Publication: Dynamics of deviations from the Gaussian state in a freely cooling homogeneous system of smooth inelastic particles
Full text at PDC
Advisors (or tutors)
The time dependence of deviations from the Gaussian state in a freely cooling homogeneous system of smooth inelastically colliding spheres is investigated by kinetic theory. We determine the full time dependence of the coefficients of an expansion around the Gaussian state in Generalized Laguerre polynomials. Approximating this system of equations to sixth order, we find that the asymptotic state, where the mean energy T follows Haff's law with time independent cooling rate, is reached within a few collisions per particle. Two-dimensional molecular dynamics stimulations confirm our results and show exponential behavior in the high-energy tails.
© Springer-Verlag 2000. The authors thank to A. Zippelius, T. Aspelmeier, P. Müller, and A. Santos for useful discussions. M. H. acknowledges financial support by the DFG through SFB 345 (Germany), and J. A. G. O. and R. B. from DGES number PB97-0076 (Spain)
1. I. Goldhirsch & G. Zanetti, Clustering instability in dissipative gases. Phys. Rev. Lett. 70 (1993), p. 1619 2. I. Goldhirsch, M.-L. Tan & G. Zanetti, Molecular dynamical study of granular fluids I: the unforced granular gas in two dimensions. J. Sci. Comput. 8 (1993), p. 1 3. P. Deltour & J. L. Barrat, Quantitative study of a freely cooling granular medium. J. Phys. I France 7 (1997), p. 137 4. S. E. Esipov & T. Pöschel, The granular phase diagram. J. Stat. Phys. 86 (1997), p. 1385 5. S. McNamara & W. R. Young, Dynamics of a freely evolving, two-dimensional granular medium. Phys. Rev. E 53 (1996), p. 5089 6. T. P. C. van Noije, M. H. Ernst, R. Brito & J. A. G. Orza, Mesoscopic theory of granular fluids. Phys. Rev. Lett. 79 (1997), p. 411 7. J. A. G. Orza, R. Brito, T. P. C. van Noije & M. H. Ernst, Patterns and long range correlations in idealized granular flows. Int. J. Mod. Phys. C 8 (1997), p. 953 8. T. P. C. van Noije, M. H. Ernst & R. Brito, Spatial correlations in compressible granular flows. Phys. Rev. E 57 (1998), R4891. T. P. C. van Noije & M. H. Ernst, Cahn- Hilliard theory for unstable granular fluids. Phys. Rev. E 61 (2000), p. 1765 9. A. Goldshtein & M. Shapiro, Mechanics of collisional motion of granular materials. Part 1. General hydrodynamic equations. J. Fluid Mech. 282 (1995), p. 75 10. T. P. C. van Noije & M. H. Ernst, Velocity distributions in homogeneously cooling and heated granular fluids. Gran. Matt. 1 (1998), p. 57 11. N. V. Brilliantov & T. Pöschel, Deviation from Maxwell distribution in granular gases with constant restitution coefficient. Phys. Rev. E 61 (2000), p. 2809 12. J. J. Brey, M. J. Ruiz-Montero & D. Cubero, Homogeneous cooling state of a low-density granular flow. Phys. Rev. E 54 (1996), p. 3664 13. N. V. Brilliantov & T. Pöschel, Velocity distribution in granular gases of viscoelastic particles. cond-mat Phys Rev E, 61, 2000, p. 5573 14. J. J. Brey, D. Cubero & M. J. Ruiz-Montero, High energy tail in the velocity distribution of a granular gas. Phys. Rev. E 59 (1999), p. 1256 15. W. Losert, D. G. W. Cooper, J. Deltour, A. Kudrolli & J. P. Gollub, Velocity statistics in excited granular media. Chaos, 9, 1999, p. 682 16. J. M. Montanero & A. Santos, Computer simulation of uniformly heated granular fluids, to be published in Granular Matter, cond-mat/0002323, (2000) 17. W. Magnus, F. Oberhettinger & R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics. Springer, 1966 18. S. Chapman & T. G. Cowling, The Mathematical Theory of Nonuniform Gases. Cambridge University Press, London, (1960) 19. P. K. Haff, Grain flow as a fluid-mechanical phenomenon. J. Fluid Mech. 134 (1983), p. 401 20. M. P. Allen & D. J. Tildesley, Computer Simulation of Liquids. Clarendon Press, Oxford, (1989) 21. B. D. Lubachevsky, How to simulate billards and similar systems. J. Comp. Phys. 94 (1991), p. 255 22. R. Brito & M. H. Ernst, Extension of Haff’s cooling law in granular flows. Europhys. Lett. 43 (1998), p. 497 23. S. Luding, M. Müller & S. McNamara, The validity of “molecular chaos” in granular flows. In World Congress on Particle Technology (Inst. of Chem. Eng., Davis Building, 165-189 Railway Terrace, Rugby CV21 3HQ, UK) (1998), ISBN 0-85295-401-9 24. J. A. G. Orza & R. Brito, (in preparation) 25. S. G. Eubank & J. D. Farmer, Introduction to nonlinear physics. Springer, New York, (1997), p. 106–151 26. S. Luding, M. Huthmann, S. McNamara & A. Zippelius, Homogeneous cooling of rough dissipative particles: theory and simulations. Phys. Rev. E 58 (1998), p. 3416 27. T. Aspelmeier, M. Huthmann & A. Zippelius, Free cooling of particles with rotational degrees of freedom, to be published in Granular Matter, Lecture Notes in Physics. Eds.: T . Pöschel and S. Luding, Springer-Verlag, (2000) 28. O. Herbst, M. Huthmann, & A. Zippelius, Dynamics of inelastically colliding spheres with Coulomb friction: relaxation of translational and rotational energy, to be published in Granular Matter, cond-mat/9911306, (2000)