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Binary particle collisions with mass exchange

Pierre Degond, Jian-Guo Liu

Abstract

We investigate a kinetic model for interacting particles whose masses are integer multiples of an elementary mass. These particles undergo binary collisions which preserve momentum and energy but during which some number of elementary masses can be exchanged between the particles. We derive a Boltzmann collision operator for such collisions and study its conservation properties. Under some adequate assumptions on the collision rates, we show that it satisfies a H-theorem and exhibit its equilibria. We formally derive the system of fluid equations that arises from the hydrodynamic limit of this Boltzmann equation. We compute the viscous corrections to the leading order hydrodynamic equations on a simplified collision operator of BGK type. We show that this diffusive system can be put in the formalism of nonequilibrium thermodynamics. In particular, it satisfies Onsager's reciprocity relation and entropy decay.

Binary particle collisions with mass exchange

Abstract

We investigate a kinetic model for interacting particles whose masses are integer multiples of an elementary mass. These particles undergo binary collisions which preserve momentum and energy but during which some number of elementary masses can be exchanged between the particles. We derive a Boltzmann collision operator for such collisions and study its conservation properties. Under some adequate assumptions on the collision rates, we show that it satisfies a H-theorem and exhibit its equilibria. We formally derive the system of fluid equations that arises from the hydrodynamic limit of this Boltzmann equation. We compute the viscous corrections to the leading order hydrodynamic equations on a simplified collision operator of BGK type. We show that this diffusive system can be put in the formalism of nonequilibrium thermodynamics. In particular, it satisfies Onsager's reciprocity relation and entropy decay.
Paper Structure (15 sections, 19 theorems, 220 equations)

This paper contains 15 sections, 19 theorems, 220 equations.

Table of Contents

  1. Introduction
  2. Boltzmann operator for mass exchange collisions
  3. The collision rule
  4. The Boltzmann operator with mass exchange (BME operator)
  5. Conservations, entropy dissipation and equilibria
  6. Hydrodynamics of mass exchange processes
  7. The Euler system for mass exchange processes
  8. The Navier-Stokes mass-exchange model
  9. Entropy and thermodynamics
  10. Kinetic entropy
  11. Thermodynamic entropy
  12. Onsager's symmetry
  13. Entropy dissipation
  14. Conclusion
  15. Remarkable formulas

Key Result

Lemma 2.1

We have with

Theorems & Definitions (24)

  • Lemma 2.1: BME operator in strong form
  • Lemma 2.2: Equivalent weak form of the BME operator
  • Proposition 2.3: Conservations
  • Lemma 2.4: Collisional invariants
  • Proposition 2.5: Entropy dissipation
  • Remark 2.1
  • Proposition 2.6: Equilibria
  • Theorem 3.1: EME system in conservative form
  • Proposition 3.2: EME system in nonconservative form
  • Lemma 3.3: Positivity of $\nu$, $\kappa$, $\mu$
  • ...and 14 more