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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

Dawan Mustafa 1, and  Bernt Wennberg 1,

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.
Keywords: relativistic Kac model, microcanonical measure., master equation, Chaotic distributions, propagation of chaos.
Mathematics Subject Classification: Primary: 60K35, 82B40, 82C40; Secondary: 60J75, 83A0.
Citation: Dawan Mustafa, Bernt Wennberg. Chaotic distributions for relativistic particles. Kinetic & Related Models, 2016, 9 (4) : 749-766. doi: 10.3934/krm.2016014
References:
[1]

E. A. Carlen, M. C. Carvalho, J. Le Roux, M. Loss and C. Villani, Entropy and chaos in the Kac model, Kinet. Relat. Models, 3 (2010), 85-122. doi: 10.3934/krm.2010.3.85.  Google Scholar

[2]

E. A. Carlen, P. Degond and B. Wennbrg, Kinetic limits for pair-interaction driven master equations and biological swarm models, Math. Models Methods Appl. Sci., 23 (2013), 1339-1376. doi: 10.1142/S0218202513500115.  Google Scholar

[3]

K. Carrapatoso, Quantitative and qualitative Kac's chaos on the Boltzmann's sphere, Ann. Inst. Henri Poincaré Probab. Stat., 51 (2015), 993-1039. doi: 10.1214/14-AIHP612.  Google Scholar

[4]

C. Cercignani and G. Medeiros Kremer, The Relativistic Boltzmann Equation: Theory and Applications, Birkhäuser Verlag, Basel, 2002. doi: 10.1007/978-3-0348-8165-4.  Google Scholar

[5]

J. T. Chang D. Pollard, Conditioning as disintegration, Statist. Neerlandica, 51 (1997), 287-317. doi: 10.1111/1467-9574.00056.  Google Scholar

[6]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Revised Edition, CRC Press, 2015.  Google Scholar

[7]

I. Gallagher, L. Saint-Raymond and B. Texier, From Newton to Boltzmann: Hard Spheres and Short-Range Potentials, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2013.  Google Scholar

[8]

R. V. Gamkrelidze, Integral representations and asymptotic methods, in Encyclopaedia of Mathematical Sciences, Springer-Verlag, 1989. doi: 10.1007/978-3-642-61310-4.  Google Scholar

[9]

M. Kac, Foundations of kinetic theory, in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 3 (1956), 171-197.  Google Scholar

[10]

O. E. Landford III, Time evolution of large classical systems, Dynamical Systems, Theory and Applications, 38 (1975), 1-111.  Google Scholar

[11]

T. Lelièvre, M. Rousset and G. Stoltz, Free Energy Computations, Imperial College Press, London, 2010. doi: 10.1142/9781848162488.  Google Scholar

[12]

M. Pulvirenti, C. Saffirio and S. Simonella, On the validity of the Boltzmann equation for short range potentials, Rev. Math. Phys., 26 (2014), 1450001, 64pp. doi: 10.1142/S0129055X14500019.  Google Scholar

[13]

A.-S. Sznitman, Topics in propagation of chaos, in École d'Été de Probabilités de Saint-Flour XIX-1989, 1464 (1991), 165-251. doi: 10.1007/BFb0085169.  Google Scholar

[14]

R. M. Strain, Coordinates in the relativistic Boltzmann theory, Kinet. Relat. Models, 4 (2011), 345-359. doi: 10.3934/krm.2011.4.345.  Google Scholar

[15]

R. M. Strain and S.-B. Yun, Spatially homogeneous Boltzmann equation for relativistic particles, SIAM J. Math. Anal., 46 (2014), 917-938. doi: 10.1137/130923531.  Google Scholar

[16]

M. Toda, R. Kubo and N. Saitô, Statistical Physics I, Equilibrium Statistical Mechanics, 2nd edition, Springer Verlag, 1992.  Google Scholar

show all references

References:
[1]

E. A. Carlen, M. C. Carvalho, J. Le Roux, M. Loss and C. Villani, Entropy and chaos in the Kac model, Kinet. Relat. Models, 3 (2010), 85-122. doi: 10.3934/krm.2010.3.85.  Google Scholar

[2]

E. A. Carlen, P. Degond and B. Wennbrg, Kinetic limits for pair-interaction driven master equations and biological swarm models, Math. Models Methods Appl. Sci., 23 (2013), 1339-1376. doi: 10.1142/S0218202513500115.  Google Scholar

[3]

K. Carrapatoso, Quantitative and qualitative Kac's chaos on the Boltzmann's sphere, Ann. Inst. Henri Poincaré Probab. Stat., 51 (2015), 993-1039. doi: 10.1214/14-AIHP612.  Google Scholar

[4]

C. Cercignani and G. Medeiros Kremer, The Relativistic Boltzmann Equation: Theory and Applications, Birkhäuser Verlag, Basel, 2002. doi: 10.1007/978-3-0348-8165-4.  Google Scholar

[5]

J. T. Chang D. Pollard, Conditioning as disintegration, Statist. Neerlandica, 51 (1997), 287-317. doi: 10.1111/1467-9574.00056.  Google Scholar

[6]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Revised Edition, CRC Press, 2015.  Google Scholar

[7]

I. Gallagher, L. Saint-Raymond and B. Texier, From Newton to Boltzmann: Hard Spheres and Short-Range Potentials, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2013.  Google Scholar

[8]

R. V. Gamkrelidze, Integral representations and asymptotic methods, in Encyclopaedia of Mathematical Sciences, Springer-Verlag, 1989. doi: 10.1007/978-3-642-61310-4.  Google Scholar

[9]

M. Kac, Foundations of kinetic theory, in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 3 (1956), 171-197.  Google Scholar

[10]

O. E. Landford III, Time evolution of large classical systems, Dynamical Systems, Theory and Applications, 38 (1975), 1-111.  Google Scholar

[11]

T. Lelièvre, M. Rousset and G. Stoltz, Free Energy Computations, Imperial College Press, London, 2010. doi: 10.1142/9781848162488.  Google Scholar

[12]

M. Pulvirenti, C. Saffirio and S. Simonella, On the validity of the Boltzmann equation for short range potentials, Rev. Math. Phys., 26 (2014), 1450001, 64pp. doi: 10.1142/S0129055X14500019.  Google Scholar

[13]

A.-S. Sznitman, Topics in propagation of chaos, in École d'Été de Probabilités de Saint-Flour XIX-1989, 1464 (1991), 165-251. doi: 10.1007/BFb0085169.  Google Scholar

[14]

R. M. Strain, Coordinates in the relativistic Boltzmann theory, Kinet. Relat. Models, 4 (2011), 345-359. doi: 10.3934/krm.2011.4.345.  Google Scholar

[15]

R. M. Strain and S.-B. Yun, Spatially homogeneous Boltzmann equation for relativistic particles, SIAM J. Math. Anal., 46 (2014), 917-938. doi: 10.1137/130923531.  Google Scholar

[16]

M. Toda, R. Kubo and N. Saitô, Statistical Physics I, Equilibrium Statistical Mechanics, 2nd edition, Springer Verlag, 1992.  Google Scholar

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