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Kinetic theory of stellar systems, two-dimensional vortices and HMF model

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Abstract

We discuss the kinetic theories of stellar systems, two-dimensional vortices and Hamiltonian mean field model, stressing their analogies and differences. We describe the evolution of the system as a whole and discuss the timescale of relaxation towards the Boltzmann distribution predicted by statistical mechanics. We also consider the relaxation of a “test” particle in a bath of “field” particles and analyze it with the aid of a Fokker–Planck equation involving a term of diffusion counterbalanced by a friction or a drift.

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References

  1. Binney J., Tremaine S.: Galactic Dynamics (Princeton Series in Astrophysics). Princeton University Press, New Jersey (1987)

    Google Scholar 

  2. Sommeria J.: Two-dimensional turbulence. In: Lesieur, M., Yaglom, A., David, F. (eds) New Trends in Turbulence, vol. 74, p. 385. Les Houches Summer School, Les Houches (2001)

    Chapter  Google Scholar 

  3. Tabeling P.: Phys. Rep. 362, 1 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chavanis P.H.: Statistical mechanics of two-dimensional vortices and stellar systems. In: Dauxois, T., Ruffo, S., Arimondo, E., Wilkens, M. (eds) Dynamics and Thermodynamics of Systems with Long Range Interactions. Lecture Notes in Physics, vol. 602, Springer, Berlin (2002) [cond-mat/0212223]

    Google Scholar 

  5. Clercx H.J.H., van Heijst G.J.F.: Appl. Mech. Rev. 62, 020802 (2009)

    Article  Google Scholar 

  6. Hénon M.: Ann. Astrophys. 27, 83 (1964)

    Google Scholar 

  7. Lynden-Bell D.: Mon. Not. R. Astron. Soc. 136, 101 (1967)

    Google Scholar 

  8. Stiavelli M., Bertin G.: Mon. Not. R. Astron. Soc. 229, 61 (1987)

    Google Scholar 

  9. Hjorth J., Madsen J.: Mon. Not. R. Astron. Soc. 253, 703 (1991)

    Google Scholar 

  10. Landau L.D.: Phys. Z. Sowj. Union 10, 154 (1936)

    MATH  Google Scholar 

  11. Chandrasekhar S.: Principles of Stellar Dynamics. University of Chicago Press, Chicago (1942)

    MATH  Google Scholar 

  12. Chandrasekhar S.: Astrophys. J. 97, 255 (1943)

    Article  MathSciNet  Google Scholar 

  13. Kandrup H.: Phys. Rep. 63, 1 (1980)

    Article  MathSciNet  Google Scholar 

  14. Kandrup H.: Astrophys. J. 244, 316 (1981)

    Article  MathSciNet  Google Scholar 

  15. Saslaw W.C.: Gravitational Physics of Stellar and Galactic Systems. Cambridge University Press, London (1985)

    Google Scholar 

  16. Spitzer L.: Dynamical Evolution of Globular Clusters (Princeton Series in Astrophysics). Princeton University Press, New Jersey (1987)

    Google Scholar 

  17. Chavanis P.H.: Eur. Phys. J. B 52, 61 (2006)

    Article  Google Scholar 

  18. Chavanis P.H.: Physica A 387, 787 (2008)

    Article  MathSciNet  Google Scholar 

  19. Ogorodnikov K.F.: Dynamics of Stellar Systems. Pergamon, Edinburg (1965)

    MATH  Google Scholar 

  20. Ambartsumian V.A.: Ann. Leningr. State U 22, 19 (1938)

    Google Scholar 

  21. Spitzer L.: Mon. not. R. astron. Soc. 100, 396 (1940)

    MATH  Google Scholar 

  22. Chandrasekhar S.: Astrophys. J. 97, 263 (1943)

    Article  MathSciNet  Google Scholar 

  23. Spitzer L., Härm R.: Astrophys. J. 127, 544 (1958)

    Article  Google Scholar 

  24. Michie R.: Mon. Not. R. Astron. Soc. 125, 127 (1963)

    MathSciNet  Google Scholar 

  25. King I.R.: Astron. J. 70, 376 (1965)

    Article  Google Scholar 

  26. Antonov V.A.: Vest. Leningr. Gos. Univ. 7, 135 (1962)

    Google Scholar 

  27. Lynden-Bell D., Wood R.: Mon. Not. R. Astron. Soc. 138, 495 (1968)

    Google Scholar 

  28. Padmanabhan T.: Phys. Rep. 188, 285 (1990)

    Article  MathSciNet  Google Scholar 

  29. de Vega H.J., Sanchez N.: Nucl. Phys. B 625, 409 (2002)

    Article  MATH  Google Scholar 

  30. Katz J.: Found. Phys. 33, 223 (2003)

    Article  MathSciNet  Google Scholar 

  31. Chavanis P.H.: Int J. Mod. Phys. B 20, 3113 (2006)

    Article  MATH  Google Scholar 

  32. Cohn H.: Astrophys. J. 242, 765 (1980)

    Article  Google Scholar 

  33. Inagaki S., Lynden-Bell D.: Mon. not. R. astron. Soc. 205, 913 (1983)

    MATH  Google Scholar 

  34. Chavanis P.H.: Astron. Astrophys. 432, 117 (2005)

    Article  MATH  Google Scholar 

  35. Pedlosky J.: Geophysical Fluid Dynamics. Springer, Berlin (1987)

    MATH  Google Scholar 

  36. Marcus P.S.: Annu. Rev. Astron. Astrophys. 31, 523 (1993)

    Article  Google Scholar 

  37. Miller J.: Phys. Rev. Lett. 65, 2137 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  38. Robert R., Sommeria J.: J. Fluid Mech. 229, 291 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  39. Chavanis P.H., Sommeria J.: J. Fluid Mech. 314, 267 (1996)

    Article  MATH  Google Scholar 

  40. Chavanis P.H., Sommeria J.: J. Fluid Mech. 356, 259 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  41. Robert R., Rosier C.: J. Stat. Phys. 86, 481 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  42. Kazantsev E., Sommeria J., Verron J.: J. Phys. Ocean. 28, 1017 (1998)

    Article  Google Scholar 

  43. Turkington B., Majda A., Haven K., Dibattista M.: Proc. Natl. Acad. Sci. 98, 12346 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  44. Bouchet F., Sommeria J.: J. Fluid Mech. 464, 165 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  45. Chavanis P.H., Sommeria J.: Phys. Rev. E 65, 026302 (2002)

    Article  MathSciNet  Google Scholar 

  46. Nezlin M.V., Snezhkin E.N.: Rossby Vortices, Spiral Structures, Solitons. Springer, Berlin (1993)

    Google Scholar 

  47. Barge P., Sommeria J.: Astron. Astrophys. 295, L1 (1995)

    Google Scholar 

  48. Bracco A., Chavanis P.H., Provenzale A., Spiegel E.: Phys. Fluids 11, 2280 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  49. Chavanis P.H., Sommeria J., Robert R.: Astrophys. J. 471, 385 (1996)

    Article  Google Scholar 

  50. Kirchhoff G.: In: Lectures in Mathematical Physics, Mechanics. Teubner, Leipzig (1877)

    Google Scholar 

  51. Aref H.: Ann. Rev. Fluid Mech. 25, 345 (1983)

    Article  MathSciNet  Google Scholar 

  52. Newton P.K.: The N-Vortex Problem: Analytical Techniques. Applied Mathematical Sciences, vol. 145. Springer, Berlin (2001)

    Google Scholar 

  53. Boatto S., Laskar J.: Chaos 13, 824 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  54. Onsager L.: Nuovo Cimento, Suppl. 6, 279 (1949)

    Article  MathSciNet  Google Scholar 

  55. Chavanis P.H.: Phys. Rev. E 64, 026309 (2001)

    Article  Google Scholar 

  56. Kawahara R., Nakanishi H.: J. Phys. Soc. Jpn. 76, 074001 (2007)

    Article  Google Scholar 

  57. Huang X.P., Driscoll C.F.: Phys. Rev. Lett. 72, 2187 (1994)

    Article  Google Scholar 

  58. Brands H., Chavanis P.H., Pasmanter R., Sommeria J.: Phys. Fluids 11, 3465 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  59. Kawahara R., Nakanishi H.: J. Phys. Soc. Jpn. 75, 054001 (2006)

    Article  Google Scholar 

  60. Chavanis P.H.: Physica A 387, 1123 (2008)

    Article  MathSciNet  Google Scholar 

  61. Montgomery D., Joyce G.: Phys. Fluids 17, 1139 (1974)

    Article  MathSciNet  Google Scholar 

  62. Lundgren T.S., Pointin Y.B.: J. Stat. Phys. 17, 323 (1977)

    Article  Google Scholar 

  63. Chavanis P.H., Lemou M.: Eur. Phys. J. B 59, 217 (2007)

    Article  MathSciNet  Google Scholar 

  64. Dauxois, T., Ruffo, S., Arimondo, E., Wilkens, M. (eds): Dynamics and thermodynamics of systems with long range interactions. Lecture Notes in Physics, vol. 602. Springer, Berlin (2002)

    Google Scholar 

  65. Antoni M., Ruffo S.: Phys. Rev. E 52, 2361 (1995)

    Article  Google Scholar 

  66. Chavanis P.H., Vatteville J., Bouchet F.: Eur. Phys. J. B 46, 61 (2005)

    Article  Google Scholar 

  67. Latora V., Rapisarda A., Tsallis C.: Phys. Rev. E 64, 056134 (2001)

    Article  Google Scholar 

  68. Yamaguchi Y., Barré J., Bouchet F., Dauxois T., Ruffo S.: Physica A 337, 36 (2004)

    Article  Google Scholar 

  69. Bouchet F., Dauxois T.: Phys. Rev. E 72, 5103 (2005)

    Article  Google Scholar 

  70. Chavanis P.H.: Eur. Phys. J. B 52, 47 (2006)

    Article  Google Scholar 

  71. Antoniazzi A., Fanelli D., Barré J., Chavanis P.H., Dauxois T., Ruffo S.: Phys. Rev. E 75, 011112 (2007)

    Article  Google Scholar 

  72. Campa A., Giansanti A., Morelli G.: Phys. Rev. E 76, 041117 (2007)

    Article  MathSciNet  Google Scholar 

  73. Campa A., Chavanis P.H., Giansanti A., Morelli G.: Phys. Rev. E 78, 040102 (2008)

    Article  Google Scholar 

  74. Chavanis P.H.: Physica A 361, 81 (2006)

    Article  MathSciNet  Google Scholar 

  75. Messer J., Spohn H.: J. Stat. Phys. 29, 561 (1982)

    Article  MathSciNet  Google Scholar 

  76. Kiessling M.: J. Stat. Phys. 55, 203 (1989)

    Article  MathSciNet  Google Scholar 

  77. Katz J.: Mon. not. R. Astron. Soc. 183, 765 (1978)

    MATH  Google Scholar 

  78. Thirring W.: Z. Phys. 235, 339 (1970)

    Article  Google Scholar 

  79. Lynden-Bell D., Lynden-Bell R.: Mon. Not. R. Astron. Soc. 181, 405 (1977)

    Google Scholar 

  80. Kadomtsev B.B., Pogutse O.P.: Phys. Rev. Lett. 25, 1155 (1970)

    Article  Google Scholar 

  81. Severne G., Luwel M.: Astrophys. Space Sci. 72, 293 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  82. Chavanis P.H.: Mon. Not. R. Astron. Soc. 300, 981 (1998)

    Article  Google Scholar 

  83. Chavanis P.H., Bouchet F.: Astron. Astrophys. 430, 771 (2005)

    Article  Google Scholar 

  84. Chavanis P.H.: Physica A 387, 1504 (2008)

    Article  MathSciNet  Google Scholar 

  85. Lenard A.: Ann. Phys. N.Y. 10, 390 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  86. Balescu R.: Phys. Fluids 3, 52 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  87. Gilbert I.H.: Astrophys. J. 152, 1043 (1968)

    Article  Google Scholar 

  88. Chandrasekhar S.: Astrophys. J. 99, 47 (1944)

    Article  MathSciNet  Google Scholar 

  89. Chandrasekhar S., von Neumann J.: Astrophys. J. 95, 489 (1942)

    Article  MATH  MathSciNet  Google Scholar 

  90. Rosenbluth M., MacDonald W., Judd D.: Phys. Rev. 107, 1 (1957)

    Article  MATH  MathSciNet  Google Scholar 

  91. Prigogine I., Severne G.: Physica 32, 1376 (1966)

    Article  MATH  Google Scholar 

  92. Severne G., Haggerty M.J.: Astrophys. Space Sci. 45, 287 (1976)

    Article  MathSciNet  Google Scholar 

  93. Kandrup H.: Astrophys. Space. Sci. 97, 435 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  94. Caglioti E., Lions P.L., Marchioro C., Pulvirenti M.: Commun. Math. Phys. 143, 501 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  95. Eyink G.L., Spohn H.: J. Stat. Phys. 70, 833 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  96. Kiessling M., Lebowitz J.: Lett. Math. Phys. 42, 43 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  97. Caglioti E., Lions P.L., Marchioro C., Pulvirenti M.: Commun. Math. Phys. 174, 229 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  98. Smith R.A., O’Neil T.M.: Phys. Fluids B 2, 2961 (1990)

    Article  Google Scholar 

  99. Robert R., Sommeria J.: Phys. Rev. Lett. 69, 2776 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  100. Chavanis P.H., Sommeria J.: Phys. Rev. Lett. 78, 3302 (1997)

    Article  Google Scholar 

  101. Chavanis P.H.: Phys. Rev. Lett. 84, 5512 (2000)

    Article  MathSciNet  Google Scholar 

  102. Dubin D., O’Neil T.M.: Phys. Rev. Lett. 60, 1286 (1988)

    Article  Google Scholar 

  103. Eyink G.L., Sreenivasan K.R.: Rev. Mod. Phys. 78, 87 (2006)

    Article  MathSciNet  Google Scholar 

  104. Khanin K.M.: Physica D 2, 261 (1982)

    Article  MathSciNet  Google Scholar 

  105. Dubin D.: Phys. Plasmas 10, 1338 (2003)

    Article  Google Scholar 

  106. Chavanis P.H.: Phys. Rev. E 58, R1199 (1998)

    Article  MathSciNet  Google Scholar 

  107. Chavanis P.H., Lemou M.: Phys. Rev. E 72, 061106 (2005)

    Article  MathSciNet  Google Scholar 

  108. Chavanis P.H.: AIP Conf. Proc. 970, 39 (2008)

    Article  MathSciNet  Google Scholar 

  109. Campa, A., Dauxois, T., Ruffo, S.: arXiv:0907.0323

  110. Ichimaru S.: Basic Principles of Plasma Physics. W.A. Benjamin, Reading (1973)

    Google Scholar 

  111. Nicholson D.R.: Introduction to Plasma Theory. Krieger Publishing Company, Florida (1992)

    Google Scholar 

  112. Balescu R.: Statistical Mechanics of Charged Particles. Interscience, New York (1963)

    MATH  Google Scholar 

  113. Lifshitz E.M., Pitaevskii L.P.: Physical Kinetics. Pergamon Press, Oxford (1981)

    Google Scholar 

  114. Benedetti C., Rambaldi S., Turchetti G.: Physica A 364, 197 (2006)

    Article  Google Scholar 

  115. Hubbard J.: Proc. R. Soc. Lond. 260, 114 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  116. Bouchet F.: Phys. Rev. E 70, 036113 (2004)

    Article  Google Scholar 

  117. Chavanis P.H.: Physica A 377, 469 (2007)

    Article  MathSciNet  Google Scholar 

  118. Marmanis H.: Proc. R. Soc. Lond. A 454, 587 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  119. Sire C., Chavanis P.H.: Phys. Rev. E 61, 6644 (2000)

    Article  MathSciNet  Google Scholar 

  120. Newton P.K., Mezic I.: J. Turbul. 3, 52 (2002)

    MathSciNet  Google Scholar 

  121. Nazarenko S., Zakharov V.E.: Physica D 56, 381 (1992)

    Article  MATH  MathSciNet  Google Scholar 

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  1. Laboratoire de Physique Théorique, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse, France

    Pierre-Henri Chavanis

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  1. Pierre-Henri Chavanis
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Chavanis, PH. Kinetic theory of stellar systems, two-dimensional vortices and HMF model. Theor. Comput. Fluid Dyn. 24, 217–239 (2010). https://doi.org/10.1007/s00162-009-0129-2

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