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Well-posedness for the Keller-Segel equation with fractional Laplacian and the theory of propagation of chaos
December  2016, 9(4): 749-766. doi: 10.3934/krm.2016014

## Chaotic distributions for relativistic particles

 1 Department of Mathematical sciences, Chalmers University of Technology and the University of Gothenburg, 412 96 GÖTEBORG, Sweden, Sweden

Received  July 2015 Revised  April 2016 Published  September 2016

We study a modified Kac model where the classical kinetic energy is replaced by an arbitrary energy function $\phi(v)$, $v \in \mathbb{R}$. The aim of this paper is to show that the uniform density with respect to the microcanonical measure is $Ce^{-z_0\phi(v)}$-chaotic, $C,z_0 \in \mathbb{R}_+$. The kinetic energy for relativistic particles is a special case. A generalization to the case $v\in \mathbb{R}^d$ which involves conservation momentum is also formally discussed.
Citation: Dawan Mustafa, Bernt Wennberg. Chaotic distributions for relativistic particles. Kinetic & Related Models, 2016, 9 (4) : 749-766. doi: 10.3934/krm.2016014
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##### References:
 [1] Eric A. Carlen, Maria C. Carvalho, Jonathan Le Roux, Michael Loss, Cédric Villani. Entropy and chaos in the Kac model. Kinetic & Related Models, 2010, 3 (1) : 85-122. doi: 10.3934/krm.2010.3.85 [2] Amit Einav. On Villani's conjecture concerning entropy production for the Kac Master equation. Kinetic & Related Models, 2011, 4 (2) : 479-497. doi: 10.3934/krm.2011.4.479 [3] Evelyne Miot, Mario Pulvirenti, Chiara Saffirio. On the Kac model for the Landau equation. Kinetic & Related Models, 2011, 4 (1) : 333-344. doi: 10.3934/krm.2011.4.333 [4] Milana Pavić-Čolić, Maja Tasković. Propagation of stretched exponential moments for the Kac equation and Boltzmann equation with Maxwell molecules. Kinetic & Related Models, 2018, 11 (3) : 597-613. doi: 10.3934/krm.2018025 [5] Kleber Carrapatoso. Propagation of chaos for the spatially homogeneous Landau equation for Maxwellian molecules. Kinetic & Related Models, 2016, 9 (1) : 1-49. doi: 10.3934/krm.2016.9.1 [6] Samir Salem. A gradient flow approach of propagation of chaos. Discrete & Continuous Dynamical Systems, 2020, 40 (10) : 5729-5754. doi: 10.3934/dcds.2020243 [7] Hui Huang, Jian-Guo Liu. Well-posedness for the Keller-Segel equation with fractional Laplacian and the theory of propagation of chaos. Kinetic & Related Models, 2016, 9 (4) : 715-748. doi: 10.3934/krm.2016013 [8] Todd Young. Asymptotic measures and distributions of Birkhoff averages with respect to Lebesgue measure. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 359-378. doi: 10.3934/dcds.2003.9.359 [9] J. Alberto Conejero, Francisco Rodenas, Macarena Trujillo. Chaos for the Hyperbolic Bioheat Equation. Discrete & Continuous Dynamical Systems, 2015, 35 (2) : 653-668. doi: 10.3934/dcds.2015.35.653 [10] Kaijen Cheng, Kenneth Palmer. Chaos in a model for masting. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1917-1932. doi: 10.3934/dcdsb.2015.20.1917 [11] Federico Bassetti, Lucia Ladelli. Large deviations for the solution of a Kac-type kinetic equation. Kinetic & Related Models, 2013, 6 (2) : 245-268. doi: 10.3934/krm.2013.6.245 [12] Kazuhisa Ichikawa, Mahemauti Rouzimaimaiti, Takashi Suzuki. Reaction diffusion equation with non-local term arises as a mean field limit of the master equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 115-126. doi: 10.3934/dcdss.2012.5.115 [13] Proscovia Namayanja. Chaotic dynamics in a transport equation on a network. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3415-3426. doi: 10.3934/dcdsb.2018283 [14] Yan Yong, Weiyuan Zou. Macroscopic regularity for the relativistic Boltzmann equation with initial singularities. Kinetic & Related Models, 2019, 12 (5) : 945-967. doi: 10.3934/krm.2019036 [15] José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic & Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401 [16] Mario Lefebvre. A stochastic model for computer virus propagation. Journal of Dynamics & Games, 2020, 7 (2) : 163-174. doi: 10.3934/jdg.2020010 [17] José Miguel Pasini, Tuhin Sahai. Polynomial chaos based uncertainty quantification in Hamiltonian, multi-time scale, and chaotic systems. Journal of Computational Dynamics, 2014, 1 (2) : 357-375. doi: 10.3934/jcd.2014.1.357 [18] Maxime Hauray, Samir Salem. Propagation of chaos for the Vlasov-Poisson-Fokker-Planck system in 1D. Kinetic & Related Models, 2019, 12 (2) : 269-302. doi: 10.3934/krm.2019012 [19] Nadia Lekrine, Chao-Jiang Xu. Gevrey regularizing effect of the Cauchy problem for non-cutoff homogeneous Kac's equation. Kinetic & Related Models, 2009, 2 (4) : 647-666. doi: 10.3934/krm.2009.2.647 [20] S. Jiménez, Pedro J. Zufiria. Characterizing chaos in a type of fractional Duffing's equation. Conference Publications, 2015, 2015 (special) : 660-669. doi: 10.3934/proc.2015.0660

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