Abstract
We discuss the asymptotic behavior of certain models of dissipative systems obtained from a suitable modification of Kac caricature of a Maxwellian gas. It is shown that global equilibria different from concentration are possible if the energy is not finite. These equilibria are distributed like stable laws, and attract initial densities which belong to the normal domain of attraction. If the initial density is assumed of finite energy, with higher moments bounded, it is shown that the solution converges for large-time to a profile with power law tails. These tails are heavily dependent of the collision rule.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
A. Baldassarri, U. Marini Bettolo Marconi, and A. Puglisi, Kinetic models of inelastic gases, Math. Models Methods Appl. Sci. 12:965–983 (2002).
N. Ben-Naim and P. Krapivski, Multiscaling in inelastic collisions, Phys. Rev. E 61: R5–R8 (2000).
A. V. Bobylev, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwellian molecules, Sov. Sci. Rev. C 7:111–233 (1988).
A. V. Bobylev, J. A. Carrillo, and I. Gamba, On some properties of kinetic and hydrodynamics equations for inelastic interactions, J. Stat. Phys. 98:743–773 (2000).
A. V. Bobylev and C. Cercignani, Moment equation for a granular material in a thermal bath, J. Stat. Phys. 106:547–567 (2002).
A. V. Bobylev and C. Cercignani, Exact eternal solutions of the Boltzmann equation, J. Stat. Phys. 106:1019–1038(2002).
A. V. Bobylev and C. Cercignani, Self-similar solutions of the Boltzmann equation and their applications, J. Stat. Phys. 106:1039–1071(2002).
A. V. Bobylev and C. Cercignani, Self-similar asymptotics for the Boltzmann equation with inelastic and elastic interactions, J. Stat. Phys. 110:333–375(2003).
A. V. Bobylev, C. Cercignani, and G. Toscani, Proof of an asymptotic property of self-similar solutions of the Boltzmann equation for granular materials, J. Stat. Phys. 111: 403–417 (2003).
E. A. Carlen, E. Gabetta, and G. Toscani, Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous Maxwellian gas, Comm. Math. Phys. 305:521–546 (1999).
E. A. Carlen, M. C. Carvalho, and E. Gabetta, Central limit theorem for Maxwellian molecules and truncation of the Wild expansion, Comm. Pure Appl. Math. 53:370–397 (2000).
J. A. Carrillo, C. Cercignani, and I. M. Gamba, Steady states of a Boltzmann equation for driven granular media, Phys. Rev. E 62:7700–7707 (2000).
C. Cercignani, Shear flow of a granular material, J. Stat. Phys. 102:1407–1415 (2001).
C. Cercignani, R. Illner, and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer Series in Applied Mathematical Sciences, Vol. 106 (Springer-Verlag, New York, 1994).
M. H. Ernst and R. Brito, High energy tails for inelastic Maxwell models, Europhys. Lett. 43:497–502 (2002).
M. H. Ernst and R. Brito, Scaling solutions of inelastic Boltzmann equation with over-populated high energy tails, J. Stat. Phys. 109:407–432 (2002).
E. Gabetta, G. Toscani, and B. Wennberg, Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation, J. Stat. Phys. 81:901–934 (1995).
T. Goudon, S. Junca, and G. Toscani, Fourier-based distances and Berry-Esseen like inequalities for smooth densities, Monatsh. Math. 135:115–136 (2002).
P. K. Haff, Grain flow as a fluid-mechanical phenomenon, J. Fluid Mech. 134:401–430 (1983).
Hailiang Li and G. Toscani, Long-time asymptotics of kinetic models of granular flows, preprint (2002).
J. Hoffmann-Jø;rgensen, Probability with a View Towards Statistics, Vol. I (Chapman & Hall, New York, 1994).
R. G. Laha and V. K. Rohatgi, Probability Theory (Wiley, New York, 1979).
M. Kac, Probability and Related Topics in the Physical Sciences (Interscience, London/ New York, 1959).
H. P. McKean, Jr., Speed of approach to equilibrium for Kac's caricature of a Maxwellian gas, Arch. Rational Mech. Anal. 21:343–367 (1966).
G. Toscani, One-dimensional kinetic models of granular flows, RAIRO Modél Math. Anal. Numér. 34:1277–1292 (2000).
G. Toscani and C. Villani, Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas, J. Stat. Phys. 94:619–637 (1999).
G. Toscani and C. Villani, Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation, Comm. Math. Phys. 203:667–706 (1999).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Pulvirenti, A., Toscani, G. Asymptotic Properties of the Inelastic Kac Model. Journal of Statistical Physics 114, 1453–1480 (2004). https://doi.org/10.1023/B:JOSS.0000013964.98706.00
Issue Date:
DOI: https://doi.org/10.1023/B:JOSS.0000013964.98706.00