Dynamics of ballistic annihilation

Piasecki, J.; Trizac, Emmanuel; Droz, Michel

doi:10.1103/physreve.66.066111

Cited by 35 publications

(113 citation statements)

References 19 publications

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“…We demonstrate that for arbitrarily short but finite times the limit of the empirical density is not consistent with the mean-field theory. This shows that further assumptions are needed in the informal justification of the gainless Boltzmann equation in [PTD02]. In the proof we insert an additional layer between the single-body densities and the Nbody evolution: The probability distribution of trees which encode the collision history of the individual particles.…”

Section: V) (4)mentioning

confidence: 99%

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Validity and Failure of the Boltzmann Approximation of Kinetic Annihilation

Matthies

Theil

2009

J Nonlinear Sci

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Abstract. This paper introduces a new method to show the validity of a continuum description for the deterministic dynamics of many interacting particles. Here the many particle evolution is analyzed for a hard sphere flow with the addition that after a collision the collided particles are removed from the system. We consider random initial configurations which are drawn from a Poisson point process with spatially homogeneous velocity density f 0 (v). Assuming that the moments of order less than three of f 0 are finite and no mass is concentrated on lines, the homogeneous Boltzmann equation without gain term is derived for arbitrary long times in the Boltzmann-Grad scaling. A key element is a characterization of the many particle flow by a hierarchy of trees which encode the possible collisions. The occurring trees are shown to have favorable properties with a high probability, allowing to restrict the analysis to a finite number of interacting particles, enabling us to extract a single-body distribution. A counter-example is given for a concentrated initial density f 0 even to short-term validity.

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Section: V) (4)mentioning

confidence: 99%

“…Kinetic annihilation dynamics can be used to model growth and coarsening of surfaces, see [KS88], and has been studied extensively in the physics literature, see [EF85,Pia95,DFPR95,PTD02,CDPTW03].…”

Section: V) (4)mentioning

confidence: 99%

Validity and Failure of the Boltzmann Approximation of Kinetic Annihilation

Matthies

Theil

2009

J Nonlinear Sci

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“…Such points, that can be coined "isobestic", have already been observed in a different context (see e.g. section IV-E in reference [41]), where their occurrence could not be rationalized.…”

Section: Discussionmentioning

confidence: 99%

“…Once Γ(θ) is known from DSMC measurements, one can construct a simple graphical method for solving (41) and classifying its possible solutions for different values of θ and α. Here we discuss only the ν models with σ = 1.…”

Section: Substitution Of F (C) = δ(C)+h(c) In the Collision Kernel Ofmentioning

confidence: 99%

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Boltzmann equation for dissipative gases in homogeneous states with nonlinear friction

2007

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Combining analytical and numerical methods, we study within the framework of the homogeneous non-linear Boltzmann equation, a broad class of models relevant for the dynamics of dissipative fluids, including granular gases. We use the new method presented in a previous paper [J. Stat. Phys. 124, 549 (2006)] and extend our results to a different heating mechanism, namely a deterministic non-linear friction force. We derive analytically the high energy tail of the velocity distribution and compare the theoretical predictions with high precision numerical simulations. Stretched exponential forms are obtained when the non-equilibrium steady state is stable. We derive sub-leading corrections and emphasize their relevance. In marginal stability cases, power-law behaviors arise, with exponents obtained as the roots of transcendental equations. We also consider some simple BGK (Bhatnagar, Gross, Krook) models, driven by similar heating devices, to test the robustness of our predictions.

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Kinetics of Fragmenting Freely Evolving Granular Gases

Pagonabarraga

Trizac

2003

Granular Gas Dynamics

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We investigate the effect of fragmentation on the homogeneous free cooling of inelastic hard spheres, using Boltzmann kinetic theory and Direct Monte Carlo simulations. We analyze in detail a model where dissipative collisions may subsequently lead to a break-up of the grains. With a given probability, two off-springs are then created from one of the two colliding partners, with conservation of mass, momentum and kinetic energy. We observe a scaling regime characterized by a single collisional average, that quantifies the deviations from Gaussian behaviour for the joint size and velocity distribution function. We also discuss the possibility of a catastrophe whereby the number of particles diverges in a finite time. This phenomenon appears correlated to a ``shattering'' transition marked by a delta singularity at vanishingly small grains for the rescaled size distribution.Comment: 22 pages, 8 figure

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Dynamics of ballistic annihilation

Cited by 35 publications

References 19 publications

Validity and Failure of the Boltzmann Approximation of Kinetic Annihilation

Validity and Failure of the Boltzmann Approximation of Kinetic Annihilation

Boltzmann equation for dissipative gases in homogeneous states with nonlinear friction

Kinetics of Fragmenting Freely Evolving Granular Gases

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