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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 and 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.

[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.

[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.

[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.

[5]

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

[6]

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

[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.

[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.

[9]

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

[10]

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

[11]

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

[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.

[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.

[14]

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

[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.

[16]

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

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.

[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.

[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.

[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.

[5]

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

[6]

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

[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.

[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.

[9]

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

[10]

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

[11]

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

[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.

[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.

[14]

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

[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.

[16]

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

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