case of a constant $\epsilon$ the particles spread logarithmically slow with time, whereas the velocity dependent coefficient yields a power law time-dependence. The impact of the difference in these time dependences on the properties of a freely cooling granular gas is discussed.Comment: 6 pages, no figure

Gaussian Kinetic Model for Granular Gases

A kinetic model for the Boltzmann equation is proposed and explored as a practical means to investigate the properties of a dilute granular gas. It is shown that all spatially homogeneous initial distributions approach a universal "homogeneous cooling solution" after a few collisions. The homogeneous cooling solution (HCS) is studied in some detail and the exact solution is compared with known results for the hard sphere Boltzmann equation. It is shown that all qualitative features of the HCS, including the nature of over population at large velocities, are reproduced semi-quantitatively by the kinetic model. It is also shown that all the transport coefficients are in excellent agreement with those from the Boltzmann equation. Also, the model is specialized to one having a velocity independent collision frequency and the resulting HCS and transport coefficients are compared to known results for the Maxwell Model. The potential of the model for the study of more complex spatially inhomogeneous states is discussed.Comment: to be submitted to 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

Transport coefficients for inelastic Maxwell mixtures

The Boltzmann equation for inelastic Maxwell models is used to determine the Navier-Stokes transport coefficients of a granular binary mixture in $d$ dimensions. The Chapman-Enskog method is applied to solve the Boltzmann equation for states near the (local) homogeneous cooling state. The mass, heat, and momentum fluxes are obtained to first order in the spatial gradients of the hydrodynamic fields, and the corresponding transport coefficients are identified. There are seven relevant transport coefficients: the mutual diffusion, the pressure diffusion, the thermal diffusion, the shear viscosity, the Dufour coefficient, the pressure energy coefficient, and the thermal conductivity. All these coefficients are {\em exactly} obtained in terms of the coefficients of restitution and the ratios of mass, concentration, and particle sizes. The results are compared with known transport coefficients of inelastic hard spheres obtained analytically in the leading Sonine approximation and by means of Monte Carlo simulations. The comparison shows a reasonably good agreement between both interaction models for not too strong dissipation, especially in the case of the transport coefficients associated with the mass flux.Comment: 9 figures, to be published in J. Stat. Phy

Multi-model simulations of the impact of international shipping on Atmospheric Chemistry and Climate in 2000 and 2030

The global impact of shipping on atmospheric chemistry and radiative forcing, as well as the associated uncertainties, have been quantified using an ensemble of ten state-of-the-art atmospheric chemistry models and a predefined set of emission data. The analysis is performed for present-day conditions ( year 2000) and for two future ship emission scenarios. In one scenario ship emissions stabilize at 2000 levels; in the other ship emissions increase with a constant annual growth rate of 2.2% up to 2030 ( termed the "Constant Growth Scenario" (CGS)). Most other anthropogenic emissions follow the IPCC ( Intergovernmental Panel on Climate Change) SRES ( Special Report on Emission Scenarios) A2 scenario, while biomass burning and natural emissions remain at year 2000 levels. An intercomparison of the model results with observations over the Northern Hemisphere (25 degrees - 60 degrees N) oceanic regions in the lower troposphere showed that the models are capable to reproduce ozone (O-3) and nitrogen oxides (NOx= NO+ NO2) reasonably well, whereas sulphur dioxide (SO2) in the marine boundary layer is significantly underestimated. The most pronounced changes in annual mean tropospheric NO2 and sulphate columns are simulated over the Baltic and North Seas. Other significant changes occur over the North Atlantic, the Gulf of Mexico and along the main shipping lane from Europe to Asia, across the Red and Arabian Seas. Maximum contributions from shipping to annual mean near-surface O-3 are found over the North Atlantic ( 5 - 6 ppbv in 2000; up to 8 ppbv in 2030). Ship contributions to tropospheric O3 columns over the North Atlantic and Indian Oceans reach 1 DU in 2000 and up to 1.8 DU in 2030. Tropospheric O-3 forcings due to shipping are 9.8 +/- 2.0 mW/m(2) in 2000 and 13.6 +/- 2.3 mW/m(2) in 2030. Whilst increasing O-3, ship NOx simultaneously enhances hydroxyl radicals over the remote ocean, reducing the global methane lifetime by 0.13 yr in 2000, and by up to 0.17 yr in 2030, introducing a negative radiative forcing. The models show future increases in NOx and O-3 burden which scale almost linearly with increases in NOx emission totals. Increasing emissions from shipping would significantly counteract the benefits derived from reducing SO2 emissions from all other anthropogenic sources under the A2 scenario over the continents, for example in Europe. Globally, shipping contributes 3% to increases in O-3 burden between 2000 and 2030, and 4.5% to increases in sulphate under A2/CGS. However, if future ground based emissions follow a more stringent scenario, the relative importance of ship emissions will increase. Inter-model differences in the simulated O-3 contributions from ships are significantly smaller than estimated uncertainties stemming from the ship emission inventory, mainly the ship emission totals, the distribution of the emissions over the globe, and the neglect of ship plume dispersion

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

Molecular dynamics simulations of vibrated granular gases

We present molecular dynamics simulations of mono- or bidisperse inelastic granular gases driven by vibrating walls, in two dimensions (without gravity). Because of the energy injection at the boundaries, a situation often met experimentally, density and temperature fields display heterogeneous profiles in the direction perpendicular to the walls. A general equation of state for an arbitrary mixture of fluidized inelastic hard spheres is derived and successfully tested against numerical data. Single-particle velocity distribution functions with non-Gaussian features are also obtained, and the influence of various parameters (inelasticity coefficients, density...) analyzed. The validity of a recently proposed Random Restitution Coefficient model is assessed through the study of projected collisions onto the direction perpendicular to that of energy injection. For the binary mixture, the non-equipartition of translational kinetic energy is studied and compared both to experimental data and to the case of homogeneous energy injection (``stochastic thermostat''). The rescaled velocity distribution functions are found to be very similar for both species

Transport Coefficients for Granular Media from Molecular Dynamics Simulations

Under many conditions, macroscopic grains flow like a fluid; kinetic theory pred icts continuum equations of motion for this granular fluid. In order to test the theory, we perform event driven molecular simulations of a two-dimensional gas of inelastic hard disks, driven by contact with a heat bath. Even for strong dissipation, high densities, and small numbers of particles, we find that continuum theory describes the system well. With a bath that heats the gas homogeneously, strong velocity correlations produce a slightly smaller energy loss due to inelastic collisions than that predicted by kinetic theory. With an inhomogeneous heat bath, thermal or velocity gradients are induced. Determination of the resulting fluxes allows calculation of the thermal conductivity and shear viscosity, which are compared to the predictions of granular kinetic theory, and which can be used in continuum modeling of granular flows. The shear viscosity is close to the prediction of kinetic theory, while the thermal conductivity can be overestimated by a factor of 2; in each case, transport is lowered with increasing inelasticity.Comment: 14 pages, 17 figures, 39 references, submitted to PRE feb 199