Fluctuation Theory in the Boltzmann–Grad Limit

Bodineau, Thierry; Gallagher, Isabelle; Saint‐Raymond, Laure; Simonella, Sergio

doi:10.1007/s10955-020-02549-5

Cited by 38 publications

(62 citation statements)

References 22 publications

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“…One can look for instance in [18] for this connection. Moreover, the large deviation Hamiltonian for the Boltzmann equation has already been obtained, for toy models which are analogue to the dilute gas dynamics [26] or for the dilute gas dynamics [3,4]. The aim of this section is to derive the large deviation Hamiltonian associated with the Landau equation from the large deviation Hamiltonian associated with the Boltzmann equation.…”

Section: Large Deviations Associated With the Landau Kinetic Theory From The Boltzmann Kinetic Theorymentioning

confidence: 99%

“…and with γ = (λ D /L) 3 , with the prescription that g Λ (τ = 0) is in the neighborhood of g(τ = 0), we have…”

Section: Deriving Landau's Large Deviation Principle From Boltzmann's Large Deviation Principlementioning

confidence: 99%

“…We already computed the first two cumulants H (1) and H (2) associated to this cumulant generating function. Now let us compute the third cumulant H (3) which can be expressed as a combination of moments of the random variable X (90):…”

Section: Computation Of Higher Order Cumulantsmentioning

confidence: 99%

“…The functional form of the large deviation Hamiltonian we deduced from Boltzmann's molecular chaos hypothesis is actually the same as Rezakhanlou's one. Moreover, for the specific case of hard spheres and in the Boltzmann-Grad limit, Bodineau, Gallagher, Saint-Raymond and Simonella [3] have rigorously proven large deviation asymptotics that give an information equivalent to the large deviation formulas (56-59), and which is valid for times of order of one collision time, as Lanford result for the kinetic equation.…”

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confidence: 90%

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Dynamical Large Deviations for Plasmas Below the Debye Length and the Landau Equation

Feliachi

Bouchet

2021

J Stat Phys

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We consider a homogeneous plasma composed of N particles of the same electric charge which interact through a Coulomb potential. In the large plasma parameter limit, classical kinetic theories justify that the empirical density is the solution of the Balescu-Guernsey-Lenard equation, at leading order. This is a law of large numbers. The Balescu-Guernsey-Lenard equation is approximated by the Landau equation for scales much smaller than the Debye length. In order to describe typical and rare fluctuations, we compute for the first time a large deviation principle for dynamical paths of the empirical density, within the Landau approximation. We obtain a large deviation Hamiltonian that describes fluctuations and rare excursions of the empirical density, in the large plasma parameter limit. We obtain this large deviation Hamiltonian either from the Boltzmann large deviation Hamiltonian in the grazing collision limit, or directly from the dynamics, extending the classical kinetic theory for plasmas within the Landau approximation. We also derive the large deviation Hamiltonian for the empirical density of N particles, each of which is governed by a Markov process, and coupled in a mean field way. We explain that the plasma large deviation Hamiltonian is not the one of N particles coupled in a mean-field way.

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Section: Large Deviations Associated With the Landau Kinetic Theory From The Boltzmann Kinetic Theorymentioning

confidence: 99%

“…and with γ = (λ D /L) 3 , with the prescription that g Λ (τ = 0) is in the neighborhood of g(τ = 0), we have…”

Section: Deriving Landau's Large Deviation Principle From Boltzmann's Large Deviation Principlementioning

confidence: 99%

Section: Computation Of Higher Order Cumulantsmentioning

confidence: 99%

mentioning

confidence: 90%

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Dynamical Large Deviations for Plasmas Below the Debye Length and the Landau Equation

Feliachi

Bouchet

2021

J Stat Phys

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“…Collecting these "fresh" particles, together with the fresh particles encountered by them, and so on, we arrive to a natural notion of "cluster of influence", to be associated to the particle 1. These clusters have both theoretical and applied interest and they are involved in the control of dynamical correlations [2,20,4]. Our goal is to derive an estimate on the cardinality of collisional clusters, uniform in the lowdensity (Boltzmann-Grad) limit.…”

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confidence: 99%

On the cardinality of collisional clusters for hard spheres at low density

Pulvirenti

Simonella

2021

DCDS

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We resume the investigation, started in [2], of the statistics of backward clusters in a gas of N hard spheres of small diameter ε. A backward cluster is defined as the group of particles involved directly or indirectly in the backwards-in-time dynamics of a given tagged sphere. We obtain an estimate of the average cardinality of clusters with respect to the equilibrium measure, global in time, uniform in ε, N for ε 2 N = 1 (Boltzmann-Grad regime).

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Long‐time correlations for a hard‐sphere gas at equilibrium

Bodineau

Gallagher

Saint‐Raymond

et al. 2023

Comm Pure Appl Math

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It has been known since Lanford that the dynamics of a hard‐sphere gas is described in the low density limit by the Boltzmann equation, at least for short times. The classical strategy of proof fails for longer times, even close to equilibrium. In this paper, we introduce a weak convergence method coupled with a sampling argument to prove that the covariance of the fluctuation field around equilibrium is governed by the linearized Boltzmann equation globally in time (including in diffusive regimes). This method is much more robust and simpler than the one devised in Bodineau et al which was specific to the 2D case.

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Fluctuation Theory in the Boltzmann–Grad Limit

Cited by 38 publications

References 22 publications

Dynamical Large Deviations for Plasmas Below the Debye Length and the Landau Equation

Dynamical Large Deviations for Plasmas Below the Debye Length and the Landau Equation

On the cardinality of collisional clusters for hard spheres at low density

Long‐time correlations for a hard‐sphere gas at equilibrium

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