<i>Probability and Related Topics in Physical Sciences</i>

Kac, Mark; Uhlenbeck, G. E.; Hibbs, A. R.; Pol, Balth. van der; Gillis, J.

doi:10.1063/1.3056918

Cited by 463 publications

(655 citation statements)

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“…In this last section we consider a simple one-dimensional model introduced in [23] which can be seen as a dissipative version of the Kac caricature of a Maxwellian gas [19,20]. Let us remark that the definition and properties of the Euclidean Wasserstein distance W 2 discussed above generalizes equally well to any dimension.…”

Section: Inelastic Kac Modelmentioning

confidence: 99%

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Tanaka Theorem for Inelastic Maxwell Models

Bolley

Carrillo

2007

Commun. Math. Phys.

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Abstract. We show that the Euclidean Wasserstein distance is contractive for inelastic homogeneous Boltzmann kinetic equations in the Maxwellian approximation and its associated Kac-like caricature. This property is as a generalization of the Tanaka theorem to inelastic interactions. Even in the elastic classical Boltzmann equation, we give a simpler proof of the Tanaka theorem than the ones in [25,27]. Consequences are drawn on the asymptotic behavior of solutions in terms only of the Euclidean Wasserstein distance.

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Section: Inelastic Kac Modelmentioning

confidence: 99%

“…Moreover, a generalization for non constant cross sections including Tanaka's theorem as a particular case will be proven in Section 4. Finally, we shall also show this generic property for the inelastic Kac model introduced in [23] as a dissipative version of Kac's caricature of Maxwellian gases [19,20].…”

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confidence: 99%

Tanaka Theorem for Inelastic Maxwell Models

Bolley

Carrillo

2007

Commun. Math. Phys.

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“…quantities which multilinearly depend on them, compare [56] where we explain in details the ingenious recognition of Newton. 59 From this we can conclude that if the algebra is non commutative and thus time arrow is unavoidable, we should observe essentially non symmetric tensor quantities corresponding to the essentially non-commutative algebra describing the classical parameters. Of course there are some ambiguities on both sides: two Morita equivalent algebras have isomorphic representation spaces, and we have some ambiguities in defining tensor quantities, for example that recognized by Casimir in transport processes, as well as some other much less easy to control in practice in extracting the relevant physical content.…”

Section: Our Hypothesis and The Onsager Principlementioning

confidence: 99%

“…The second reason comes from the results of Kac [59]. Namely he devoted almost all his life in examining equivalence between the stochastic method of Smoluchowski and the method of Gibbs.…”

Section: Our Hypothesis and The Onsager Principlementioning

confidence: 99%

Index Theory and Adiabatic Limit in QFT

Wawrzycki

2013

Int J Theor Phys

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The paper has the form of a proposal concerned with the relationship between the three mathematically rigorous approaches to quantum field theory: (1) local algebraic formulation of Haag, (2) Wightman formulation and (3) the perturbative formulation based on the microlocal renormalization method. In this project we investigate the relationship between (1) and (3) and utilize the known relationships between (1) and (2). The main goal of the proposal lies in obtaining obstructions for the existence of the adiabatic limit (confinement problem in the phenomenological standard model approach). We extend the method of deformation of Dütsch and Fredenhagen (in the Bordeman-Waldmann sense) and apply Fedosov construction of the formal index-an analog of the index for deformed symplectic manifolds, generalizing the Atiyah-Singer index. We present some first steps in realization of the proposal.

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“…For example, in some scaling limit for small time scales it should give thermodynamic formulas. In this paper we give examples of such "thermo" dynamics, which, from one side, generalize the classical Kac [15] model (for convergence to Boltzman equation) and stochastic chemical kinetics processes (see [13,14]). From another side, it adds energy parameter to Streater's statistical dynamics [18] and Othmer's complex reaction networks theory [16,17].…”

Section: Introductionmentioning

confidence: 99%

Microscopic Models for Chemical Thermodynamics

Малышев

2005

J Stat Phys

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We introduce an infinite particle system dynamics, which includes stochastic chemical kinetics models, the classical Kac model and free space movement. We study energy redistribution between two energy types (kinetic and chemical) in different time scales, similar to energy redistribution in the living cell. One example is considered in great detail, where the model provides main formulas of chemical thermodynamics.

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Probability and Related Topics in Physical Sciences

Cited by 463 publications

References 0 publications

Tanaka Theorem for Inelastic Maxwell Models

Tanaka Theorem for Inelastic Maxwell Models

Index Theory and Adiabatic Limit in QFT

Microscopic Models for Chemical Thermodynamics

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