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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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