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

Simulation of tropospheric chemistry and aerosols with the climate model EC-Earth

We have integrated the atmospheric chemistry and transport model TM5 into the global climate model EC-Earth version 2.4. We present an overview of the TM5 model and the two-way data exchange between TM5 and the IFS model from the European Centre for Medium-Range Weather Forecasts (ECMWF), the atmospheric general circulation model of EC-Earth. In this paper we evaluate the simulation of tropospheric chemistry and aerosols in a one-way coupled configuration. We have carried out a decadal simulation for present-day conditions and calculated chemical budgets and climatologies of tracer concentrations and aerosol optical depth. For comparison we have also performed offline simulations driven by meteorological fields from ECMWF's ERA-Interim reanalysis and output from the EC-Earth model itself. Compared to the offline simulations, the online-coupled system produces more efficient vertical mixing in the troposphere, which reflects an improvement of the treatment of cumulus convection. The chemistry in the EC-Earth simulations is affected by the fact that the current version of EC-Earth produces a cold bias with too dry air in large parts of the troposphere. Compared to the ERA-Interim driven simulation, the oxidizing capacity in EC-Earth is lower in the tropics and higher in the extratropics. The atmospheric lifetime of methane in EC-Earth is 9.4 years, which is 7% longer than the lifetime obtained with ERA-Interim but remains well within the range reported in the literature. We further evaluate the model by comparing the simulated climatologies of surface radon-222 and carbon monoxide, tropospheric and surface ozone, and aerosol optical depth against observational data. The work presented in this study is the first step in the development of EC-Earth into an Earth system model with fully interactive atmospheric chemistry and aerosols

Exact steady state solution of the Boltzmann equation: A driven 1-D inelastic Maxwell gas

The exact nonequilibrium steady state solution of the nonlinear Boltzmann equation for a driven inelastic Maxwell model was obtained by Ben-Naim and Krapivsky [Phys. Rev. E 61, R5 (2000)] in the form of an infinite product for the Fourier transform of the distribution function $f(c)$. In this paper we have inverted the Fourier transform to express $f(c)$ in the form of an infinite series of exponentially decaying terms. The dominant high energy tail is exponential, $f(c)\simeq A_0\exp(-a|c|)$, where $a\equiv 2/\sqrt{1-\alpha^2}$ and the amplitude $A_0$ is given in terms of a converging sum. This is explicitly shown in the totally inelastic limit ($\alpha\to 0$) and in the quasi-elastic limit ($\alpha\to 1$). In the latter case, the distribution is dominated by a Maxwellian for a very wide range of velocities, but a crossover from a Maxwellian to an exponential high energy tail exists for velocities $|c-c_0|\sim 1/\sqrt{q}$ around a crossover velocity $c_0\simeq \ln q^{-1}/\sqrt{q}$, where $q\equiv (1-\alpha)/2\ll 1$. In this crossover region the distribution function is extremely small, $\ln f(c_0)\simeq q^{-1}\ln q$.Comment: 11 pages, 4 figures; a table and a few references added; to be published in PR

Collision statistics of driven granular materials

We present an experimental investigation of the statistical properties of spherical granular particles on an inclined plane that are excited by an oscillating side-wall. The data is obtained by high-speed imaging and particle tracking techniques. We identify all particles in the system and link their positions to form trajectories over long times. Thus, we identify particle collisions to measure the effective coefficient of restitution and find a broad distribution of values for the same impact angles. We find that the energy inelasticity can take on values greater than one, which implies that the rotational degrees play an important role in energy transfer. We also measure the distance and the time between collision events in order to directly determine the distribution of path lengths and the free times. These distributions are shown to deviate from expected theoretical forms for elastic spheres, demonstrating the inherent clustering in this system. We describe the data with a two-parameter fitting function and use it to calculated the mean free path and collision time. We find that the ratio of these values is consistent with the average velocity. The velocity distribution are observed to be strongly non-Gaussian and do not demonstrate any apparent universal behavior. We report the scaling of the second moment, which corresponds to the granular temperature, and higher order moments as a function of distance from the driving wall. Additionally, we measure long time correlation functions in both space and in the velocities to probe diffusion in a dissipative gas.Comment: 12 pages, 4 figures, uses revtex

Hydrodynamic theory for granular gases

A granular gas subjected to a permanent injection of energy is described by means of hydrodynamic equations derived from a moment expansion method. The method uses as reference function not a Maxwellian distribution $f_{\sf M}$ but a distribution $f_0 = \Phi f_{\sf M}$, such that $\Phi$ adds a fourth cumulant $\kappa$ to the velocity distribution. The formalism is applied to a stationary conductive case showing that the theory fits extraordinarily well the results coming from our molecular dynamic simulations once we determine $\kappa$ as a function of the inelasticity of the particle-particle collisions. The shape of $\kappa$ is independent of the size $N$ of the system.Comment: 10 pages, 9 figures, more about our research in http://www.cec.uchile.cl/cinetica

Granular clustering in a hydrodynamic simulation

We present a numerical simulation of a granular material using hydrodynamic equations. We show that, in the absence of external forces, such a system phase-separates into high density and low density regions. We show that this separation is dependent on the inelasticity of collisions, and comment on the mechanism for this clustering behavior. Our results are compatible with the granular clustering seen in experiments and molecular dynamic simulations of inelastic hard disks.Comment: 4 pages, 5 figure

Granular fluid thermostatted by a bath of elastic hard spheres

The homogeneous steady state of a fluid of inelastic hard spheres immersed in a bath of elastic hard spheres kept at equilibrium is analyzed by means of the first Sonine approximation to the (spatially homogeneous) Enskog--Boltzmann equation. The temperature of the granular fluid relative to the bath temperature and the kurtosis of the granular distribution function are obtained as functions of the coefficient of restitution, the mass ratio, and a dimensionless parameter $\beta$ measuring the cooling rate relative to the friction constant. Comparison with recent results obtained from an iterative numerical solution of the Enskog--Boltzmann equation [Biben et al., Physica A 310, 308 (202)] shows an excellent agreement. Several limiting cases are also considered. In particular, when the granular particles are much heavier than the bath particles (but have a comparable size and number density), it is shown that the bath acts as a white noise external driving. In the general case, the Sonine approximation predicts the lack of a steady state if the control parameter $\beta$ is larger than a certain critical value $\beta_c$ that depends on the coefficient of restitution and the mass ratio. However, this phenomenon appears outside the expected domain of applicability of the approximation.Comment: 16 pages, 7 figures; minor changes; to be published in Phys. Rev.

Dynamics and stress in gravity driven granular flow

We study, using simulations, the steady-state flow of dry sand driven by gravity in two-dimensions. An investigation of the microscopic grain dynamics reveals that grains remain separated but with a power-law distribution of distances and times between collisions. While there are large random grain velocities, many of these fluctuations are correlated across the system and local rearrangements are very slow. Stresses in the system are almost entirely transfered by collisions and the structure of the stress tensor comes almost entirely from a bias in the directions in which collisions occur.Comment: 4 pages, 3 eps figures, RevTe

Velocity Fluctuations in Electrostatically Driven Granular Media

We study experimentally the particle velocity fluctuations in an electrostatically driven dilute granular gas. The experimentally obtained velocity distribution functions have strong deviations from Maxwellian form in a wide range of parameters. We have found that the tails of the distribution functions are consistent with a stretched exponential law with typical exponents of the order 3/2. Molecular dynamic simulations shows qualitative agreement with experimental data. Our results suggest that this non-Gaussian behavior is typical for most inelastic gases with both short and long range interactions.Comment: 4 pages, 4 figure