10 research outputs found
Probabilistic ballistic annihilation with continuous velocity distributions
We investigate the problem of ballistically controlled reactions where
particles either annihilate upon collision with probability , or undergo an
elastic shock with probability . Restricting to homogeneous systems, we
provide in the scaling regime that emerges in the long time limit, analytical
expressions for the exponents describing the time decay of the density and the
root-mean-square velocity, as continuous functions of the probability and
of a parameter related to the dissipation of energy. We work at the level of
molecular chaos (non-linear Boltzmann equation), and using a systematic Sonine
polynomials expansion of the velocity distribution, we obtain in arbitrary
dimension the first non-Gaussian correction and the corresponding expressions
for the decay exponents. We implement Monte-Carlo simulations in two
dimensions, that are in excellent agreement with our analytical predictions.
For , numerical simulations lead to conjecture that unlike for pure
annihilation (), the velocity distribution becomes universal, i.e. does
not depend on the initial conditions.Comment: 10 pages, 9 eps figures include
Hydrodynamics of probabilistic ballistic annihilation
We consider a dilute gas of hard spheres in dimension that upon
collision either annihilate with probability or undergo an elastic
scattering with probability . For such a system neither mass, momentum,
nor kinetic energy are conserved quantities. We establish the hydrodynamic
equations from the Boltzmann equation description. Within the Chapman-Enskog
scheme, we determine the transport coefficients up to Navier-Stokes order, and
give the closed set of equations for the hydrodynamic fields chosen for the
above coarse grained description (density, momentum and kinetic temperature).
Linear stability analysis is performed, and the conditions of stability for the
local fields are discussed.Comment: 19 pages, 3 eps figures include
On the first Sonine correction for granular gases
We consider the velocity distribution for a granular gas of inelastic hard
spheres described by the Boltzmann equation. We investigate both the free of
forcing case and a system heated by a stochastic force. We propose a new method
to compute the first correction to Gaussian behavior in a Sonine polynomial
expansion quantified by the fourth cumulant . Our expressions are compared
to previous results and to those obtained through the numerical solution of the
Boltzmann equation. It is numerically shown that our method yields very
accurate results for small velocities of the rescaled distribution. We finally
discuss the ambiguities inherent to a linear approximation method in .Comment: 9 pages, 8 eps figures include
Stationary state of a heated granular gas: fate of the usual H-functional
We consider the characterization of the nonequilibrium stationary state of a
randomly-driven granular gas in terms of an entropy-production based
variational formulation. Enforcing spatial homogeneity, we first consider the
temporal stability of the stationary state reached after a transient. In
connection, two heuristic albeit physically motivated candidates for the
non-equilibrium entropy production are put forward. It turns out that none of
them displays an extremum for the stationary velocity distribution selected by
the dynamics. Finally, the relevance of the relative Kullbach entropy is
discussed.Comment: 17 pages, 2 figures, to be published in Physica
Sur une classe de systèmes dissipatifs hors d'équilibre
Membres du jury:Prof. Emmanuel Trizac (Université de Paris-Sud, Orsay, France)Prof. Philippe-André Martin (Ecole Polytechnique Fédérale de Lausanne, Lausanne, Suisse)We consider dissipative, out-of-equilibrium, and low density systems made of many interacting classical particles. In a first part, we study probabilistic ballistic annihilation, where particles have a ballistic trajectory except upon binary contact where they annihilate with probability p and undergo an elastic scattering with probability (1-p). We establish for this system with no conservation law a hydrodynamic description that stems from kinetic theory. The linear stability analysis of the homogeneous state shows then that the amplification of the fluctuations by the dynamics is a transient effect. In a second part, we present a mesoscopic model that reproduces a spontaneous symmetry breaking observed in some experiments on granular gases.Nous considérons des systèmes dissipatifs, hors d'équilibre, de faible densité, et constitués d'un grand nombre de particules classiques en interaction. Dans une première partie, nous étudions l'annihilation balistique probabiliste, où les particules ont une trajectoire balistique sauf lorsqu'elles entrent en contact, s'annihilant alors avec probabilité p et subissant une collision élastique avec probabilité (1-p). Nous établissons pour ce système sans loi de conservation une description hydrodynamique résultant de la théorie cinétique. L'analyse de stabilité linéaire de l'état homogène montre alors que l'amplification des fluctuations par la dynamique est un phénomène transitoire. Dans la seconde partie, nous présentons un modèle mésoscopique décrivant le phénomène de brisure spontanée de symétrie observé dans certaines expériences sur la matière granulaire vibrée
On a class of nonequilibrium dissipative systems
Nous considérons des systèmes dissipatifs, hors d'équilibre, de faible densité, et constitués d'un grand nombre de particules classiques en interaction. Dans une première partie, nous étudions l'annihilation balistique probabiliste, où les particules ont une trajectoire balistique sauf lorsqu'elles entrent en contact, s'annihilant alors avec probabilité p et subissant une collision élastique avec probabilité (1-p). Nous établissons pour ce système sans loi de conservation une description hydrodynamique résultant de la théorie cinétique. L'analyse de stabilité linéaire de l'état homogène montre alors que l'amplification des fluctuations par la dynamique est un phénomène transitoire. Dans la seconde partie, nous présentons un modèle mésoscopique décrivant le phénomène de brisure spontanée de symétrie observé dans certaines expériences sur la matière granulaire vibrée
Some exact results for Boltzmann's annihilation dynamics
The problem of ballistic annihilation for a spatially homogeneous system is revisited within Boltzmann's kinetic theory in two and three dimensions. Analytical results are derived for the time evolution of the particle density for some isotropic discrete bimodal velocity modulus distributions. According to the allowed values of the velocity modulus, different behaviors are obtained: power law decay with nonuniversal exponents depending continuously upon the ratio of the two velocities, or exponential decay. When one of the two velocities is equal to zero, the model describes the problem of ballistic annihilation in the presence of static traps. The analytical predictions are shown to be in agreement with the results of two-dimensional molecular dynamics simulations