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

Quasi-T\"oplitz functions in KAM theorem

We define and describe the class of Quasi-T\"oplitz functions. We then prove an abstract KAM theorem where the perturbation is in this class. We apply this theorem to a Non-Linear-Scr\"odinger equation on the torus $T^d$, thus proving existence and stability of quasi-periodic solutions and recovering the results of [10]. With respect to that paper we consider only the NLS which preserves the total Momentum and exploit this conserved quantity in order to simplify our treatment.Comment: 34 pages, 1 figur

KAM for the quantum harmonic oscillator

In this paper we prove an abstract KAM theorem for infinite dimensional Hamiltonians systems. This result extends previous works of S.B. Kuksin and J. P\"oschel and uses recent techniques of H. Eliasson and S.B. Kuksin. As an application we show that some 1D nonlinear Schr\"odinger equations with harmonic potential admits many quasi-periodic solutions. In a second application we prove the reducibility of the 1D Schr\"odinger equations with the harmonic potential and a quasi periodic in time potential.Comment: 54 pages. To appear in Comm. Math. Phy

Linear Response for Granular Fluids

The linear response of an isolated, homogeneous granular fluid to small spatial perturbations is studied by methods of non-equilibrium statistical mechanics. The long wavelength linear hydrodynamic equations are obtained, with formally exact expressions for the susceptibilities and transport coefficients. The latter are given in equivalent Einstein-Helfand and Green-Kubo forms. The context of these results and their contrast with corresponding results for normal fluids are discussed.Comment: Submitted to PR

Movers and shakers: Granular damping in microgravity

The response of an oscillating granular damper to an initial perturbation is studied using experiments performed in microgravity and granular dynamics mulations. High-speed video and image processing techniques are used to extract experimental data. An inelastic hard sphere model is developed to perform simulations and the results are in excellent agreement with the experiments. The granular damper behaves like a frictional damper and a linear decay of the amplitude is bserved. This is true even for the simulation model, where friction forces are absent. A simple expression is developed which predicts the optimal damping conditions for a given amplitude and is independent of the oscillation frequency and particle inelasticities.Comment: 9 pages, 9 figure

KAM \`{a} la R

Recently R\"ussmann proposed a new new variant of KAM theory based on a slowly converging iteration scheme. It is the purpose of this note to make this scheme accessible in an even simpler setting, namely for analytic perturbations of constant vector fields on a torus. As a side effect the result may be the shortest complete KAM proof for perturbations of integrable vector fields available so far.Comment: 11 pages, version 2.

Avalanche statistics of sand heaps

Large scale computer simulations are presented to investigate the avalanche statistics of sand piles using molecular dynamics. We could show that different methods of measurement lead to contradicting conclusions, presumably due to avalanches not reaching the end of the experimental table.Comment: 6 pages, 4 figure

Towards a continuum theory of clustering in a freely cooling inelastic gas

We performed molecular dynamics simulations to investigate the clustering instability of a freely cooling dilute gas of inelastically colliding disks in a quasi-one-dimensional setting. We observe that, as the gas cools, the shear stress becomes negligibly small, and the gas flows by inertia only. Finite-time singularities, intrinsic in such a flow, are arrested only when close-packed clusters are formed. We observe that the late-time dynamics of this system are describable by the Burgers equation with vanishing viscosity, and predict the long-time coarsening behavior.Comment: 7 pages, 5 eps figures, to appear in Europhys. Let

Angular velocity distribution of a granular planar rotator in a thermalized bath

The kinetics of a granular planar rotator with a fixed center undergoing inelastic collisions with bath particles is analyzed both numerically and analytically by means of the Boltzmann equation. The angular velocity distribution evolves from quasi-gaussian in the Brownian limit to an algebraic decay in the limit of an infinitely light particle. In addition, we compare this model with a planar rotator with a free center. We propose experimental tests that might confirm the predicted behaviors.Comment: 10 Pages, 9 Figure

A note on the violation of the Einstein relation in a driven moderately dense granular gas

The Einstein relation for a driven moderately dense granular gas in $d$-dimensions is analyzed in the context of the Enskog kinetic equation. The Enskog equation neglects velocity correlations but retains spatial correlations arising from volume exclusion effects. As expected, there is a breakdown of the Einstein relation $\epsilon=D/(T_0\mu)\neq 1$ relating diffusion $D$ and mobility $\mu$, $T_0$ being the temperature of the impurity. The kinetic theory results also show that the violation of the Einstein relation is only due to the strong non-Maxwellian behavior of the reference state of the impurity particles. The deviation of $\epsilon$ from unity becomes more significant as the solid volume fraction and the inelasticity increase, especially when the system is driven by the action of a Gaussian thermostat. This conclusion qualitatively agrees with some recent simulations of dense gases [Puglisi {\em et al.}, 2007 {\em J. Stat. Mech.} P08016], although the deviations observed in computer simulations are more important than those obtained here from the Enskog kinetic theory. Possible reasons for the quantitative discrepancies between theory and simulations are discussed.Comment: 6 figure