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Non-Equilibrium Steady States

  • Dario VillamainaEmail author
Chapter
Part of the Springer Theses book series (Springer Theses)

Abstract

In this chapter a brief collection of the results present in literature and used in this work is described. We starts with a derivation of the Langevin equation in a way that makes clear the assumptions on the basis of equilibrium dynamics. Then, generalized response relations are presented and the role of entropy production is discussed. Since a large part of this work regards the study of granular gases, the second part of the chapter is entirely devoted to them, paying attention to the still open problems in dense regimes. This is not a chapter of a review article, and for this reason it could appear incomplete. However, it must be seen as an occasion to present the common ground where there are the basis of our research, and it proposes some questions which are developed and, at least partially, solved in the rest of the work.

Keywords

Granular Material Entropy Production Effective Temperature Langevin Equation Time Reversal Symmetry 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Andrieux, D., Gaspard, P.: Fluctuation theorem for currents and schnakenberg network theory. J. Stat. Phys. 127, 107 (2007)MathSciNetADSCrossRefzbMATHGoogle Scholar
  2. 2.
    Astumian, R.D.: The unreasonable effectiveness of equilibrium theory for interpreting nonequilibrium experiments. Am. J. Phys 74, 683 (2006)CrossRefGoogle Scholar
  3. 3.
    Baiesi, M., Maes, C., Wynants, B.: Nonequilibrium linear response for markov dynamics, i: jump processes and overdamped diffusions. J. Stat. Phys. 137, 1094 (2009)MathSciNetADSCrossRefzbMATHGoogle Scholar
  4. 4.
    Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality. Phys. Rev. A 38, 364 (1988)MathSciNetADSCrossRefzbMATHGoogle Scholar
  5. 5.
    Baldassarri, A., Barrat, A., D’Anna, G., Loreto, V., Mayor, P., Puglisi, A.: What is the temperature of a granular medium? J. Phys. Condens. Matter 17, S2405 (2005)ADSCrossRefGoogle Scholar
  6. 6.
    Barrat, A., Trizac, E.: Lack of energy equipartition in homogeneous heated binary granular mixtures. Granular Matter 4, 57 (2002)CrossRefzbMATHGoogle Scholar
  7. 7.
    Barrat, A., Loreto, V., Puglisi, A.: Temperature probes in binary granular gases. Phys. A. 334, 513 (2004)CrossRefGoogle Scholar
  8. 8.
    Berthier, L., Barrat, J.L.: Shearing a glassy material: numerical tests of nonequilibrium mode-coupling approaches and experimental proposals. Phys. Rev. Lett. 89, 095702 (2002)ADSCrossRefGoogle Scholar
  9. 9.
    Boffetta, G., Lacorata, G., Musacchio, S., Vulpiani, A.: Relaxation of finite perturbations: Beyond the fluctuation-response relation. Chaos 13, 806 (2003)ADSCrossRefGoogle Scholar
  10. 10.
    Boksenbojm, E., Wynants, B., Jarzynski, C.: Nonequilibrium thermodynamics at the microscale: work relations and the second law. Stat. Mech. Appl. Phys A 389, 4406 (2010)Google Scholar
  11. 11.
    Bouchaud, J.P., Cugliandolo, L.F., Kurchan, J., Mezard. M.: Spin Glasses and Random Fields. World Scientific, singapore (1998)Google Scholar
  12. 12.
    Bouchaud, J., Dean, D.: Aging on parisi’s tree. J. Phy. I(5), 265 (1995)ADSGoogle Scholar
  13. 13.
    Brilliantov, N.K., Poschel, T.: Kinetic Theory of Granular Gases. Oxford University Press, Oxford (2004)Google Scholar
  14. 14.
    Brilliantov, N.V., Poschel, T.: Self-diffusion in granular gases: green-kubo versus chapman-enskog. Chaos 15, 026108 (2005)MathSciNetADSCrossRefGoogle Scholar
  15. 15.
    Van den Broeck, C., Kawai, R., Meurs, P.: Microscopic analysis of a thermal brownian motor. Phys. Rev. Lett. 93, 90601 (2004)CrossRefGoogle Scholar
  16. 16.
    Brown, R.: A brief account of microscopical observations made.. on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies. Phil. Mag. Ser. 2 4, 161 (1828)Google Scholar
  17. 17.
    Brown, R.: Additional remarks on active molecules. Phil. Mag Ser. 2 6, 161 (1829)Google Scholar
  18. 18.
    Brown’s microscopical observations on the particles of bodies. Philos. Mag. N. S., 8, 296 (1830)Google Scholar
  19. 19.
    Castellani, T., Cavagna, A.: Spin-glass theory for pedestrians. J. Stat. Mech. Theor. Exp. 2005, P05012 (2005)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Cavagna, A.: Supercooled liquids for pedestrians. Phys. Rep. 476, 51 (2009)ADSCrossRefGoogle Scholar
  21. 21.
    Chapman, S., Cowling, T.: The mathematical theory of non-uniform gases. The mathematical theory of non-uniform gases, vol. 1. Cambridge University Press, Cambridge (1991).Google Scholar
  22. 22.
    Corberi, F., Lippiello, E., Sarracino, A., Zannetti, M.: Fluctuation-dissipation relations and field-free algorithms for the computation of response functions. Phys. Rev. E 81, 011124 (2010)ADSCrossRefGoogle Scholar
  23. 23.
    Costantini, G., Marconi, U., Puglisi, A.: Granular brownian ratchet model. Phys. Rev. E 75, 061124 (2007)ADSCrossRefGoogle Scholar
  24. 24.
    Crisanti, A., Ritort, F.: Violation of the fluctuation-dissipation theorem in glassy systems: basic notions and the numerical evidence. J. Phys. A 36, R181 (2003)MathSciNetADSCrossRefzbMATHGoogle Scholar
  25. 25.
    Cugliandolo, L.: Disordered systems. Lecture notes, Cargese (2011)Google Scholar
  26. 26.
    Cugliandolo, L.F.: Weak-ergodicity breaking in mean-field spin-glass models. Phil. Mag. 71, 501–514 (1995)CrossRefGoogle Scholar
  27. 27.
    Cugliandolo, L., Kurchan, J., Peliti, L.: Energy flow, partial equilibration, and effective temperatures in systems with slow dynamics. Phys. Rev. E 55, 3898 (1997)ADSCrossRefGoogle Scholar
  28. 28.
    Dunkel, J., Hänggi, P.: Relativistic brownian motion. Phys. Rep. 471, 1 (2009)MathSciNetADSCrossRefGoogle Scholar
  29. 29.
    Einstein, A.: On the movement of small particles suspended in a stationary liquid demanded by the molecular-kinetic theory of heat. Ann. d. Phys. 17, 549 (1905)ADSCrossRefzbMATHGoogle Scholar
  30. 30.
    Evans, D.J., Searles, D.J.: Equilibrium microstates which generate second law violating steady states. Phys. Rev. E 50, 1645 (1994)ADSCrossRefGoogle Scholar
  31. 31.
    Evans, D.J., Searles, D.J.: The fluctuation theorem. Adv. Phys. 52, 1529 (2002)ADSCrossRefGoogle Scholar
  32. 32.
    Falcioni, M., Isola, S., Vulpiani, A.: Correlation functions and relaxation properties in chaotic dynamics and statistical mechanics. Phys. Lett. A 144, 341 (1990)MathSciNetADSCrossRefGoogle Scholar
  33. 33.
    Feitosa, K., Menon, N.: Fluidized granular medium as an instance of the fluctuation theorem. Phys. Rev. Lett. 92, 164301 (2004)ADSCrossRefGoogle Scholar
  34. 34.
    Feynman, R., Leighton, R., Sands, M., et al.: The Feynman lectures on physics, vol. 2. Addison-Wesley Reading, MA (1964)Google Scholar
  35. 35.
    Fielding, S., Sollich, P.: Observable dependence of fluctuation-dissipation relations and effective temperatures. Phys. Rev. Lett. 88, 50603 (2002)ADSCrossRefGoogle Scholar
  36. 36.
    Gallavotti, G., Cohen, E.G.D.: Dynamical ensembles in stationary states. J. Stat. Phys. 80, 931 (1995)MathSciNetADSCrossRefzbMATHGoogle Scholar
  37. 37.
    Goldhirsch, I.: Rapid granular flows. Ann. Rev. Fluid Mech. 35, 267 (2003)MathSciNetADSCrossRefGoogle Scholar
  38. 38.
    Hänggi, P.: Generalized langevin equations: a useful tool for the perplexed modeller of nonequilibrium fluctuations? Stochastic, dynamics, p. 15. Springer, Berlin (1997)Google Scholar
  39. 39.
    Hänggi, P., Ingold, G.: Fundamental aspects of quantum brownian motion. Chaos: an Interdisciplinary. J. Nonlinear Sci. 15, 026105 (2005)Google Scholar
  40. 40.
    Hänggi, P., Marchesoni, F., Nori, F.: Brownian motors. Ann. Phys. 14, 51 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Hänggi, P., Marchesoni, F.: Artificial brownian motors: controlling transport on the nanoscale. Rev. Mod. Phys. 81, 387 (2009)ADSCrossRefGoogle Scholar
  42. 42.
    Hatano, T., Sasa, S.: Steady-state thermodynamics of Langevin systems. Phys. Rev. Lett. 86, 3463 (2001)ADSCrossRefGoogle Scholar
  43. 43.
    Jaeger, H.M., Nagel, S.R.: Physics of the granular state. Science 255, 1523 (1992)ADSCrossRefGoogle Scholar
  44. 44.
    Janssen, H. Versuche uber getreidedruck in silozellen. z. ver deut. Ing. 39 1045 (1895)Google Scholar
  45. 45.
    Jarzynski, C.: Nonequilibrium equality for free energy differences. Phys. Rev. Lett. 78, 2690 (1997)ADSCrossRefGoogle Scholar
  46. 46.
    Jarzynski, C.: Nonequilibrium work relations: foundations and applications. Eur. Phys. J. B Condens. Matter Complex Syst. 64, 331 (2008)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Kadanoff, L.P.: Built upon sand: theoretical ideas inspired by granular flows. Rev. Mod. Phys. 71, 435 (1999)ADSCrossRefGoogle Scholar
  48. 48.
    Kawai, R., Parrondo, J., den Broeck, C.: Dissipation: the phase-space perspective. Phys. Rev. Lett. 98, 80602 (2007)ADSCrossRefGoogle Scholar
  49. 49.
    Kob, W., Barrat, J., Sciortino, F., Tartaglia, P.: Aging in a simple glass former. J. Phys. Condens. Matter 12, 6385 (2000)ADSCrossRefGoogle Scholar
  50. 50.
    Kubo, R.: The fluctuation-dissipation theorem. Rep. Prog. Phys. 29, 255 (1966)ADSCrossRefGoogle Scholar
  51. 51.
    Kubo, R.: Brownian motion and nonequilibrium statistical mechanics. Science 32, 2022 (1986)Google Scholar
  52. 52.
    Kubo, R., Toda, M., Hashitsume, N.: Statistical physics II Nonequilibrium Stastical Mechanics. Springer, Berlin (1991)CrossRefGoogle Scholar
  53. 53.
    Kullback, S., Leibler, R.: On information and sufficiency. Ann. Math. Stat. 22, 79 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Kumaran, V.: Temperature of a granular material “fluidized” by external vibrations. Phys. Rev. E 57, 5660 (1998)ADSCrossRefGoogle Scholar
  55. 55.
    Kurchan, J.: Fluctuation theorem for stochastic dynamics. J. Phys. A 31, 3719 (1998)MathSciNetADSCrossRefzbMATHGoogle Scholar
  56. 56.
    Kurchan, J.: Non-equilibrium work relations. J. Stat. Mech. Theor. Exp. 2007, P07005 (2007)CrossRefGoogle Scholar
  57. 57.
    Lacorata, G., Puglisi, A., Vulpiani, A.: On the fluctuation-response relation in geophysical systems. Int. J. Mod. Phys. B 23, 5515 (2009)ADSCrossRefzbMATHGoogle Scholar
  58. 58.
    Langevin, P.: Sur la theorie du mouvement brownien. C. R. Acad. Sci. (Paris) 146 530 (1908) (Translated in. Am. J. Phys. 65, 1079 (1997))Google Scholar
  59. 59.
    Lebowitz, J.L., Spohn, H.: A Gallavotti-Cohen-type symmetry in the large deviation functional for stochastic dynamics. J. Stat. Phys. 95, 333 (1999)MathSciNetADSCrossRefzbMATHGoogle Scholar
  60. 60.
    Di Leonardo, R., et al.: Bacterial ratchet motors. Proc. Nat. Acad. Sci. 107, 9541 (2010)ADSCrossRefGoogle Scholar
  61. 61.
    Leuzzi, L., Nieuwenhuizen, T.M.: Thermodynamics of the Glassy State. Taylor & Francis, New York (2007)Google Scholar
  62. 62.
    Leuzzi, L.: A stroll among effective temperatures in aging systems: limits and perspectives. J. Non-Cryst. Solids 355, 686 (2009)ADSCrossRefGoogle Scholar
  63. 63.
    Lippiello, E., Corberi, F., Zannetti, M.: Off-equilibrium generalization of the fluctuation dissipation theorem for Ising spins and measurement of the linear response function. Phys. Rev. E 71, 036104 (2005)MathSciNetADSCrossRefGoogle Scholar
  64. 64.
    Marconi, U., Puglisi, A., Rondoni, L., Vulpiani, A.: Fluctuation-dissipation: response theory in statistical physics. Phys. Rep. 461, 111 (2008)ADSCrossRefGoogle Scholar
  65. 65.
    Van Der Meer, D., Reimann, P., Van Der Weele, K., Lohse, D.: Spontaneous ratchet effect in a granular gas. Phys. Rev. Lett. 92, 184301 (2004)ADSCrossRefGoogle Scholar
  66. 66.
    Mori, H.: Transport, collective motion, and brownian motion. Progress Theoret. Phys. 33, 423 (1965)ADSCrossRefzbMATHGoogle Scholar
  67. 67.
    Nelson, E. Dynamical theories of Brownian motion. Citeseer 17 (1967)Google Scholar
  68. 68.
    Nicodemi, M.: Dynamical response functions in models of vibrated granular media. Phys. Rev. Lett. 82, 3734 (1999)ADSCrossRefGoogle Scholar
  69. 69.
    van Noije, T.P.C., Ernst, M.H., Trizac, E., Pagonabarraga, I.: Randomly driven granular fluids: large-scale structure. Phys. Rev. E 59, 4326 (1999)ADSCrossRefGoogle Scholar
  70. 70.
    Onsager, L.: Reciprocal relations in irreversible processes. I. Phys. Rev. 37, 405 (1931)ADSCrossRefGoogle Scholar
  71. 71.
    Onsager, L., Machlup, S.: Fluctuations and irreversible processes. Phys. Rev. 91, 1505 (1953)MathSciNetADSCrossRefzbMATHGoogle Scholar
  72. 72.
    Perez-Espigares, C., Kolton, A. B., Kurchan, J.: An infinite family of second law-like inequalities. Phys. Rev. 85(3), 031135 (2012)Google Scholar
  73. 73.
    Perrin, J.: Les Atomes. Alcan, Paris (1913)zbMATHGoogle Scholar
  74. 74.
    Puglisi, A., Baldassarri, A., Vulpiani, A.: Violations of the einstein relation in granular fluids: the role of correlations. J. Stat. Mech. P08016 (2007)Google Scholar
  75. 75.
    Puglisi, A.: Granular fluids, a short walkthrough (2010)Google Scholar
  76. 76.
    Puglisi, A., Loreto, V., Marconi, U.M.B., Petri, A., Vulpiani, A.: Clustering and non-gaussian behavior in granular matter. Phys. Rev. Lett. 81, 3848 (1998)ADSCrossRefGoogle Scholar
  77. 77.
    Puglisi, A., Loreto, V., Marconi, U.M.B., Vulpiani, A.: A kinetic approach to granular gases. Phys. Rev. E 59, 5582 (1999)ADSCrossRefGoogle Scholar
  78. 78.
    Puglisi, A., Baldassarri, A., Loreto, V.: Fluctuation-dissipation relations in driven granular gases. Phys. Rev. E 66, 061305 (2002)ADSCrossRefGoogle Scholar
  79. 79.
    Puglisi, A., Visco, P., Barrat, A., Trizac, E., van Wijland, F.: Fluctuations of internal energy flow in a vibrated granular gas. Phys. Rev. Lett. 95, 110202 (2005)ADSCrossRefGoogle Scholar
  80. 80.
    Seifert, U., Speck, T.: Fluctuation-dissipation theorem in nonequilibrium steady states. EPL (Europhys. Lett.) 89, 10007 (2010)ADSCrossRefGoogle Scholar
  81. 81.
    Smoluchowski, M.: Zur kinetischen theorie der brownschen molekularbewegung und der suspensionen. Ann. d. Phys. 21, 756 (1906)CrossRefzbMATHGoogle Scholar
  82. 82.
    Smoluchowski, M.: Experimentell nachweisbare, der üblichen thermodynamik widersprechende molekularphänomene. Physik. Zeitschr 13, 1069 (1912)zbMATHGoogle Scholar
  83. 83.
    Struik, L.: Physical Aging in Amorphous Polymers and Other Materials. Elsevier, Amsterdam (1978)Google Scholar
  84. 84.
    Villamaina, D., Puglisi, A., Vulpiani, A.: The fluctuation-dissipation relation in sub-diffusive systems: the case of granular single-file diffusion. J. Stat. Mech. L10001 (2008)Google Scholar
  85. 85.
    Williams, D.R.M., MacKintosh, F.C.: Driven granular media in one dimension: correlations and equation of state. Phys. Rev. E 54, R9 (1996)ADSCrossRefGoogle Scholar
  86. 86.
    Zamponi, F., Bonetto, F., Cugliandolo, L. F., Kurchan, J.: A fluctuation theorem for non-equilibrium relaxational systems driven by external forces. J. Stat. Mech. P09013 (2005)Google Scholar
  87. 87.
    Zamponi, F.: Is it possible to experimentally verify the fluctuation relation? a review of theoretical motivations and numerical evidence. J. Stat. Mech. P02008 (2007)Google Scholar
  88. 88.
    Zwangzig, R.: Nonequilibrium statistical mechanics. Oxford University Press, Oxford (2001)Google Scholar
  89. 89.
    Zwanzig, R.: Time-correlation functions and transport coefficients in statistical mechanics. Ann. Rev. Phys. Chem. 16, 67 (1965)ADSCrossRefGoogle Scholar

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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.LPTMSUniversité Paris Sud and CNRSOrsay CedexFrance

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