Probabilistic Study of the Speed of Approach to Equilibrium for an Inelastic Kac Model

Bassetti, Federico; Ladelli, Lucia; Regazzini, Eugenio

doi:10.1007/s10955-008-9630-z

Cited by 18 publications

(57 citation statements)

References 31 publications

(37 reference statements)

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“…Clearly, S(α) = 0. A probability measure μ 0 on R is prescribed as initial condition for (1)- (2). Denote by F 0 its probability distribution function, by φ 0 =μ 0 its Fourier-Stieltjes transform, and by X 0 some random variable (independent of everything else) with law μ 0 .…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…Denote by F 0 its probability distribution function, by φ 0 =μ 0 its Fourier-Stieltjes transform, and by X 0 some random variable (independent of everything else) with law μ 0 . Let μ t be the corresponding solution to (1)- (2), which is shown to be unique and global in time below. Finally, recall that φ(t; ·) denotes the Fourier-Stieltjes transform of μ t .…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…A representation of the form (11) has already been successfully employed in studying the long-time behavior of solutions to the classical and the inelastic Kac equation, for instance, in [2,12], respectively.…”

Section: Probabilistic Representation Of the Solutionmentioning

confidence: 99%

“…The latter constitutes the solution to (1)- (2). Recall also that M (α) ∞ has been defined in Proposition 2.…”

Section: Convergence In Distributionmentioning

confidence: 99%

“…For the original Kac equation, probabilistic methods have been used to estimate the approximation error of truncated Wild sums in [3], to study necessary and sufficient conditions for the convergence to a steady state in [12], to study the blow-up behavior of solutions of infinite energy in [4], to obtain rates of convergence to equilibrium of the solutions both in strong and weak metrics, [7,8,13]. Also the inelastic Kac model has been studied by probabilistic methods, see [2].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Central limit theorem for a class of one-dimensional kinetic equations

Bassetti

Ladelli

Matthes

2010

Probab. Theory Relat. Fields

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We introduce a class of kinetic-type equations on the real line, which constitute extensions of the classical Kac caricature. The collisional gain operators are defined by smoothing transformations with rather general properties. By establishing a connection to the central limit problem, we are able to prove long-time convergence of the equation's solutions toward a limit distribution. For example, we prove that if the initial condition belongs to the domain of normal attraction of a certain stable law ν α , then the limit is a scale mixture of ν α . Under some additional assumptions, explicit exponential rates for the convergence to equilibrium in Wasserstein metrics are calculated, and strong convergence of the probability densities is shown.

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Section: Preliminary Resultsmentioning

confidence: 99%

Section: Preliminary Resultsmentioning

confidence: 99%

Section: Probabilistic Representation Of the Solutionmentioning

confidence: 99%

“…The latter constitutes the solution to (1)- (2). Recall also that M (α) ∞ has been defined in Proposition 2.…”

Section: Convergence In Distributionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Central limit theorem for a class of one-dimensional kinetic equations

Bassetti

Ladelli

Matthes

2010

Probab. Theory Relat. Fields

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135

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Proof of a McKean conjecture on the rate of convergence of Boltzmann-equation solutions

Dolera

Regazzini

2013

Probab. Theory Relat. Fields

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The present work provides a definitive answer to the problem of quantifying relaxation to equilibrium of the solution to the spatially homogeneous Boltzmann equation for Maxwellian molecules. The beginning of the story dates back to a pioneering work by Hilbert, who first formalized the concept of linearization of the collision operator and pointed out the importance of its eigenvalues with respect to a certain asymptotic behavior of the Boltzmann equation. Under really mild conditions on initial data -close to being necessary -and a weak, physically consistent, angular cutoff hypothesis, our main result (Theorem 1.1) contains the first precise statement that the total variation distance between the solution and the limiting Maxwellian distribution admits an upper bound of the form Ce Λ b t , Λ b being the least negative of the aforesaid eigenvalues and C a constant which depends only on a few simple numerical characteristics (e.g. moments) of the initial datum. The validity of this quantification was conjectured, about fifty years ago, in a paper by Henry P. McKean but, in spite of several attempts, the best answer known up to now consists in a bound with a rate which can be made arbitrarily close to Λ b , to the cost of the "explosion" of the constant * Supported in part by MIUR-2008MK3AFZ † Also affiliated with CNR-IMATI, Milano, Italy.C. Moreover, its deduction is subject to restrictive hypotheses on the initial datum, besides the Grad angular cutoff condition. As to the proof of our results, we have taken as point of reference an analogy between the problem of convergence to equilibrium and the central limit theorem of probability theory, highlighted by McKean. Our work represents in fact a confirmation of this analogy, since the techniques we develop here crucially rely on certain formulations of the central limit theorem. The proof of Theorem 1.1 starts by assuming the Grad angular cutoff and proceeds with these steps: 1) A new representation, in Theorem 1.2, for the solution of the Boltzmann equation as expectation of a random probability distribution of a weighted random sum of independent and identically distributed random vectors. 2) An upper bound for the total variation distance of interest expressed as sum of expectations of the total variation distance, between the aforesaid random probability distribution and the limiting Maxwellian law, over two appropriate events U and U c . 3) The proof that the probability of U approaches zero, as time goes to infinity, at an exponential rate equal to Λ b . 4) An extension of a classical Beurling inequality which, combined with newBerry-Esseen-like inequalities, leads to the validity of the desired exponential rate Λ b of decay also for the expectation over U c . Then, the conclusion can be extended to the case of weak cutoff hypothesis by a standard truncation argument. To complete this description of the paper, we mention the use of the aforesaid representation to characterize, in Theorem 1.3, the domain of attraction of the Maxwellian limit.Mathematics subj...

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Kinetic Models with Randomly Perturbed Binary Collisions

2011

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We introduce a class of Kac-like kinetic equations on the real line, with general random collisional rules which, in some special cases, identify models for granular gases with a background heat bath (Carrillo et al. in Discrete Contin. Dyn. Syst. 24(1):59–81, 2009), and models for wealth redistribution in an agent-based market (Bisi et al. in Commun. Math. Sci. 7:901–916, 2009). Conditions on these collisional rules which guarantee both the existence and uniqueness of equilibrium profiles and their main properties are found. The characterization of these stationary states is of independent interest, since we show that they are stationary solutions of different evolution problems, both in the kinetic theory of rarefied gases (Cercignani et al. in J. Stat. Phys. 105:337–352, 2001; Villani in J. Stat. Phys. 124:781–822, 2006) and in the econophysical context (Bisi et al. in Commun. Math. Sci. 7:901–916, 2009)

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Probabilistic Study of the Speed of Approach to Equilibrium for an Inelastic Kac Model

Cited by 18 publications

References 31 publications

Central limit theorem for a class of one-dimensional kinetic equations

Central limit theorem for a class of one-dimensional kinetic equations

Proof of a McKean conjecture on the rate of convergence of Boltzmann-equation solutions

Kinetic Models with Randomly Perturbed Binary Collisions

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