In this chapter we consider generalizations of kinetic granular gas models given by Boltzmann equations of Maxwell type. These type of models for nonlinear elastic or inelastic interactions, have many applications in physics, dynamics of granular gases, economy, etc. We present the problem and develop its form in the space of characteristic functions, i.e., Fourier transforms of probability measures, from a very general point of view, including those with arbitrary polynomial nonlinearities and in any dimension space. We find a whole class of generalized Maxwell models that satisfy properties that characterize the existence and asymptotic of dynamically scaled or self-similar solutions, often referred as homogeneous cooling states. Of particular interest is a concept interpreted as an operator generalization of usual Lipschitz conditions which allows to describe the behavior of solutions to the corresponding initial value problem. In particular, we present, in the most general case, existence of self similar solutions and study, in the sense of probability measures, the convergence of dynamically scaled solutions associated with the Cauchy problem to those self-similar solutions, as time goes to infinity. In addition we show that the properties of these self-similar solutions lead to non classical equilibrium stable states exhibiting power tails. These results apply to different specific problems related to the Boltzmann equation (with elastic and inelastic interactions) and show that all physically relevant properties of solutions follow directly from the general theory developed in this presentation.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. Baldassarri, U.M.B. Marconi, and A. Puglisi; Influence of correlations on the velocity statistics of scalar granular gases, Europhys. Lett., 58 (2002), pp. 14–20.
Ben-Abraham D., Ben-Naim E., Lindenberg K., Rosas A.; Self-similarity in random collision processes, Phys. Review E, 68, R050103 (2003).
Ben-Naim E., Krapivski P.; Multiscaling in inelastic collisions, Phys. Rev. E, R5–R8 (2000).
Bisi M., Carrillo J.A., Toscani G.; Decay rates in probability metrics towards homogeneous cooling states for the inelastic Maxwell model, J. Stat. Phys. 118 (2005), no. 1–2, 301–331.
A.V. Bobylev; The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules. Mathematical Physics Reviews, Vol. 7, 111–233, Soviet Sci. Rev. Sect. C Math. Phys. Rev., 7, Harwood Academic Publ., Chur, (1988).
A.V. Bobylev; The Fourier transform method for the Boltzmann equation for Maxwell molecules, Sov. Phys. Dokl. 20: 820–822 (1976).
A.V. Bobylev, J.A. Carrillo, and I.M. Gamba; On some properties of kinetic and hydrodynamic equations for inelastic interactions. J. Statist. Phys. 98 (2000), no. 3–4, 743–773.
A.V. Bobylev and C. Cercignani; Self-similar solutions of the Boltzmann equation and their applications, J. Statist. Phys. 106 (2002), 1039–1071.
A.V. Bobylev and C. Cercignani; Self-similar asymptotics for the Boltzmann equation with inelastic and elastic interactions, J. Statist. Phys. 110 (2003), 333–375.
A.V. Bobylev, C. Cercignani, and I.M. Gamba; On the self-similar asymptotics for generalized non-linear kinetic Maxwell models, To appear in Comm. Math. Physics. http://arxiv.org/abs/math-ph/0608035 (2006).
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. Statist. Phys. 111 (2003), 403–417.
A.V. Bobylev and I.M. Gamba; Boltzmann equations for mixtures of Maxwell gases: exact solutions and power like tails, J. Stat. Phys. 124, no. 2–4, 497–516. (2006).
N. Brilliantov and T. Poschel (Eds.); Granular Gas Dynamics, Springer, Berlin, 2003.
C. Cercignani; The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules, Sov. Sci. Rev. C. Math. Phys., 7, 111–233 (1988).
S. Cordier, L. Pareschi, and G. Toscani; On a kinetic model for a simple market economy. J. Statist. Phys. 120 (2005), 253–277.
M.H. Ernst and R.Brito; Scaling solutions of inelastic Boltzmann equations with overpopulated high energy tails; J. Stat. Phys. 109 (2002), 407–432.
M. Ernst, E. Trizac, and A. Barrat, Granular gases: Dynamics and collective effects, Journal of Physics C: Condensed Matter. condmat-0411435. (2004).
W. Feller; An Introduction to Probability Theory and Applications, Vol. 2, Wiley, N.-Y., 1971.
E. Gabetta, G. Toscani, and W. Wennberg; Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation, J. Stat. Phys. 81:901 (1995).
E. Lukacs; Characteristic Functions, Griffin, London 1970.
G. Menon and R. Pego; Approach to self-similarity in Smoluchowski’s coagulation equations. Comm. Pure Appl. Math. 57 (2004), no. 9, 1197–1232.
G. Menon and R. Pego; Dynamical scaling in Smoluchowski’s coagulation equations: uniform convergence, no. 5, 1629–1651, SIAM Applied Math, 2005.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Bobylev, A.V., Cercignani, C., Gamba, I.M. (2008). Generalized Kinetic Maxwell Type Models of Granular Gases. In: Capriz, G., Mariano, P.M., Giovine, P. (eds) Mathematical Models of Granular Matter. Lecture Notes in Mathematics, vol 1937. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78277-3_2
Download citation
DOI: https://doi.org/10.1007/978-3-540-78277-3_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-78276-6
Online ISBN: 978-3-540-78277-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)