Ballistic aggregation: a solvable model of irreversible many particles dynamics

The adhesive dynamics of a one-dimensional aggregating gas of point particles is rigorously described. The infinite hierarchy of kinetic equations for the distributions of clusters of nearest neighbours is shown to be equivalent to a system of two coupled equations for a large class of initial conditions. The solution to these nonlinear equations is found by a direct construction of the relevant probability distributions in the limit of a continuous initial mass distribution. We show that those limiting distributions are identical to those of the statistics of shocks in the Burgers turbulence. The analysis relies on a mapping on a Brownian motion problem with parabolic constraints.Comment: 23 pages, 6 figures include

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

Granular Rough Sphere in a Low-Density Thermal Bath

We study the stationary state of a rough granular sphere immersed in a thermal bath composed of point particles. When the center of mass of the sphere is fixed the stationary angular velocity distribution is shown to be Gaussian with an effective temperature lower than that of the bath. For a freely moving rough sphere coupled to the thermostat via inelastic collisions we find a condition under which the joint distribution of the translational and rotational velocities is a product of Gaussian distributions with the same effective temperature. In this rather unexpected case we derive a formula for the stationary energy flow from the thermostat to the sphere in accordance with Fourier law

Prediction of a Structural Transition in the Hard Disk Fluid

Starting from the second equilibrium equation in the BBGKY hierarchy under the Kirkwood superposition closure, we implement a new method for studying the asymptotic decay of correlations in the hard disk fluid in the high density regime. From our analysis and complementary numerical studies, we find that exponentially damped oscillations can occur only up to a packing fraction {\eta}*~0.718, a value which is in substantial agreement with the packing fraction, {\eta}~0.723, believed to characterize the transition from the ordered solid phase to a dense fluid phase, as inferred from Mak's Monte Carlo simulations [Phys. Rev. E 73, 065104 (2006)]. We next show that the same method of analysis predicts that exponential damping of oscillations in the hard sphere fluid becomes impossible when \lambda = 4n\pi {\sigma}^3 [1 + H(1)]>/- 34.81, where H(1) is the contact value of the correlation function, n is the number density and {\sigma} is the sphere diameter, in exact agreement with the condition, \lambda >/- 34.8, first reported in a numerical study of the Kirkwood equation by Kirkwood et al. [J. Chem. Phys. 18, 1040 (1950)]. Finally, we show that our method confirms the absence of any structural transition in hard rods for the entire range of densities below close packing.Comment: to be published in J. Chem. Phy

Power law in the angular velocity distribution of a granular needle

We show how inelastic collisions induce a power law with exponent -3 in the decay of the angular velocity distribution of anisotropic particles with sufficiently small moment of inertia. We investigate this question within the Boltzmann kinetic theory for an elongated granular particle immersed in a bath. The power law persists so long as the collisions are inelastic for a large range of angular velocities provided the mass ratio of the anisotropic particle and the bath particles remains small. Suggestions for observing this peculiar feature are made.Comment: 8 pages, 4 figure

Ballistic annihilation kinetics for a multi-velocity one-dimensional ideal gas

Ballistic annihilation kinetics for a multi-velocity one-dimensional ideal gas is studied in the framework of an exact analytic approach. For an initial symmetric three-velocity distribution, the problem can be solved exactly and it is shown that different regimes exist depending on the initial fraction of particles at rest. Extension to the case of a n-velocity distribution is discussed.Comment: 19 pages, latex, uses Revtex macro

Search for universality in one-dimensional ballistic annihilation kinetics

We study the kinetics of ballistic annihilation for a one-dimensional ideal gas with continuous velocity distribution. A dynamical scaling theory for the long time behavior of the system is derived. Its validity is supported by extensive numerical simulations for several velocity distributions. This leads us to the conjecture that all the continuous velocity distributions \phi(v) which are symmetric, regular and such that \phi(0) does not vanish, are attracted in the long time regime towards the same Gaussian distribution and thus belong to the same universality class. Moreover, it is found that the particle density decays as n(t)~t^{-\alpha}, with \alpha=0.785 +/- 0.005.Comment: 8 pages, needs multicol, epsf and revtex. 8 postscript figures included. Submitted to Phys. Rev. E. Also avaiable at http://mykonos.unige.ch/~rey/publi.html#Secon

Field induced stationary state for an accelerated tracer in a bath

Our interest goes to the behavior of a tracer particle, accelerated by a constant and uniform external field, when the energy injected by the field is redistributed through collision to a bath of unaccelerated particles. A non equilibrium steady state is thereby reached. Solutions of a generalized Boltzmann-Lorentz equation are analyzed analytically, in a versatile framework that embeds the majority of tracer-bath interactions discussed in the literature. These results --mostly derived for a one dimensional system-- are successfully confronted to those of three independent numerical simulation methods: a direct iterative solution, Gillespie algorithm, and the Direct Simulation Monte Carlo technique. We work out the diffusion properties as well as the velocity tails: large v, and either large -v, or v in the vicinity of its lower cutoff whenever the velocity distribution is bounded from below. Particular emphasis is put on the cold bath limit, with scatterers at rest, which plays a special role in our model.Comment: 20 pages, 6 figures v3:minor corrections in sec.III and added reference