Journal of Statistical Physics

, Volume 125, Issue 3, pp 533–568 | Cite as

Fluctuations of Power Injection in Randomly Driven Granular Gases

  • Paolo Visco
  • Andrea Puglisi
  • Alain Barrat
  • Emmanuel Trizac
  • Frédéric van Wijland


We investigate the large deviation function π(w) for the fluctuations of the power W(t) = wt, integrated over a time t, injected by a homogeneous random driving into a granular gas, in the infinite time limit. Our analytical study starts from a generalized Liouville equation and exploits a Molecular Chaos-like assumption. We obtain an equation for the generating function of the cumulants μ(λ) which appears as a generalization of the inelastic Boltzmann equation and has a clear physical interpretation. Reasonable assumptions are used to obtain μ(λ) in a closed analytical form. A Legendre transform is sufficient to get the large deviation function π(w). Our main result, apart from an estimate of all the cumulants of W(t) at large times t, is that π has no negative branch. This immediately results in the inapplicability of the Gallavotti-Cohen Fluctuation Relation (GCFR), that in previous studies had been suggested to be valid for injected power in driven granular gases. We also present numerical results, in order to discuss the finite time behavior of the fluctuations of W (t) . We discover that their probability density function converges extremely slowly to its asymptotic scaling form: the third cumulant saturates after a characteristic time τ larger than ∼50 mean free times and the higher order cumulants evolve even slower. The asymptotic value is in good agreement with our theory. Remarkably, a numerical check of the GCFR is feasible only at small times (at most τ/10), since negative events disappear at larger times. At such small times this check leads to the misleading conclusion that GCFR is satisfied for π(w). We offer an explanation for this remarkable apparent verification. In the inelastic Maxwell model, where a better statistics can be achieved, we are able to numerically observe the “failure” of GCFR.


Granular gases nonequilibrium steady-states large deviation fluctuation-dissipation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    N. van Kampen, Stochastic processes in physics and chemistry, North-Holland, 1992.Google Scholar
  2. 2.
    A. Einstein, Ann. d. Phys. 17:549 (1905).CrossRefADSGoogle Scholar
  3. 3.
    L. Onsager, Phys. Rev. 37:405 (1931); Phys. Rev. 38:2265 (1931).Google Scholar
  4. 4.
    M. S. Green, J. Chem. Phys. 22:398 (1954).CrossRefMathSciNetADSGoogle Scholar
  5. 5.
    R. Kubo, J. Phys. Soc. Japan 12:570 (1957).CrossRefMathSciNetADSGoogle Scholar
  6. 6.
    S. R. de Groot and P. Mazur, Non-equilibrium thermodynamics, North-Holland, 1969.Google Scholar
  7. 7.
    S. T. Bramwell, P. C. W. Holdsworth and J.-F. Pinton, Nature 396:552–554, (1998).CrossRefADSGoogle Scholar
  8. 8.
    J. Javier Brey, M. I. García de Soria, P. Maynar, and M. J. Ruiz-Montero, Phys. Rev. Lett. 94:098001 (2005).CrossRefADSGoogle Scholar
  9. 9.
    D. J. Evans, E. G. D. Cohen and G. P. Morriss, Phys. Rev. Lett. 71:2401 (1993).zbMATHCrossRefADSGoogle Scholar
  10. 10.
    G. Gallavotti and E.G.D. Cohen, Phys. Rev. Lett. 74:2694 (1995).CrossRefADSGoogle Scholar
  11. 11.
    D. J. Evans and G.P. Morriss, Statistical Mechanics of Nonequilibrium Liquids, Academic Press, London, 1990; G. P. Morriss and C. P. Dettmann, Chaos 8:321 (1998).Google Scholar
  12. 12.
    J. Kurchan, J. Phys. A 31:3719 (1998).zbMATHCrossRefMathSciNetADSGoogle Scholar
  13. 13.
    J. L. Lebowitz and H. Spohn, J. Stat. Phys. 95:333 (1999).zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    S. Ciliberto and C. Laroche, J. de Physique IV, 8, 215 (1998); N. Garnier and S. Ciliberto, Phys. Rev. E 71:060101(R) (2005); S. Ciliberto, N. Garnier, S. Hernandez, C. Lacpatia, J.-F. Pinton and G. Ruiz Chavarria, Physica A 340(1–3) pp. 240–250 (2004).Google Scholar
  15. 15.
    F. Bonetto, N. I. Chernov, and J. L. Lebowitz, Chaos 8:823 (1998); M. Dolowschiák and Z. Kovács Phys. Rev. E 71:025202 (2005).Google Scholar
  16. 16.
    S. Aumaître, S. Fauve, S. McNamara and P. Poggi, Eur. Phys. J. B 19:449 (2001).CrossRefADSGoogle Scholar
  17. 17.
    J. Farago, J. Stat. Phys. 107:781 (2002).zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    K. Feitosa and N. Menon, Phys. Rev. Lett. 92: 164301 (2004).CrossRefADSGoogle Scholar
  19. 19.
    P. Visco, A. Puglisi, A. Barrat, E. Trizac and F. van Wijland, Europhys. Lett. 72:55 (2005).CrossRefADSGoogle Scholar
  20. 20.
    A. Puglisi, P. Visco, A. Barrat, E. Trizac and F. van Wijland, Phys. Rev. Lett. 95:110202 (2005).CrossRefADSGoogle Scholar
  21. 21.
    D. R. M. Williams and F. C. MacKintosh, Phys. Rev. E 54: R9 (1996); G. Peng and T. Ohta, Phys. Rev. E 58:4737 (1998); T. P. C. van Noije, M. H. Ernst, E. Trizac and I. Pagonabarraga, Phys. Rev E 59:4326 (1999); C. Henrique, G. Batrouni and D. Bideau, Phys. Rev. E 63:011304 (2000); S. J. Moon, M. D. Shattuck and J. B. Swift, Phys. Rev. E 64:031303 (2001); I. Pagonabarraga, E. Trizac, T. P. C. van Noije and M. H. Ernst, Phys. Rev. E 65:011303 (2002).Google Scholar
  22. 22.
    A. Prevost, D. A. Egolf and J. S. Urbach, Phys. Rev. Lett. 89:084301 (2002).CrossRefADSGoogle Scholar
  23. 23.
    T. P. C. van Noije and M.H. Ernst, Granular Matter 1:57 (1998).CrossRefGoogle Scholar
  24. 24.
    J. M. Montanero and A. Santos, Granular Matter 2:53 (2000).CrossRefGoogle Scholar
  25. 25.
    F. Coppex, M. Droz, J. Piasecki and E. Trizac, Physica A 329:114 (2003).zbMATHCrossRefADSGoogle Scholar
  26. 26.
    G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Clarendon 1994 (Oxford).Google Scholar
  27. 27.
    J. J. Brey, J. W. Dufty and A. Santos, J. Stat. Phys. 87:1051 (1997); T. P. C. van Noije, M. H. Ernst and R. Brito, Physica A 251:266 (1998).Google Scholar
  28. 28.
    J. Farago, Physica A 331:69 (2004).CrossRefADSGoogle Scholar
  29. 29.
    R. van Zon and E. G. D. Cohen, Phys. Rev. Lett. 91: 110601 (2003).CrossRefADSGoogle Scholar
  30. 30.
    F. Bonetto, G. Gallavotti, A. Giuliani and F. Zamponi, cond-mat/0507672.Google Scholar
  31. 31.
    W. Feller, An Introduction to Probability Theory and Its Applications, John Wiley & Sons 1966.Google Scholar
  32. 32.
    J. C. Maxwell, Phil. Trans. 157:49 (1867).CrossRefGoogle Scholar
  33. 33.
    A. Baldassarri, U. Marini Bettolo Marconi and A. Puglisi, Europhys. Lett. 58:14–20 (2002).CrossRefADSGoogle Scholar
  34. 34.
    E. Ben-Naim and P. L. Krapivsky, J. Phys. A 35:L147 (2002); Lecture Notes in Physics 624:65 (2003).Google Scholar
  35. 35.
    M. H. Ernst and R. Brito, Europhys. Lett. 58:182 (2002).CrossRefADSGoogle Scholar
  36. 36.
    A. V. Bobylev, J. A. Carrillo and I. M. Gamba, J. Stat. Phys. 98:743 (2000).zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    A. Santos, Physica A 321:442 (2003).zbMATHCrossRefMathSciNetADSGoogle Scholar
  38. 38.
    A. Puglisi, A. Baldassarri and V. Loreto, Phys. Rev. E 66:061305 (2002).CrossRefADSGoogle Scholar
  39. 39.
    V. Garzó, Physica A 343:105 (2004).CrossRefADSGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  • Paolo Visco
    • 1
    • 2
  • Andrea Puglisi
    • 1
  • Alain Barrat
    • 1
  • Emmanuel Trizac
    • 2
  • Frédéric van Wijland
    • 1
    • 3
  1. 1.Laboratoire de Physique Théorique (CNRS UMR 8627), Bâtiment 210Université Paris-SudOrsay cedexFrance
  2. 2.Laboratoire de Physique Théorique et Modèles Statistiques (CNRS UMR 8626), Bâtiment 100Université Paris-SudOrsay cedexFrance
  3. 3.Laboratoire Matière et Systèmes Complexes (CNRS UMR 7057)Université Denis Diderot (Paris VII)Paris cedex 05France

Personalised recommendations