Hydrodynamics of probabilistic ballistic annihilation

Coppex, François; Droz, Michel; Trizac, Emmanuel

doi:10.1103/physreve.70.061102

Cited by 11 publications

(43 citation statements)

References 41 publications

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“…In order for the transport coefficients to reach their τ → ∞ limit faster than any of the hydrodynamic time scales, we need moreover the more stringent condition that the fastest kinetic mode is at least separated by a pζ T gap from λ 2 = −pζ T − 2pζ n : under this condition, the time dependence of the exponential term in the integral giving F 2 is fast enough so that the transport coefficients, that depend on the F i functions through (72)-(74), can be considered as constants on the hydrodynamic time scale. With this proviso in mind, it is possible to set τ → ∞ in the integrals (75)-(77) and the time-independent transport coefficients obtained in this section are then equivalent to those calculated in reference [12] by the Chapman-Enskog method. We recall in Appendix A their expressions in the first order Sonine approximation.…”

Section: Linear Hydrodynamic Equations In Navier-stokes Ordermentioning

confidence: 99%

“…(59) of Ref. [12], where the analysis amounts to overlooking the time dependence of the mean free path, so that all entries of the hydrodynamic matrix exhibit the same time dependence. The different time dependences present in Eq.…”

Section: Linear Hydrodynamic Equations In Navier-stokes Ordermentioning

confidence: 99%

“…By taking moments in the Boltzmann equation and using the scaling (9), it can be seen that the homogeneous density and temperature obey the equations [12] ∂n H (t) ∂t = −pν H (t)ζ n n H (t),…”

Section: The Boltzmann Equation Approach To the Homogeneous Decaymentioning

confidence: 99%

“…B) Evolution of δu k⊥ (τ )/δu k⊥ (0) as a function of n 2 (0)/n 2 (τ ). n 2 (0)/n 2 (τ ) = 400 corresponds to τ ≃ 3. standard tools of kinetic theory (here, the first Sonine approximation [12], see Appendix A). An excellent agreement is obtained without any adjustable parameter, including the predicted increase of the perturbation at short times, and as also observed for a larger annihilation probability p = 0.5 (see Fig.…”

Section: B)mentioning

confidence: 99%

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Dynamics of annihilation. I. Linearized Boltzmann equation and hydrodynamics

et al. 2008

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We study the non-equilibrium statistical mechanics of a system of freely moving particles, in which binary encounters lead either to an elastic collision or to the disappearance of the pair. Such a system of ballistic annihilation therefore constantly looses particles. The dynamics of perturbations around the free decay regime is investigated from the spectral properties of the linearized Boltzmann operator, that characterize linear excitations on all time scales. The linearized Boltzmann equation is solved in the hydrodynamic limit by a projection technique, which yields the evolution equations for the relevant coarse-grained fields and expressions for the transport coefficients. We finally present the results of Molecular Dynamics simulations that validate the theoretical predictions.

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Section: Linear Hydrodynamic Equations In Navier-stokes Ordermentioning

confidence: 99%

Section: Linear Hydrodynamic Equations In Navier-stokes Ordermentioning

confidence: 99%

Section: The Boltzmann Equation Approach To the Homogeneous Decaymentioning

confidence: 99%

Section: B)mentioning

confidence: 99%

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Dynamics of annihilation. I. Linearized Boltzmann equation and hydrodynamics

et al. 2008

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“…However, it is possible to consider models in which these quantities are no longer conserved. An example is the case of probabilistic ballistic annihilation, in which particles either collide elastically or annihilate [105,106,107]. This annihilation process makes the number of particles non-conserved, which in turn implies a nonconservation of momentum and energy.…”

Section: Driven Granular Gasmentioning

confidence: 99%

Theoretical approaches to the steady-state statistical physics of interacting dissipative units

Bertin¹

2017

J. Phys. A: Math. Theor.

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Abstract. The aim of this review is to provide a concise overview of some of the generic approaches that have been developed to deal with the statistical description of large systems of interacting dissipative 'units'. The latter notion includes, e.g., inelastic grains, active or self-propelled particles, bubbles in a foam, low-dimensional dynamical systems like driven oscillators, or even spatially extended modes like Fourier modes of the velocity field in a fluid for instance. We first review methods based on the statistical properties of a single unit, starting with elementary mean-field approximations, either static or dynamic, that describe a unit embedded in a 'selfconsistent' environment. We then discuss how this basic mean-field approach can be extended to account for spatial dependences, in the form of space-dependent mean-field Fokker-Planck equations for example. We also briefly review the use of kinetic theory in the framework of the Boltzmann equation, which is an appropriate description for dilute systems. We then turn to descriptions in terms of the full N -body distribution, starting from exact solutions of one-dimensional models, using a Matrix Product Ansatz method when correlations are present. Since exactly solvable models are scarce, we also present some approximation methods that can be used to determine the N -body distribution in a large system of dissipative units. These methods include the Edwards approach for dense granular matter and the approximate treatment of multiparticle Langevin equations with coloured noise, which models systems of self-propelled particles. Throughout this review, emphasis is put on methodological aspects of the statistical modeling and on formal similarities between different physical problems, rather than on the specific behaviour of a given system.

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A Kac Model for Kinetic Annihilation

2020

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In this paper we consider the stochastic dynamics of a finite system of particles in a finite volume (Kac-like particle system) which annihilate with probability α ∈ (0, 1) or collide elastically with probability 1 − α. We first establish the well-posedness of the particle system which exhibits no conserved quantities. We rigorously prove that, in some thermodynamic limit, a suitable hierarchy of kinetic equations is recovered for which tensorized solution to the homogenous Boltzmann with annihilation is a solution. For bounded collision kernels, this shows in particular that propagation of chaos holds true. Furthermore, we make conjectures about the limit behaviour of the particle system when hard-sphere interactions are taken into account.

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Hydrodynamics of probabilistic ballistic annihilation

Cited by 11 publications

References 41 publications

Dynamics of annihilation. I. Linearized Boltzmann equation and hydrodynamics

Dynamics of annihilation. I. Linearized Boltzmann equation and hydrodynamics

Theoretical approaches to the steady-state statistical physics of interacting dissipative units

A Kac Model for Kinetic Annihilation

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