Rigorous derivation of a binary-ternary Boltzmann equation for a dense gas of hard spheres

Abstract

This paper provides the first rigorous derivation of a binary-ternary Boltzmann equation describing the kinetic properties of a dense hard-spheres gas, where particles undergo either binary or ternary instantaneous interactions, while preserving momentum and energy. An important challenge we overcome in deriving this equation is related to providing a mathematical framework that allows us to detect both binary and ternary interactions. Furthermore, this paper introduces new algebraic and geometric techniques in order to eventually decouple binary and ternary interactions and understand the way they could succeed one another in time. IOAKEIM AMPATZOGLOU AND NATA ŠA PAVLOVI Ć 8. Geometric estimates 39 8.1. Cylinder-Sphere estimates 40 8.2. Estimates relying on the (2d − 1)-sphere representation 40 8.2.1. Truncation of impact directions 40 8.2.2. Conic estimates 41 8.2.3. Annuli estimates 42 9. Good configurations and stability 45 9.1. Adjunction of new particles 45 9.2. Stability under binary adjunction 46 9.2.1. Binary adjunction 46 9.2.2. Measure estimate for binary adjunction 48 9.3. Stability under ternary adjunction 51 9.3.1. Ternary adjunction 51 9.3.2. Measure estimate for ternary adjunction 54 10. Elimination of recollisions 61 10.1. Restriction to good configurations 61 10.2. Reduction to elementary observables 62 10.3. Boltzmann hierarchy pseudo-trajectories 64 10.4. Reduction to truncated elementary observables 65 11. Convergence proof 67 11.1. BBGKY hierarchy pseudo-trajectories and proximity to the Boltzmann hierarchy pseudo-trajectories 67 11.2. Reformulation in terms of pseudo-trajectories 71 11.3. Proof of Theorem 6.5 74 12. Appendix 74 12.1. Calculation of Jacobians 74 12.2. The binary transition map 74 References 75

Derivation of the kinetic wave equation for quadratic dispersive problems in the inhomogeneous setting

“…Interpolating between this bound and the X s, 1 2 to X s,− 1 2 bound, we obtain a bound from X s, 1 2 −δ to X s, 1 2 +δ with a loss −k , where k can be made arbitrarily small choosing δ sufficiently small. Finally, we choose b > 1 2 such that b − 1 < − 1 2 − δ.…”

“…Short range potentials were also discussed in [34]. A few articles deal with the derivation of kinetic models for higher order interactions [1,2], mixtures [3] and quantum particles [4,5,6]. The derivation of the Boltzmann equation is closely related to the derivation of the kinetic wave equation, but possibly more challenging, since dispersive equations can be thought of as an intermediary step between a quantum mechanical model with a large number of particles, and kinetic theory.…”

Global well-posedness of a binary–ternary Boltzmann equation

“…dω dv 1,2 dv (A.17) Although we will not use (A.17), it is worth mentioning it, since it implies entropy dissipation for the ternary collisional operator, see [8] for more details.…”

“…For example, in the case of a colloid 1 it was noted by Russ -Von-Günberg [38] that multi interactions among particles significantly contribute to the grand potential of a colloidal gas and are modeled by a sum of higher order interaction terms. Motivated in part by this observation, in a sequence of works [9,8,7] Ampatzoglou and Pavlović proposed a toy model for a non-ideal gas of the form:…”

Moment estimates and well-posedness of the binary-ternary Boltzmann equation

“…For example, in the case of a colloid (which is a homogeneous noncrystalline substance consisting of ultramicroscopic particles of one substance dispersed through a second substance) it was noted by Russ -Von-Günberg [40] that multi interactions among particles significantly contribute to the grand potential of a colloidal gas and are modeled by a sum of higher order interaction terms. Motivated in part by this observation, in a sequence of works [9,8,7, 10] Ampatzoglou, Pavlović and Warner proposed a toy model capturing higher order collisions of the form:…”