Skip to main content
SpringerLink
Log in

Moment Inequalities and High-Energy Tails for Boltzmann Equations with Inelastic Interactions

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We study high-energy asymptotics of the steady velocity distributions for model kinetic equations describing various regimes in dilute granular flows. The main results obtained are integral estimates of solutions of the Boltzmann equation for inelastic hard spheres, which imply that steady velocity distributions behave in a certain sense as C exp(−r∣v∣s), for ∣v∣ large. The values of s, which we call the orders of tails, range from s = 1 to s = 2, depending on the model of external forcing. To obtain these results we establish precise estimates for exponential moments of solutions, using a sharpened version of the Povzner-type inequalities.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, A Wiley-Interscience Publication, John Wiley & Sons Inc., New York, 1984. Reprint of the 1972 edition.

    Google Scholar 

  2. A. Baldassarri, U. M. B. Marconi, and A. Puglisi, Influence of Correlations on the Velocity Statistics of Scalar Granular Gases. Europhys. Lett. 58:14–20 (2002).

    Google Scholar 

  3. C. Bizon, M. D. Shattuck, J. B. Swift, and H. L. Swinney, Transport Coefficients for Granular Media from Molecular Dynamics Simulations, Phys. Rev. E, 60 (1999), pp. 4340–4351, ArXiv:cond-mat/9904132.

    Google Scholar 

  4. A. V. Bobylev, Moment Inequalities for the Boltzmann Equation and Applications to Spatially Homogeneous Problems. J. Statist. Phys. 88:1183–1214 (1997).

    Google Scholar 

  5. A. V. Bobylev, J. A. Carrillo, and I. M. Gamba, On Some Properties of Kinetic and Hydrodynamic Equations for Inelastic Interactions. J. Statist. Phys. 98:743–773 (2000).

    Google Scholar 

  6. A. V. Bobylev and C. Cercignani, Moment Equations for a Granular Material in a Thermal Bath. J. Stat.Phys. 106:547–567 (2002).

    Google Scholar 

  7. A. V. Bobylev and C. Cercignani, Self-similar Asymptotics for the Boltzmann Equation with Inelastic and Elastic Interactions. J. Statist. Phys. 110:333–375 (2003)

    Google Scholar 

  8. 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. 111:403–417 (2003).

    Google Scholar 

  9. A. V. Bobylev, M. Groppi, and G. Spiga, Approximate Solutions to the Problem of Stationary Shear Flow of Smooth Granular Materials. Eur. J. Mech. B Fluids 21:91–103 (2002).

  10. C. Cercignani, Recent Developments in the Mechanics of Granular Materials, in sica matematica e ingegneria delle strutture, Pitagora Editrice, Bologna, 1995, pp. 119–132.

  11. C. Cercignani, Shear Flow of a Granular Material.J. Statist. Phys. 102:1407–1415 (2001).

    Google Scholar 

  12. L. Desvillettes, Some Applications of the Method of Moments for the Homogeneous Boltzmann and Kac Equations. Arch. Rational Mech. Anal. 123:387–404 (1993).

    Google Scholar 

  13. T. Elmroth, Global Boundedness of Moments of Solutions of the Boltzmann Equation for Forces of In nite Range. Arch. Rational Mech. Anal. 82:1–12 (1983).

    Google Scholar 

  14. M. H. Ernst and R. Brito, Velocity Tails for Inelastic Maxwell Models,cond-mat/0111093, (2001).

  15. M. H. Ernst and R. Brito, Driven Inelastic Maxwell Models with High Energy Tails. Phys. Rev. E 65:040301 (1–4) (2002).

    Google Scholar 

  16. M. H. Ernst and R. Brito, Scaling Solutions of Inelastic Boltzmann Equations with Over-Populated High Energy Tails, J. Stat. Phys. to appear (2002), ArXiv:cond-mat/0112417.

  17. S. E. Esipov and T. Pöschel, The Granular Phase Diagram. J. Stat. Phys. 86:1385 (1997).

    Google Scholar 

  18. I. M. Gamba, V. Panferov, and C.Villani, work in progress.

  19. I. M. Gamba, V. Panferov, and C. Villani, On the Boltzmann Equations for Diffusively Excited Granular Media. Comm. Math. Phys. 246:503–541 (2004).

    Google Scholar 

  20. I. M. Gamba, S. Rjasanow, and W. Wagner, Direct Simulation of the Uniformly Heated Granular Boltzmann Equation. Submitted, (2003).

  21. I. Goldhirsch, Scales and Kinetics of Granular Flows.Chaos,9:659–672 (1999).

    Google Scholar 

  22. I. Goldhirsch, Rapid Granular. ows:Kinetics and Hydrodynamics. In Modeling in applied sciences: A kinetic theory approach, Birkhäuser Boston, Boston, MA, 2000, pp. 21–79.

    Google Scholar 

  23. T. Gustafsson, Global L P -properties for the Spatially Homogeneous Boltzmann Equation, Arch. Rational Mech. Anal. 103:1–38 (1988).

  24. P. L. Krapivsky and E. Ben-Naim, Multiscaling in In nite Dimensional Collision Processes. Phys. Rev. E, 61:R5 (2000)ArXiv:cond-mat/9909176.

    Google Scholar 

  25. P. L. Krapivsky and E. Ben-Naim, Nontrivial Velocity Distributions in Inelastic Gases. J. Phys. A 35:p. L147 (2002) ArXiv:cond-mat/011104.

    Google Scholar 

  26. P. L. Krapivsky and E. Ben-Naim, The Inelastic Maxwell Model, cond-mat/0301238, (2003).

  27. X. Lu, Conservation of Energy,Entropy Identity,and Local Stability for the Spatially Homogeneous Boltzmann equation, J. Statist. Phys. 96:765–796 (1999).

    Google Scholar 

  28. S. Mischler and B. Wennberg, On the Spatially Homogeneous Boltzmann Equation. Ann. Inst.H.Poincaré Anal.Non Linéaire 16:467–501 (1999).

    Google Scholar 

  29. J. M. Montanero and A. Santos, Computer Simulation of Uniformly Heated Granular Fluids. Gran. Matt. 2:53–64 (2000)ArXiv:cond-mat/0002323.

    Google Scholar 

  30. S. J. Moon, M. D. Shattuck, and J. B. Swift, Velocity Distributions and Correlations in Homogeneously Heated Granular Media, Phys. Rev. E 64:031303(1–10)(2001) ArXiv:cond-mat/0105322.

  31. A. Y. Povzner, On the Boltzmann Equation in the Kinetic Theory of Gases. Mat. Sb. (N.S.) 58 (100):65–86 (1962).

    Google Scholar 

  32. R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1997.

    Google Scholar 

  33. T. P. C. van Noije and M. H. Ernst, Velocity Distributions in Homogeneously Cooling and Heated Granular Fluids. Gran. Matt. 1:57 (1998) ArXiv:cond-mat/9803042.

    Google Scholar 

  34. C. Villani, A Survey of Mathematical Topics in Collisional Kinetic Theory. In Handbook of Mathematical Fluid Mechanics, S. Friedlander, and D. Serre, eds., vol. 1, Elsevier, 2002, ch. 2.

  35. B. Wennberg, Entropy Dissipation and Moment Production for the Boltzmann Equation. J. Statist. Phys. 86:1053–1066 (1997).

    Google Scholar 

  36. D. R. M. Williams and F. C. MacKintosh, Driven Dranular Media in One Dimension: Correlations and Equation of State. Phys. Rev. E 54:9–12 (1996).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Karlstad University, Karlstad, SE-651 88, Sweden

    A. V. Bobylev

  2. Department of Mathematics, The University of Texas at Austin, Austin, Texas, 78712, USA

    I. M. Gamba & V. A. Panferov

  3. Department of Mathematics and Statistics, University of Victoria, Victoria B.C., Canada V8W 3P4; e-mail:

    V. A. Panferov

Authors
  1. A. V. Bobylev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. M. Gamba
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. V. A. Panferov
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bobylev, A.V., Gamba, I.M. & Panferov, V.A. Moment Inequalities and High-Energy Tails for Boltzmann Equations with Inelastic Interactions. Journal of Statistical Physics 116, 1651–1682 (2004). https://doi.org/10.1023/B:JOSS.0000041751.11664.ea

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:JOSS.0000041751.11664.ea

Search

Navigation