Rigorous Derivation of a Ternary Boltzmann Equation for a Classical System of Particles

Ampatzoglou, Ioakeim; Pavlović, Nataša

doi:10.1007/s00220-021-04202-y

Cited by 15 publications

(36 citation statements)

References 20 publications

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“…• Many-Particle Systems. The mathematical modelling of classical many-particle systems has led to the use of higher arity devices such as higher generalization of the Boltzmann equation [AP21] or multi-particle instantaneous interactions [Dob+02]. On the quantum side, the most prominent appearance of higher arity ideas is in the phenomenon of higher order entanglement [ZHG92; YBP20].…”

Section: Biologymentioning

confidence: 99%

An Invitation to Higher Arity Science

Zapata-Carratalá¹,

Arsiwalla²

2022

Preprint

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Analytical thinking is dominated by binary ideas. From pair-wise interactions, to algebraic operations, to compositions of processes, to network models, binary structures are deeply ingrained in the fabric of most current scientific paradigms. In this article we introduce arity as the generic conceptualization of the order of an interaction between a discrete collection of entities and argue that there is a rich universe of higher arity ideas beyond binarity waiting to be explored. To illustrate this we discuss several higher order phenomena appearing in a wide range of research areas, paying special attention to instances of ternary interactions. From the point of view of formal sciences and mathematics, higher arity thinking opens up new paradigms of algebra, symbolic calculus and logic. In particular, we delve into the special case of ternary structures, as that itself reveals ample surprises: new notions of associativity (or lack thereof) in ternary operations of cubic matrices, ternary isomorphisms and ternary relations, the integration problem of 3-Lie algebras, and generalizations of adjacency in 3-uniform hypergraphs. All these are open problems that strongly suggest the need to develop new ternary mathematics. Finally, we comment on potential future research directions and remark on the transdisciplinary nature of higher arity science.

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Section: Biologymentioning

confidence: 99%

An Invitation to Higher Arity Science

Zapata-Carratalá¹,

Arsiwalla²

2022

Preprint

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“…Motivated by the observations of [34], in [4] we suggested a model which goes beyond binary interactions incorporating sums of higher order interaction terms. In particular, we introduced the generalized equation…”

Section: Introductionmentioning

confidence: 99%

“…[31,30,11,22,33,35,39] for results in this direction. A relevant step towards rigorously deriving (1.1) for m = 3 has been recently obtained in [4], where we considered a certain type of three particle interactions that lead us to derive a purely ternary kinetic equation, which we called a ternary Boltzmann equation. However, the derivation of (1.1) for m = 3 has not been addressed yet, and that is exactly what we do in this paper.…”

Section: Introductionmentioning

confidence: 99%

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

Ampatzoglou¹,

Pavlović²

2020

Preprint

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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

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“…This binary-ternary equation could serve as a correction to the classical Boltzmann equation (1.1) in the description of denser gases in nonequilibrium since it takes into account both binary and ternary interactions between particles. The derivation of this equation in the case of hard spheres is a work in progress [3].…”

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confidence: 99%

Global well-posedness of a binary-ternary Boltzmann equation

Ampatzoglou,

Gamba,

Pavlovic

et al. 2019

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In this paper we prove global well-posedness for small initial data for the binaryternary Boltzmann equation. The binary-ternary Boltzmann equation provides a correction term to the classical Boltzmann equation, taking into account both binary and ternary interactions of particles, and could possibly serve as a more accurate description model for denser gases in non-equilibrium. To prove global well-posedness we use a Kaniel-Shinbrot iteration and related work to approximate the solution of the nonlinear equation by monotone sequences of solutions to appropriate linear problems. We show that the ternary operator allows consideration of softer potentials than the binary operator, consequently our solution to the ternary correction of the Boltzmann equation preserves all the properties of the binary interactions solution.

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Rigorous Derivation of a Ternary Boltzmann Equation for a Classical System of Particles

Cited by 15 publications

References 20 publications

An Invitation to Higher Arity Science

An Invitation to Higher Arity Science

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

Global well-posedness of a binary-ternary Boltzmann equation

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