Kac’s chaos and Kac’s program

Mischler, Stéphane

doi:10.5802/slsedp.48

Cited by 5 publications

(8 citation statements)

References 42 publications

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“…This result deserves particular attention as it shows, unambiguously, that the Boltzmann equation admits a fully stochastic explanation. In some sense, this result completes the original Kac's program in kinetic theory [8,9,10,11], originated from the article [12] aimed at providing an extended Markov model for interpreting the celebrated Boltzmann equation of kinetic theory. For a discussion on extended Markov models see [13].…”

Section: Introductionsupporting

confidence: 61%

“…and this can be viewed as a corollary of the Wong-Zakai theorem. In this case, the Poisson-Kac process is a stochastic mollification of the Langevin-Stratonovich equation (9). Case (B): λ(x, t) depends explicitly on x.…”

Section: Nonlinear Gpk Processes and Poisson Fieldsmentioning

confidence: 99%

“…In some sense this result is surprising for its simplicity, and supports what is usually referred to as the Kac's program in kinetic theory originated by the 1954-paper by Kac on the stochastic foundations of the kinetic theory [12], in which a simple one-dimensional Markovian toy-model is introduced to describe the relaxation properties of a rarefied particle gas. For further details on the Kac's program, and on related recent advances, the reader is referred to [8,9,10,11].…”

Section: The Boltzmann Equationmentioning

confidence: 99%

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Stochastic foundations of undulatory transport phenomena: generalized Poisson–Kac processes—part III extensions and applications to kinetic theory and transport

Giona¹,

Brasiello²,

Crescitelli³

2017

J. Phys. A: Math. Theor.

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This third part extends the theory of Generalized Poisson-Kac (GPK) processes to nonlinear stochastic models and to a continuum of states. Nonlinearity is treated in two ways: (i) as a dependence of the parameters (intensity of the stochastic velocity, transition rates) of the stochastic perturbation on the state variable, similarly to the case of nonlinear Langevin equations, and (ii) as the dependence of the stochastic microdynamic equations of motion on the statistical description of the process itself (nonlinear Fokker-Planck-Kac models). Several numerical and physical examples illustrate the theory. Gathering nonlinearity and a continuum of states, GPK theory provides a stochastic derivation of the nonlinear Boltzmann equation, furnishing a positive answer to the Kac's program in kinetic theory. The transition from stochastic microdynamics to transport theory within the framework of the GPK paradigm is also addressed.

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Section: Introductionsupporting

confidence: 61%

Section: Nonlinear Gpk Processes and Poisson Fieldsmentioning

confidence: 99%

Section: The Boltzmann Equationmentioning

confidence: 99%

See 1 more Smart Citation

Stochastic foundations of undulatory transport phenomena: generalized Poisson–Kac processes—part III extensions and applications to kinetic theory and transport

Giona¹,

Brasiello²,

Crescitelli³

2017

J. Phys. A: Math. Theor.

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“…Remarkably, the generalization of this class of models to include nonlinear effects, and a continuum of stochastic states leads directly to a stochastic model corresponding to the classical collisional Boltzmann equation for a particle gas. In this sense, GPK theory strengthens and completes the original program by M. Kac in kinetic theory [84,85,86], to provide a Markovian stochastic model for the Boltzmann kinetic equation without using simplifying assumption, or toy-model representation for the collisions amongst the molecules. Stemming from GPK theory, several byproducts follow, e.g., the Poisson-Kac mollification of Wiener processes that can be useful in analysis of stochastic partial differential equations and field theory.…”

Section: Introductionmentioning

confidence: 83%

Stochastic foundations of undulatory transport phenomena: generalized Poisson–Kac processes—part I basic theory

Giona¹,

Brasiello²,

Crescitelli³

2017

J. Phys. A: Math. Theor.

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This article introduces the notion of Generalized Poisson-Kac (GPK) processes which generalize the class of "telegrapher's noise dynamics" introduced by Marc Kac [39] in 1974, using Poissonian stochastic perturbations. In GPK processes the stochastic perturbation acts as a switching amongst a set of stochastic velocity vectors controlled by a Markov-chain dynamics. GPK processes possess trajectory regularity (almost everywhere) and asymptotic Kac limit, namely the convergence towards Brownian motion (and to stochastic dynamics driven by Wiener perturbations), which characterizes also the long-term/longdistance properties of these processes. In this article we introduce the structural properties of GPK processes, leaving all the physical implications to part II and part III [80,81].In statistical physics of gases, liquids, polymers and colloids (soft-matter physics, for short), the use of Brownian motion and Wiener processes can be regarded as the legacy of a large-number ansatz, wherein the influence of a manifold of small contributions, uniquely characterized by their mean and variance justifies the application of stochastic perturbations characterized by uncorrelated increments distributed in a normal way, which is precisely the definition of a Wiener process [6,7,8].The statistical characterization of microdynamic equations written in the form of Wiener-Langevin stochastic dynamics leads (using any representation of the stochastic integrals in the meaning of Ito, Stratonovich, Klimontovich, etc.) to parabolic (forward Fokker-Planck) equations for the probability density function, in which a deterministic drift v(x), and a tensorial diffusivity D can be always identified, possessing the structure of a second-order advection-diffusion equation.The analogy between Fokker-Planck equations and advection-diffusion problems permits to identify the overall probability flux J p (x, t) associated with the probability density p(x, t) as Basic principlesThe structure of one-dimensional Poisson-Kac processes, reviewed in the Appendix, contains the germs for its generalization in higher dimensions. Three basic principles can be enucleated out of it, that can guide the development of a simple and consistent stochastic theory of undulatory transport phenomena. These principles are:• The principle of stochastic reality;• The principle of the primitive variables;• The principle of the asymptotic Kac convergence.Below, their meaning and importance is outlined and discussed. 7

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“…We refer the reader to [Car+08;Mis12] for a thorough analysis and discussion of the Kac model and its generalisations in kinetic theory. Keeping a collision rate λ constant the authors of [CDW13] generalised the arguments of the proof of the propagation of chaos to a larger class of models.…”

Section: Classical Models In Collisional Kinetic Theorymentioning

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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Kac’s chaos and Kac’s program

Cited by 5 publications

References 42 publications

Stochastic foundations of undulatory transport phenomena: generalized Poisson–Kac processes—part III extensions and applications to kinetic theory and transport

Stochastic foundations of undulatory transport phenomena: generalized Poisson–Kac processes—part III extensions and applications to kinetic theory and transport

Stochastic foundations of undulatory transport phenomena: generalized Poisson–Kac processes—part I basic theory

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

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