Derivation of the Fick’s Law for the Lorentz Model in a Low Density Regime

Basile, Giada; Nota, Alessia; Pezzotti, Federica; Pulvirenti, Mario

doi:10.1007/s00220-015-2306-z

Cited by 17 publications

(40 citation statements)

References 13 publications

(2 reference statements)

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“…Moreover, for this model it is possible to look at a longer time scale, in which the diffusive behaviour of the classical ideal Rayleigh gas is described by the heat equation for the mass density. We can recover the same behaviour, if we consider the following rescaling (in the same spirit of the one proposed for the Lorentz model in [9,11]):…”

Section: Outlook On the Long-time Behaviour Of The Boltzmann-rayleighmentioning

confidence: 53%

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Kinetic Description of a Rayleigh Gas with Annihilation

2019

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In this paper, we consider the dynamics of a tagged point particle in a gas of moving hard-spheres that are non-interacting among each other. This model is known as the ideal Rayleigh gas. We add to this model the possibility of annihilation (ideal Rayleigh gas with annihilation), requiring that each obstacle is either annihilating or elastic, which determines whether the tagged particle is elastically reflected or removed from the system. We provide a rigorous derivation of a linear Boltzmann equation with annihilation from this particle model in the Boltzmann-Grad limit. Moreover, we give explicit estimates for the error in the kinetic limit by estimating the contributions of the configurations which prevent the Markovianity. The estimates show that the system can be approximated by the Boltzmann equation on an algebraically long time scale in the scaling parameter.

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Section: Outlook On the Long-time Behaviour Of The Boltzmann-rayleighmentioning

confidence: 53%

“…The first step of the proof is similar to the one in [11]. We estimate the total probability of each pathological event (recollision or interference) by estimating the probability of a pathology for each possible pair (i, j) of obstacles, i.e.…”

Section: Proof Of Lemma 43mentioning

confidence: 99%

Kinetic Description of a Rayleigh Gas with Annihilation

2019

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“…Since 1 is a simple eigenvalue of K, then the modified chain has spectral gap. Moreover, as shown in [6], Lemma 4.1, L −1 v is bounded. Hence Assumptions 2.3, 3.1 and the alternative (i) of Assumption 3.2 hold.…”

Section: Specific Examplesmentioning

confidence: 75%

A gradient flow approach to linear Boltzmann equations

Basile¹,

Benedetto²,

Bertini³

2020

Annali Scuola Normale Superiore - Classe Di Scienze

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“…V n ∼ n as n → ∞ which combined with (1.5) gives t n ∼ n 1 3 as n → ∞ , whence V n ∼ (t n ) 3 as n → ∞ if σ > 2.…”

Section: Introductionmentioning

confidence: 91%

Self-similar asymptotic behavior for the solutions of a linear coagulation equation

Niethammer

Nota

Throm

et al. 2019

Journal of Differential Equations

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In this paper we consider the long time asymptotics of a linear version of the Smoluchowski equation which describes the evolution of a tagged particle moving at constant speed in a random distribution of fixed particles. The volumes v of the particles are independently distributed according to a probability distribution which decays asymptotically as a power law v −σ . The validity of the equation has been rigorously proved in [19] for values of the exponent σ > 3. The solutions of this equation display a rich structure of different asymptotic behaviours according to the different values of the exponent σ. Here we show that for 5 3 < σ < 2 the linear Smoluchowski equation is well posed and that there exists a unique self-similar profile which is asymptotically stable.

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Derivation of the Fick’s Law for the Lorentz Model in a Low Density Regime

Cited by 17 publications

References 13 publications

Kinetic Description of a Rayleigh Gas with Annihilation

Kinetic Description of a Rayleigh Gas with Annihilation

A gradient flow approach to linear Boltzmann equations

Self-similar asymptotic behavior for the solutions of a linear coagulation equation

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