Journal of Statistical Physics

, Volume 154, Issue 5, pp 1292–1324 | Cite as

Probabilistic View of Explosion in an Inelastic Kac Model

  • Andrea Bonomi
  • Eleonora Perversi
  • Eugenio Regazzini


Let \(\{\mu (\cdot ,t):t\ge 0\}\) be the family of probability measures corresponding to the solution of the inelastic Kac model introduced in Pulvirenti and Toscani (J Stat Phys 114:1453–1480, 2004). It has been proved by Gabetta and Regazzini (J Stat Phys 147:1007–1019, 2012) that the solution converges weakly to equilibrium if and only if a suitable symmetrized form of the initial data belongs to the standard domain of attraction of a specific stable law. In the present paper it is shown that, for initial data which are heavier-tailed than the aforementioned ones, the limiting distribution is improper in the sense that it has probability \(1/2\) “adherent” to \(-\infty \) and probability \(1/2\) “adherent” to \(+\infty \). It is explained in which sense this phenomenon is amenable to a sort of explosion, and the main result consists in an explicit expression of the rate of such an explosion. The presentation of these statements is preceded by a discussion about the necessity of the assumption under which their validity is proved. This gives the chance to make an adjustment to a portion of a proof contained in the above-mentioned paper by Gabetta and Regazzini.


Central limit theorem Explosion of solution Inelastic Kac model Skorokhod representation theorem 

Mathematics Subject Classification

60F05 82C40 60B10 



We would like to thank two anonymous referees for giving several valuable comments regarding the presentation of the main result. The research of Eleonora Perversi and Eugenio Regazzini has been partially supported by MIUR-2012AZS52J-003.


  1. 1.
    Bassetti, F., Ladelli, L.: Self similar solutions in one-dimensional kinetic models: a probabilistic view. Ann. Appl. Probab. 22, 1928–1961 (2012)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bassetti, F., Perversi, E.: Speed of convergence to equilibrium in Wasserstein metrics for Kac-like kinetic equations. Electron. J. Probab. 18(6), 1–35 (2013)MathSciNetGoogle Scholar
  3. 3.
    Bassetti, F., Toscani, G.: Explicit equilibria in a kinetic model of gambling. Phys. Rev. E 81, 066115 (2010)ADSCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bassetti, F., Ladelli, L., Matthes, D.: Central limit theorem for a class of one-dimensional kinetic equations. Probab. Theory Relat. Fields 150, 77–109 (2011)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Bassetti, F., Ladelli, L., Regazzini, E.: Probabilistic study of the speed of approach to equilibrium for an inelastic Kac model. J. Stat. Phys. 133, 683–710 (2008)ADSCrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Bassetti, F., Ladelli, L., Toscani, G.: Kinetic models with randomly perturbed binary collisions. J. Stat. Phys. 142, 686–709 (2011)ADSCrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley, New York (1999)CrossRefMATHGoogle Scholar
  8. 8.
    Bolley, F., Carrillo, J.A.: Tanaka theorem for inelastic Maxwell models. Commun. Math. Phys. 276, 287314 (2007)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Bobylev, A.V., Cercignani, C.: Self-similar asymptotics for the Boltzmann equation with inelastic and elastic interactions. J. Stat. Phys. 1(10), 333–375 (2003)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Bobylev, A.V., Carrillo, J.A., Gamba, I.M.: On some properties of kinetic and hydrodynamic equations for inelastic interactions. J. Stat. Phys. 98, 743–773 (2000)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Carlen, E., Carvalho, M.C., Gabetta, E.: Central limit theorem for Maxwellian molecules and truncation of the Wild expansion. Commun. Pure Appl. Math. 53, 370–397 (2000)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Carlen, E., Carvalho, M.C., Gabetta, E.: On the relation between rates of relaxation and convergence of Wild sums for solutions of the Kac equation. J. Funct. Anal. 220, 362–387 (2005)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Carlen, E., Gabetta, E., Regazzini, E.: Probabilistic investigation on the explosion of solutions of the Kac equation with infinite energy initial distribution. J. Appl. Probab. 45, 95–106 (2008)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Carrillo, J.A., Toscani, G.: Contractive probability metrics and asymptotic behavior of dissipative kinetic equations. Riv. Mat. Univ. Parma 6, 75198 (2007)MathSciNetGoogle Scholar
  15. 15.
    Carrillo, J.A., Cordier, S., Toscani, G.: Over-populated tails for conservative-in-the-mean inelastic Maxwell models. Discrete Contin. Dyn. Syst. 24, 57–81 (2009)MathSciNetGoogle Scholar
  16. 16.
    Chow, Y.S., Teicher, H.: Probability Theory, 3rd edn. Springer, New York (1997)CrossRefMATHGoogle Scholar
  17. 17.
    Dolera, E., Regazzini, E.: The role of the central limit theorem in discovering sharp rates of convergence to equilibrium for the solution of the Kac equation. Ann. Appl. Probab. 20, 430–461 (2010)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Dolera, E., Regazzini, E.: Proof of a McKean conjecture on the rate of convergence of Boltzmann-equation, solutions. arXiv:1206.5147v1[] (2012)Google Scholar
  19. 19.
    Dolera, E., Gabetta, E., Regazzini, E.: Reaching the best possible rate of convergence to equilibrium for solutions of Kac’s equation via central limit theorem. Ann. Appl. Probab. 19, 186–209 (2009)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Fortini, S., Ladelli, L., Regazzini, E.: A central limit problem for partially exchangeable random variables. Theory Probab. Appl. 41, 224–246 (1996)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Fristedt, B., Gray, L.: A Modern Approach to Probability Theory. Birkhäuser, Boston (1997)CrossRefMATHGoogle Scholar
  22. 22.
    Gabetta, E., Regazzini, E.: Some new results for McKean’s graphs with applications to Kac’s equation. J. Stat. Phys. 125, 947–974 (2006)ADSCrossRefMathSciNetGoogle Scholar
  23. 23.
    Gabetta, E., Regazzini, E.: Central limit theorem for the solution of the Kac equation. Ann. Appl. Probab. 18, 2320–2336 (2008)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Gabetta, E., Regazzini, E.: Central limit theorem for the solution of the Kac equation: speed of approach to equilibrium in weak metrics. Probab. Theory Relat. Fields 146, 451–480 (2010)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Gabetta, E., Regazzini, E.: Complete characterization of convergence to equilibrium for an inelastic Kac model. J. Stat. Phys. 147, 1007–1019 (2012)ADSCrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Ibragimov, I.A., Linnik, Y.V.: Independent and Stationary Sequences of Random Variables. Wolters–Noordhoff Publishing, Groningen (1971)MATHGoogle Scholar
  27. 27.
    Loève, M.: Probability Theory I, 4th edn. Springer, New York (1977)MATHGoogle Scholar
  28. 28.
    Matthes, D., Toscani, G.: On steady distributions of kinetic models of conservative economies. J. Stat. Phys. 130, 1087–1117 (2008)ADSCrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Matthes, D., Toscani, G.: Propagation of Sobolev regularity for a class of random kinetic models on the real line. Nonlinearity 23, 2081–2100 (2010)ADSCrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    McKean Jr, H.P.: Speed of approach to equilibrium for Kac’s caricature of Maxwellian gas. Arch. Ration. Mech. Anal. 21, 343–367 (1966)CrossRefMathSciNetGoogle Scholar
  31. 31.
    McKean Jr, H.P.: An exponential formula for solving Boltzmann’s equation for a Maxwellian gas. J. Combinatorial Theory 2, 358–382 (1967)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Parthasarathy, K.R.: Probability Measures on Metric Spaces. Academic Press, New York (1967)MATHGoogle Scholar
  33. 33.
    Pulvirenti, A., Toscani, G.: Asymptotic properties of the inelastic Kac model. J. Stat. Phys. 114, 1453–1480 (2004)ADSCrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Villani, C.: A review of mathematical topics in collisional theory. In: Friedlander, S., Serre, D. (eds.) Handbook of Mathematical Fluid Dynamics, vol. I, pp. 71–305. North Holland, Amsterdam (2002)Google Scholar
  35. 35.
    Villani, C.: Mathematics of granular materials. J. Stat. Phys. 124, 781–822 (2006)ADSCrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Andrea Bonomi
    • 1
  • Eleonora Perversi
    • 1
  • Eugenio Regazzini
    • 1
    • 2
  1. 1.Dipartimento di MatematicaUniversità degli Studi di PaviaPaviaItaly
  2. 2.CNR-IMATIMilanItaly

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