Derivation of the Boltzmann Equation: Hard Spheres, Short-Range Potentials and Beyond

Saffirio, Chiara

doi:10.1007/978-3-319-32144-8_15

Cited by 6 publications

(6 citation statements)

References 31 publications

(82 reference statements)

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“…This section presents a quite general and very brief overview of Lanford's theorem and its proof. In addition to the reference article [GST14], we also refer the interested reader to the reviews [Saf16] and [Gol15].…”

Section: Lanford's Theorem For the Deterministic Hard-sphere Systemmentioning

confidence: 99%

Propagation of chaos: a review of models, methods and applications. II. Applications

Chaintron,

Diez

2021

Preprint

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The notion of propagation of chaos for large systems of interacting particles originates in statistical physics and has recently become a central notion in many areas of applied mathematics. The present review describes old and new methods as well as several important results in the field. The models considered include the McKean-Vlasov diffusion, the meanfield jump models and the Boltzmann models. The first part of this review is an introduction to modelling aspects of stochastic particle systems and to the notion of propagation of chaos. The second part presents concrete applications and a more detailed study of some of the important models in the field.

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Section: Lanford's Theorem For the Deterministic Hard-sphere Systemmentioning

confidence: 99%

Propagation of chaos: a review of models, methods and applications. II. Applications

Chaintron,

Diez

2021

Preprint

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“…. We note that assumption (34) with respect the Boltzmann-Grad limit of initial states holds true for the equilibrium state [40].…”

Section: The Relationship Of the Kinetic Evolution Of Observables And...mentioning

confidence: 99%

“…where the momenta p * 1 and p * 2 are pre-collision momenta of hard spheres (3). Thus, the hierarchy of evolution equations ( 29) for additive-type marginal observables and initial state (34) describe the evolution of a hard sphere system just as the Boltzmann kinetic equation (36).…”

Section: The Relationship Of the Kinetic Evolution Of Observables And...mentioning

confidence: 99%

The Boltzmann-Grad asymptotic behavior of collisional dynamics: a brief survey

Gerasimenko,

Gapyak

2020

Preprint

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“…It is a result of the validity of the following equality for the limit -ary marginal observables (24); that is,…”

Section: On Propagation Of Initial Chaos In a Low-density Limitmentioning

confidence: 99%

“…The rigorous results on the derivation of the Boltzmann equation with hard sphere collisions by methods of perturbation theory of the BBGKY hierarchy was proved in [7][8][9][10]. The most recent advances on the lowdensity (Boltzmann-Grad) scaling asymptotic behavior [11] of many-particle systems, in particular, systems with shortrange interaction potentials, came in [12][13][14][15][16][17][18][19][20][21][22][23][24].…”

Section: Introductionmentioning

confidence: 99%

Low-Density Asymptotic Behavior of Observables of Hard Sphere Fluids

Gerasimenko

Gapyak

2018

Advances in Mathematical Physics

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The paper deals with a rigorous description of the kinetic evolution of a hard sphere system in the low-density (BoltzmannGrad) scaling limit within the framework of marginal observables governed by the dual BBGKY (Bogolyubov-BornGreen-Kirkwood-Yvon) hierarchy. For initial states specified by means of a one-particle distribution function, the link between the Boltzmann-Grad asymptotic behavior of a nonperturbative solution of the Cauchy problem of the dual BBGKY hierarchy for marginal observables and a solution of the Boltzmann kinetic equation for hard sphere fluids is established. One of the advantages of such an approach to the derivation of the Boltzmann equation is an opportunity to describe the process of the propagation of initial correlations in scaling limits.

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Derivation of the Boltzmann Equation: Hard Spheres, Short-Range Potentials and Beyond

Abstract: We review some results concerning the derivation of the Boltzmann equation starting from the many-body classical Hamiltonian dynamics. In particular, the celebrated paper by O. E. Lanford III [21] and the more recent papers [13,23] are discussed.

Cited by 6 publications

References 31 publications

Propagation of chaos: a review of models, methods and applications. II. Applications

Propagation of chaos: a review of models, methods and applications. II. Applications

The Boltzmann-Grad asymptotic behavior of collisional dynamics: a brief survey

Low-Density Asymptotic Behavior of Observables of Hard Sphere Fluids

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