Ballistic-annihilation kinetics for a multivelocity one-dimensional ideal gas

Droz, Michel; Rey, P.; Frachebourg, L.; Piasecki, J.

doi:10.1103/physreve.51.5541

Cited by 37 publications

(48 citation statements)

References 4 publications

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“…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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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%

Validity and Failure of the Boltzmann Approximation of Kinetic Annihilation

Matthies

Theil

2009

J Nonlinear Sci

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“…However, the case that will interest us in this section is that of random initial conditions whereby particles are distributed upon a line with each particles' velocity drawn from the same distribution. A broad variety of initial-velocity distributions have been studied [8,28,29] and universal effects analyzed for the continuous case [30]. However, in these cases the systems that have been examined have been infinite and translationally invariant.…”

Section: Ballistic Annihilation Near a Boundary In One Dimensionmentioning

confidence: 99%

Reaction-diffusion and ballistic annihilation near an impenetrable boundary

Kafri¹,

Richardson²

1999

J. Phys. A: Math. Gen.

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The behavior of the single-species reaction process A + A → O is examined near an impenetrable boundary, representing the flask containing the reactants. Two types of dynamics are considered for the reactants: diffusive and ballistic propagation. It is shown that the effect of the boundary is quite different in both cases: diffusion-reaction leads to a density excess, whereas ballistic annihilation exhibits a density deficit, and in both cases the effect is not localized at the boundary but penetrates into the system. The field-theoretic renormalization group is used to obtain the universal properties of the density excess in two dimensions and below for the reactiondiffusion system. In one dimension the excess decays with the same exponent as the bulk and is found by an exact solution. In two dimensions the excess is marginally less relevant than the bulk decay and the density profile is again found exactly for late times from the RG-improved field theory. The results obtained for the diffusive case are relevant for Mg 2+ or Cd 2+ doping in the TMMC crystal's exciton coalescence process and also imply a surprising result for the dynamic magnetization in the critical one-dimensional Ising model with a fixed spin. For the case of ballistic reactants, a model is introduced and solved exactly in one dimension. The density-deficit profile is obtained, as is the density of left and right moving reactants near the impenetrable boundary.

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“…In contrast, when µ → −1, the size remains roughly constant, since the velocity distribution becomes effectively unimodal and collisions become exceedingly rare. This qualitative dependence on the form of the initial velocity distribution is reminiscent of ballistic annihilation processes, where ballistically moving particles annihilate upon collision [35][36][37][38]. The above clustering process can be viewed as a ballistic aggregation process that possesses a single mass conservation law.…”

Section: The Velocity Distributionmentioning

confidence: 99%

Kinetic Theory of Traffic Flows

Ben‐Naim

Krapivsky

2003

Traffic and Granular Flow’01

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Abstract. We describe traffic flows in one lane roadways using kinetic theory, with special emphasis on the role of quenched randomness in the velocity distributions. When passing is forbidden, growing clusters are formed behind slow cars and the cluster velocity distribution is governed by an exact Boltzmann equation which is linear and has an infinite memory. The distributions of the cluster size and the cluster velocity exhibit scaling behaviors, with exponents dominated solely by extremal characteristics of the intrinsic velocity distribution. When passing is allowed, the system approaches a steady state, whose nature is determined by a single dimensionless number, the ratio of the passing time to the collision time, the two time scales in the problem. The flow exhibits two regimes, a laminar flow regime, and a congested regime where large slow clusters dominate the flow. A phase transition separates these two regimes when only the next-to-leading car can pass.

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Ballistic-annihilation kinetics for a multivelocity one-dimensional ideal gas

Cited by 37 publications

References 4 publications

Validity and Failure of the Boltzmann Approximation of Kinetic Annihilation

Validity and Failure of the Boltzmann Approximation of Kinetic Annihilation

Reaction-diffusion and ballistic annihilation near an impenetrable boundary

Kinetic Theory of Traffic Flows

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