Abstract
This paper deals with solutions of the nonlinear Boltzmann equation for spatially uniform freely cooling inelastic Maxwell models for large times and for large velocities, and the nonuniform convergence to these limits. We demonstrate how the velocity distribution approaches in the scaling limit to a similarity solution with a power law tail for general classes of initial conditions and derive a transcendental equation from which the exponents in the tails can be calculated. Moreover on the basis of the available analytic and numerical results for inelastic hard spheres and inelastic Maxwell models we formulate a conjecture on the approach of the velocity distribution function to a scaling form.
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REFERENCES
G. P. Collins, A Gas of Steel Balls, Sci. Am. 284:17 (2001); F. Rouyer and N. Menon, Phys. Rev. Lett. 85:3676 (2000).
C.S. Campbell, Annu. Rev. Fluid Mech. 22:57 (1990).
N. Sela and I. Goldhirsch, Phys. Fluids 7:507 (1995).
T. P. C. van Noije and M. H. Ernst, Granular Matter 1:57 (1998), and cond-mat/9803042.
Th. Biben, Ph. A. Martin, and J. Piasecki, preprint July 2001.
A. Barrat, T. Biben, Z. Racz, E. Trizac, and F. van Wijland, cond-mat/0110345.
J. J. Brey, M. J. Ruiz-Montero, and D. Cubero, Phys. Rev. E 54:3664 (1996).
J. M. Montanero and A. Santos, Granular Matter 2:53 (2000), and cond-mat/0002323.
M. Huthmann, J. A. G. Orza, and R. Brito, Granular Matter 2:189 (2000), and condmat/ 0004079.
T. P. C. van Noije, M. H. Ernst, E. Trizac, and I. Pagonabarraga, Phys. Rev. E 59:4326 (1999). I. Pagonabarraga, E. Trizac, T. P. C. van Noije, and M. H. Ernst, Phys. Rev E 65:011303 (2002).
S. E. Esipov and T. Pöschel, J. Stat. Phys. 86:1385 (1997).
E. Ben-Naim and P. Krapivsky, Phys. Rev. E 61:R5 (2000).
A. V. Bobylev, J. A. Carrillo, I. M. Gamba, J. Stat. Phys. 98:743 (2000).
J. A. Carrillo, C. Cercignani, and I. M. Gamba, Phys. Rev. E 62:7700 (2000).
C. Cercignani, J. Stat. Phys. 102:1407 (2001).
A. Bobylev and C. Cercignani, J. Stat. Phys. 106:547 (2002).
R. S. Krupp, A Non-Equilibrium Solution of the Fourier Transformed Boltzmann Equation, M.S. dissertation (M.I.T., Cambridge, MA, 1967).
A. V. Bobylev, Sov. Phys. Dokl. 20:820 (1975). A. V. Bobylev, Sov. Phys. Dokl. 21:632 (1976).
T. Krook and T. T. Wu, Phys. Rev. Lett. 36:1107 (1976).
M. H. Ernst, Phys. Rep. 78:1 (1981).
A. Baldassarri, U. Marini Bettolo Marconi, and A. Puglisi, Europhys. Lett. 58:14 (2002).
A. Baldassarri, private communication.
P. Krapivsky and E. Ben-Naim, cond-mat/0111044, 2 Nov 2001, and J. Phys. A 35:L147 (2002) and Phys. Rev. E 66:011309 (2002).
M. H. Ernst and R. Brito, cond-mat/0111093, 6 November 2001, and Europhys. Lett. 58:182 (2002).
M. H. Ernst and R. Brito, Phys. Rev. E 65:040301(R) (2002).
B. Nienhuis, private communication, September 2001.
I. Goldhirsch and G. Zanetti, Phys. Rev. Lett. 70:1619 (1993).
P. K. Haff, J. Fluid Mech. 134:40 (1983).
J. J. Brey, J. W. Dufty, and A. Santos, J. Stat. Phys. 87:1051 (1997).
A. Baldassarri, U. Marini Bettolo Marconi, and A. Puglisi, Phys. Rev. E 65:051301 (2002).
A. Bobylev and C. Cercignani, Exact Eternal Solutions of the Boltzmann Equation (see http://www.math.tu-berlin.de/ ' tmr/preprint/d.html).
A. Bobylev and C. Cercignani, J. Stat. Phys. 106:1039 (2002).
P. Résibois and M. de Leener, Classical Kinetic Theory of Fluids (Wiley, New York, 1977).
C. Cercignani, The Boltzmann Equation and Its Applications (Springer Verlag, New York, 1988).
D. Ruelle, J. Stat. Phys. 95:393 (1999).
D. J. Evans and L. Rondoni, Comments on the entropy of nonequilibrium steady states, J. Stat. Phys, this issue.
J. R. Dorfman, An Introduction to Chaos in Nonequilibrium Statistical Mechanics, Cambridge Lecture Notes in Physics, Vol. 14,Section 13.4 (Cambridge University Press, 1999).
A. Bobylev and C. Cercignani, private communication.
M. H. Ernst and R. Brito, in preparation.
E. W. Montroll and B. J. West, On an enriched collection of Stochastic processes, Fluctuation Phenomena, E. W. Montroll and J. L. Lebowitz, eds. (North Holland, Amsterdam, 1979).
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Ernst, M.H., Brito, R. Scaling Solutions of Inelastic Boltzmann Equations with Over-Populated High Energy Tails. Journal of Statistical Physics 109, 407–432 (2002). https://doi.org/10.1023/A:1020437925931
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DOI: https://doi.org/10.1023/A:1020437925931