Spectral gap for Kac's model of Boltzmann equation

Janvresse, Elise

doi:10.1214/aop/1008956330

Cited by 67 publications

(70 citation statements)

References 8 publications

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“…Kac's conjecture in this form, for the special case t~= 1/27r considered explicitly by Kac, was recently proved by Janvresse [9] using Yau's martingale inethod [13], [14]. Her proof gives no information on tile value of C. One result, already proved in [3], is that in the case t~=l/27r, 1 N+2 AN--2 N-l' (1.12) and hence…”

Section: F(~ T)=gtfo(~) Solves Kac's Master Equation O F(~ T) = N(qmentioning

confidence: 99%

Determination of the spectral gap for Kac's master equation and related stochastic evolution

Cárlen¹,

Carvalho²,

Loss³

2003

Acta Math.

152

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We present a method for bounding, and in some cases computing, the spectral gap for systems of many particles evolving under the influence of a random collision mechanism. In particular, the method yields the exact spectral gap in a model due to Mark Kac of energy conserving collisions with one dimensional velocities. It is also sufficiently robust to provide qualitatively sharp bounds also in the case of more physically realistic momentum and energy conserving collisions in three dimensions, as well as a range of related models.

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Section: F(~ T)=gtfo(~) Solves Kac's Master Equation O F(~ T) = N(qmentioning

confidence: 99%

Determination of the spectral gap for Kac's master equation and related stochastic evolution

Cárlen¹,

Carvalho²,

Loss³

2003

Acta Math.

152

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“…As a first step, he suggested to study the behavior of the spectral gap in L 2 (S N −1 ( √ N )) of the Markov process as N goes to infinity and conjectured it to be bounded away from zero uniformly in terms of N . This question has been answered only recently in [40,11] (see also [50,13]). However the L 2 norm behaves geometrically in terms of N for tensorized data; this leave no hope to use it for estimating the long-time behavior as N goes to infinity, as the time-decay estimates degenerate beyond times of order O(1/N ).…”

Section: 3mentioning

confidence: 99%

“…In his mind this program was to be achieved by understanding dissipativity at the level of the linear many-particle jump process and he insisted on the importance of estimating how its relaxation rate depends on the number of particles. This has motivated beautiful works on the "Kac spectral gap problem" [40,50,11,13,9], i.e. the study of this relaxation rate in a L 2 setting.…”

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confidence: 99%

Kac’s program in kinetic theory

Mischler

Mouhot²

2012

Invent. math.

153

292

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Abstract. This paper is devoted to the study of propagation of chaos and mean-field limits for systems of indistinguable particles, undergoing collision processes. The prime examples we will consider are the many-particle jump processes of Kac and McKean [42,53] giving rise to the Boltzmann equation. We solve the conjecture raised by Kac [42], motivating his program, on the rigorous connection between the long-time behavior of a collisional many-particle system and the one of its mean-field limit, for bounded as well as unbounded collision rates.Motivated by the inspirative paper by Grünbaum [35], we develop an abstract method that reduces the question of propagation of chaos to that of proving a purely functional estimate on generator operators (consistency estimates), along with differentiability estimates on the flow of the nonlinear limit equation (stability estimates). This allows us to exploit dissipativity at the level of the mean-field limit equation rather than the level of the particle system (as proposed by Kac).Using this method we show: (1) Quantitative estimates, that are uniform in time, on the chaoticity of a family of states. (2) Propagation of entropic chaoticity, as defined in [10]. (3) Estimates on the time of relaxation to equilibrium, that are independent of the number of particles in the system. Our results cover the two main Boltzmann physical collision processes with unbounded collision rates: hard spheres and true Maxwell molecules interactions. The proof of the stability estimates for these models requires significant analytic efforts and new estimates.

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“…While the conjecture was proven to be true (see [2,5,10,12]), the choice of L 2 as a reference distance is catastrophic when considering chaotic families. Intuitively speaking, one would suspect that chaoticity means (in some sense) that F N ≈ f ⊗N .…”

Section: Introductionmentioning

confidence: 99%

“…The relative entropy has some useful properties. In our context, the most important one is the Csiszar-Kullback-Leibler-Pinsker inequality: 10) which gives us a way to measure distance between measures (and in particular between probability densities). Notice that much like the log-Sobolev inequality, the constant appearing in (1.10) is independent of the dimension, giving us a way to uniformly control the distance!…”

Section: Introductionmentioning

confidence: 99%