Advertisement

Gravitational phase transitions with an exclusion constraint in position space

  • Pierre-Henri Chavanis
Regular Article

Abstract

We discuss the statistical mechanics of a system of self-gravitating particles with an exclusion constraint in position space in a space of dimension d. The exclusion constraint puts an upper bound on the density of the system and can stabilize it against gravitational collapse. We plot the caloric curves giving the temperature as a function of the energy and investigate the nature of phase transitions as a function of the size of the system and of the dimension of space in both microcanonical and canonical ensembles. We consider stable and metastable states and emphasize the importance of the latter for systems with long-range interactions. For d ≤ 2, there is no phase transition. For d > 2, phase transitions can take place between a “gaseous” phase unaffected by the exclusion constraint and a “condensed” phase dominated by this constraint. The condensed configurations have a core-halo structure made of a “rocky core” surrounded by an “atmosphere”, similar to a giant gaseous planet. For large systems there exist microcanonical and canonical first order phase transitions. For intermediate systems, only canonical first order phase transitions are present. For small systems there is no phase transition at all. As a result, the phase diagram exhibits two critical points, one in each ensemble. There also exist a region of negative specific heats and a situation of ensemble inequivalence for sufficiently large systems. We show that a statistical equilibrium state exists for any values of energy and temperature in any dimension of space. This differs from the case of the self-gravitating Fermi gas for which there is no statistical equilibrium state at low energies and low temperatures when d ≥ 4. By a proper interpretation of the parameters, our results have application for the chemotaxis of bacterial populations in biology described by a generalized Keller-Segel model including an exclusion constraint in position space. They also describe colloids at a fluid interface driven by attractive capillary interactions when there is an excluded volume around the particles. Connexions with two-dimensional turbulence are also mentioned.

Keywords

Statistical and Nonlinear Physics 

References

  1. 1.
    Dynamics and thermodynamics of systems with long range interactions, edited by T. Dauxois, S. Ruffo, E. Arimondo, M. Wilkens, Lecture Notes in Physics (Springer, 2002), Vol. 602Google Scholar
  2. 2.
    A. Campa, T. Dauxois, S. Ruffo, Phys. Rep. 480, 57 (2009)ADSCrossRefMathSciNetGoogle Scholar
  3. 3.
    D. Lynden-Bell, R. Wood, Mon. Not. R. Astron. Soc. 138, 495 (1968)ADSGoogle Scholar
  4. 4.
    W. Thirring, Z. Phys. 235, 339 (1970)ADSCrossRefGoogle Scholar
  5. 5.
    T. Padmanabhan, Phys. Rep. 188, 285 (1990)ADSCrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    P.H. Chavanis, Int. J. Mod. Phys. B 20, 3113 (2006)ADSCrossRefMATHGoogle Scholar
  7. 7.
    C. Sire, P.H. Chavanis, Phys. Rev. E 66, 046133 (2002)ADSCrossRefMathSciNetGoogle Scholar
  8. 8.
    J. Binney, S. Tremaine, Galactic Dynamics (Princeton Series in Astrophysics, 1987)Google Scholar
  9. 9.
    P.H. Chavanis, Astron. Astrophys. 556, A93 (2013)ADSCrossRefGoogle Scholar
  10. 10.
    R.W. Michie, Mon. Not. R. Astron. Soc. 125, 127 (1963)ADSMathSciNetGoogle Scholar
  11. 11.
    I.R. King, Astron. J. 70, 376 (1965)ADSCrossRefGoogle Scholar
  12. 12.
    D. Lynden-Bell, Mon. Not. R. Astron. Soc. 136, 101 (1967)ADSGoogle Scholar
  13. 13.
    V.A. Antonov, Vest. Leningr. Gos. Univ. 7, 135 (1962)Google Scholar
  14. 14.
    M. Kiessling, J. Stat. Phys. 55, 203 (1989)ADSCrossRefMathSciNetGoogle Scholar
  15. 15.
    R. Emden, Gaskugeln (Teubner Verlag, Leipzig, 1907)Google Scholar
  16. 16.
    J. Katz, Mon. Not. R. Astron. Soc. 183, 765 (1978)ADSMATHGoogle Scholar
  17. 17.
    P.H. Chavanis, Astron. Astrophys. 381, 340 (2002)ADSCrossRefMATHGoogle Scholar
  18. 18.
    P.H. Chavanis, Astron. Astrophys. 432, 117 (2005)ADSCrossRefMATHGoogle Scholar
  19. 19.
    H. Cohn, Astrophys. J. 242, 765 (1980)ADSCrossRefGoogle Scholar
  20. 20.
    C. Sire, P.H. Chavanis, Phys. Rev. E 69, 066109 (2004)ADSCrossRefGoogle Scholar
  21. 21.
    R.H. Fowler, Mon. Not. R. Astron. Soc. 87, 114 (1926)ADSGoogle Scholar
  22. 22.
    E.B. Aronson, C.J. Hansen, Astrophys. J. 177, 145 (1972)ADSCrossRefGoogle Scholar
  23. 23.
    B. Stahl, M.K.H. Kiessling, K. Schindler, Planet. Space Sci. 43, 271 (1994)ADSCrossRefGoogle Scholar
  24. 24.
    P.H. Chavanis, Phys. Rev. E 65, 056123 (2002)ADSCrossRefMathSciNetGoogle Scholar
  25. 25.
    M. Champion, A. Alastuey, T. Dauxois, S. Ruffo, arXiv:1210.5592Google Scholar
  26. 26.
    E. Follana, V. Laliena, Phys. Rev. E 61, 6270 (2000)ADSCrossRefGoogle Scholar
  27. 27.
    P.H. Chavanis, I. Ispolatov, Phys. Rev. E 66, 036109 (2002)ADSCrossRefGoogle Scholar
  28. 28.
    C. Destri, H.J. de Vega, Nucl. Phys. B 763, 309 (2007)ADSCrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    L. Casetti, C. Nardini, Phys. Rev. E 85, 061105 (2012)ADSCrossRefGoogle Scholar
  30. 30.
    P. Hertel, W. Thirring, Commun. Math. Phys. 24, 22 (1971)ADSCrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    P. Hertel, W. Thirring, in Quanten und Felder, edited by H.P. Dürr (Vieweg, Braunschweig, 1971)Google Scholar
  32. 32.
    N. Bilic, R. Viollier, Phys. Lett. B 408, 75 (1997)ADSCrossRefGoogle Scholar
  33. 33.
    P.H. Chavanis, in Proceedings of the Fourth International Heidelberg Conference on Dark Matter in Astro and Particle Physics, edited by Klapdor-Kleingrothaus, H.V. (Springer, New-York, 2002) [astro-ph/0205426].Google Scholar
  34. 34.
    P.H. Chavanis, M. Rieutord, Astron. Astrophys. 412, 1 (2003)ADSCrossRefGoogle Scholar
  35. 35.
    P.H. Chavanis, Phys. Rev. E 69, 066126 (2004)ADSCrossRefMathSciNetGoogle Scholar
  36. 36.
    P.H. Chavanis, C. R. Physique 7, 331 (2006)ADSCrossRefGoogle Scholar
  37. 37.
    P.H. Chavanis, Phys. Rev. D 76, 023004 (2007)ADSCrossRefGoogle Scholar
  38. 38.
    P.H. Chavanis, Int J. Mod. Phys. B 26, 1241002 (2012)ADSCrossRefMathSciNetGoogle Scholar
  39. 39.
    E. Keller, L.A. Segel, J. Theor. Biol. 26, 399 (1970)CrossRefMATHGoogle Scholar
  40. 40.
    P.H. Chavanis, Physica A 384, 392 (2007)ADSCrossRefMathSciNetGoogle Scholar
  41. 41.
    P.H. Chavanis, Eur. Phys. J. B 62, 179 (2008)ADSCrossRefMATHGoogle Scholar
  42. 42.
    J. Miller, Phys. Rev. Lett. 65, 2137 (1990)ADSCrossRefMATHMathSciNetGoogle Scholar
  43. 43.
    R. Robert, J. Sommeria, J. Fluid. Mech. 229, 291 (1991)ADSCrossRefMATHMathSciNetGoogle Scholar
  44. 44.
    A. Dominguez, M. Oettel, S. Dietrich, Phys. Rev. E 82, 011402 (2010)ADSCrossRefGoogle Scholar
  45. 45.
    P.H. Chavanis, preprintGoogle Scholar
  46. 46.
    R. Ellis, K. Haven, B. Turkington, J. Stat. Phys. 101, 999 (2000)CrossRefMATHMathSciNetGoogle Scholar
  47. 47.
    F. Bouchet, J. Barré, J. Stat. Phys. 118, 1073 (2005)ADSCrossRefMATHMathSciNetGoogle Scholar
  48. 48.
    H. Poincaré, Acta Math. 7, 259 (1885)CrossRefMathSciNetGoogle Scholar
  49. 49.
    J. Katz, Found. Phys. 33, 223 (2003)CrossRefMathSciNetGoogle Scholar
  50. 50.
    S. Chandrasekhar, An Introduction to the Theory of Stellar Structure (Dover, 1942)Google Scholar
  51. 51.
    H.J. de Vega, N. Sanchez, Nucl. Phys. B 625, 409 (2002)ADSCrossRefMATHMathSciNetGoogle Scholar
  52. 52.
    I. Ispolatov, M. Karttunen, Phys. Rev. E 68, 036117 (2003)ADSCrossRefGoogle Scholar
  53. 53.
    I. Ispolatov, M. Karttunen, Phys. Rev. E 70, 026102 (2004)ADSCrossRefGoogle Scholar
  54. 54.
    D.H.E. Gross, Microcanonical Thermodynamics: Phase Transitions in “Small” Systems, Lecture Notes in Physics (World Scientific, Singapore, 2001), Vol. 66Google Scholar
  55. 55.
    P.H. Chavanis, M. Ribot, C. Rosier, C. Sire, Banach Center Publ. 66, 103 (2004)CrossRefMathSciNetGoogle Scholar
  56. 56.
    H. Risken, The Fokker-Planck Equation (Springer, 1989)Google Scholar
  57. 57.
    J. Katz, I. Okamoto, Mon. Not. R. astron. Soc. 317, 163 (2000)ADSCrossRefGoogle Scholar
  58. 58.
    M. Antoni, S. Ruffo, A. Torcini, Europhys. Lett. 66, 645 (2004)ADSCrossRefGoogle Scholar
  59. 59.
    R. Robert, J. Sommeria, Phys. Rev. Lett. 69, 2776 (1992)ADSCrossRefMATHMathSciNetGoogle Scholar
  60. 60.
    P.H. Chavanis, J. Sommeria, R. Robert, Astrophys. J. 471, 385 (1996)ADSCrossRefGoogle Scholar
  61. 61.
    P.H. Chavanis, Eur. Phys. J. B 70, 73 (2009)ADSCrossRefMATHMathSciNetGoogle Scholar
  62. 62.
    P.H. Chavanis, Physica A 387, 5716 (2008)ADSCrossRefMathSciNetGoogle Scholar
  63. 63.
    P.H. Chavanis, L. Delfini, arXiv:1309.2872 (2013)Google Scholar
  64. 64.
    E.V. Votyakov, H. Hidmi, A. De Martino, D.H.E. Gross, Phys. Rev. Lett. 89, 031101 (2002)ADSCrossRefGoogle Scholar
  65. 65.
    E.V. Votyakov, A. De Martino, D.H.E. Gross, Eur. Phys. J. B 29, 593 (2002)ADSCrossRefGoogle Scholar
  66. 66.
    D.H.E. Gross, arXiv:cond-mat/0307535 (2003)Google Scholar
  67. 67.
    D.H.E. Gross, arXiv:cond-mat/0403582 (2004)Google Scholar
  68. 68.
    R.A. James, Astrophys. J. 140, 552 (1964)ADSCrossRefMathSciNetGoogle Scholar
  69. 69.
    M. Campisi, F. Zhan, P. Hänggi, Europhys. Lett. 99, 60004 (2012)ADSCrossRefGoogle Scholar
  70. 70.
    P.H. Chavanis, C. Sire, Phys. Rev. E 73, 066103 (2006)ADSCrossRefMathSciNetGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Laboratoire de Physique Théorique (IRSAMC), CNRS and UPS, Université de ToulouseToulouseFrance

Personalised recommendations