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Equilibrium properties of the Vlasov functional: The generalized Poisson-Boltzmann-Emden equation

François Bavaud*

François Bavaud*

  • Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, Scotland

  • *New address: Institut de Mathématiques Appliquées, Université de Lausanne, BFSH2, CH-1015 Lausanne, Switzerland.
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Rev. Mod. Phys. 63, 129 – Published 1 January, 1991

DOI: https://doi.org/10.1103/RevModPhys.63.129

Abstract

This article investigates in a systematic way the properties of the classical continuous mean-field theory governed by the generalized Poisson-Boltzmann-Emden equation ρ(x)=Aexp[−β∫Λdyρ(y)V(x−y)] together with the associated variational problem infopρ12∫Λdx∫Λdyρ(x)ρ(y)V(x−y)+kT∫Λdxρ(x)lnρ(x). Origins of the theory are traced back. Past studies (freezing theories, electrostatic and self-gravitating systems) are relocated in a broader framework. New results concerning the thermodynamic limit, phase transitions, metastability, and the shape of density profiles are provided. In particular, the question of ground states (in relationship to condensation and wetting phenomena) is illustrated by numerous explicit solutions.

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References (95)

  1. Amann, H., 1976a, SIAM Rev. 18, 620
  2. Amann, H., 1976b, J. Diff. Eq. 21, 363
  3. Bandle, C., 1975, Arch. Rat. Mech. Anal. 58, 219
  4. Batt, J., W. Faltenbacher, and E. Horst, 1986, Arch. Rat. Mech. Anal. 93, 159
  5. Battle, G. A., 1977, Commun. Math. Phys. 55, 299
  6. Baumgartner, B., 1976, Commun. Math. Phys. 48, 207
  7. Baus, M., 1987, J. Stat. Phys. 48, 1129
  8. Bavaud, F., 1986, J. Stat. Phys. 45, 171
  9. Bavaud, F., Ph. Choquard, and J. R. Fontaine, 1986, J. Stat. Phys. 42, 621
  10. Bouchut, F., 1990, "Global weak solutions of the Vlasov-Poisson system for small electron mass," Université d'Orléans, preprint
  11. Braun, W., and K. Hepp, 1977, Commun. Math. Phys. 56, 101
  12. Brout, R., 1963, Physica 29, 1041
  13. Cally, P., and J. J. Monaghan, 1981, J. Math. Phys. 22, 348
  14. Camm, G. L., 1950, Mon. Not. R. Astron. Soc. 110, 305
  15. Chandrasekhar, S., 1967, An Introduction to the Study of Stellar Structure (Dover, New York)
  16. Choquard, Ph., and R. Rentsch, 1986, Physica A 138, 125
  17. Chow, S. N., and J. K. Hale, 1982, Methods of Bifurcation Theory (Springer-Verlag, Berlin/Vienna/New York)
  18. Co, K. U., J. J. Kozak, and K. D. Lucks, 1976, J. Chem. Phys. 64, 2197
  19. Conlon, J. G., 1984, Commun. Math. Phys. 94, 439
  20. Dancer, E. N., 1988, J. Diff. Eq. 74, 120
  21. Davidson, N., 1962, Statistical Mechanics (McGraw-Hill, New York)
  22. Debye, P., and E. Hückel, 1923, Phys. Z. 24, 185
  23. Desvillettes, L., and J. Dolbeault, 1990, "On long-time asymptotics of the Vlasov-Poisson-Boltzmann equation," Université de Paris VII, preprint
  24. Devroye, L., 1987, A Course in Density Estimation (Birkhäuser, Cambridge, Mass.)
  25. Dixmier, J., 1964, Les C*-Algebres et leurs Représentations (Gauthier-Villars, Paris)
  26. Dobrushin, R. L., 1964, Th. Prob. Appl. (U.S.S.R.) 9, 646
  27. Dolph, C. L., 1949, Trans. Am. Math. Soc. 66, 289
  28. Doob, J. L., 1984, Classical Potential Theory and its Probabilistic Counterpart (Springer, New York)
  29. Ekeland, I., and R. Temam, 1974, Analyse Convexe et Problèmes Variationnels (Bordas, Paris)
  30. Emden, R., 1907, Gaskugeln (Teubner, Leipzig)
  31. Falk, H., 1969, Am. J. Phys. 38, 858
  32. Feijoo, L., and A. Rahman, 1982, J. Chem. Phys. 77, 5687
  33. Fisher, I. Z., T. M. Switz1964, The Statistical Theory of Liquids, translated by (University of Chicago, Chicago)
  34. Fisher, M. E., and D. Ruelle, 1965, J. Math. Phys. 7, 260
  35. Garrett, A. J. M., and L. Poladian, 1988, Ann. Phys. (N.Y.) 188, 386
  36. Gel'fand, I. M., 1963, Trans. Am. Math. Soc. 29, 295
  37. Gielerak, R., 1989, J. Math. Phys. 30, 115
  38. Gilbarg, D., and N. S. Trudinger, 1983, Elliptic Partial Differential Equations of Second Order (Springer, New York)
  39. Gogny, D., and P. L. Lions, 1989, Modélisation mathématique et analyse numérique 23, 137
  40. Hardy, G., J. Littelwood, and G. Pólya, 1967, Inequalities (Cambridge University, Cambridge, England)
  41. Helms, L. L., 1975, Introduction to Potential Theory (R. E. Krieger, New York)
  42. Hill, T. L., 1956, Statistical Mechanics (McGraw-Hill, New York)
  43. Horwitz, G., and D. Katz, 1977, Astrophys. J. 211, 226
  44. Huang, K., 1963, Statistical Mechanics (Wiley, New York)
  45. Hudson, V., and J. S. Pym, 1980, Applications of Functional Analysis and Operator Theory (Academic, New York)
  46. Jacobs, R. L., 1983, J. Phys. C 16, 273
  47. Jackson, J. D., 1962, Classical Electrodynamics (Wiley, New York)
  48. Jancovici, B., 1965, Physica 31, 1017
  49. Jones, R. O., and O. Gunnarsson, 1989, Rev. Mod. Phys. 61, 689
  50. Joseph, D. D., and T. S. Lundgren, 1973, Arch. Rat. Mech. Anal. 49, 242
  51. Kawohl, B., 1985, Rearrangements and Convexity of Level Sets in PDE, Lecture Notes in Mathematics No. 1150 (Springer, New York)
  52. Kennedy, T., 1984, J. Stat. Phys. 37, 529
  53. Kiessling, M. K.-H., 1989, J. Stat. Phys. 55, 203
  54. Kiessling, M. K.-H., 1990, J. Stat. Phys. 59, 1157
  55. Kirkwood, J. G., and E. M. Boggs, 1942, J. Chem. Phys. 10, 394
  56. Kirkwood, J. G., and E. Monroe, 1941, J. Chem. Phys. 9, 514
  57. Krasnosel'skii, M. A., 1963, Topological Methods in the Theory of Nonlinear Integral Equations (Pergamon, New York)
  58. Kullback, S., 1968, Information Theory and Statistics (Dover, New York)
  59. Kurth, R., 1978, Q. J. Appl. Math. 36, 325
  60. Lebowitz, J. L., 1960, Phys. Fluids 3, 64
  61. Lewis, J. T., J. V. Pulé, and Ph. de Smedt, 1984, J. Stat. Phys. 35, 381
  62. Lewis, J. T. and T. C. Dorlas, 1990, unpublished
  63. Lieb, E. H., 1977, Stud. Appl. Math. 57, 93
  64. Lieb, E. H., and H.-T. Yau, 1987, Commun. Math. Phys. 112, 147
  65. Liouville, J., 1853, J. Mathématiques 18, 71
  66. Lovett, R., 1977, J. Chem. Phys. 66, 1225
  67. Lovett, R., and F. P. Buff, 1980, J. Chem. Phys. 72, 2425
  68. Luttinger, J., 1973, J. Math. Phys. 14, 586
  69. Lynden-Bell, D., and R. M. Lynden-Bell, 1977, Mon. Not. R. Astron. Soc. 181, 405
  70. Lynden-Bell, D., and R. Wood, 1968, Mon. Not. R. Astron. Soc. 138, 495
  71. Messer, J., 1981, J. Math. Phys. 22, 2910
  72. Messer, J., and H. Spohn, 1982, J. Stat. Phys. 29, 561
  73. Neunzert, H., 1978, Fluid Dyn. Trans. 9, 229
  74. Ostriker, J., 1964, Astrophys. J. 140, 1056
  75. Padmanabhan, T., 1990, Phys. Rep. 188, 285
  76. Penrose, O., and P. C. Fife, 1990, Physica D 43, 44
  77. Posch, H. A., H. Narnhofer, and W. Thirring, 1990, Phys. Rev. A 42, 1880
  78. Raveché, H. J., and R. F. Kayser, 1978, J. Chem. Phys. 68, 3632
  79. Raveché, H. J., and C. A. Stuart, 1975, J. Chem. Phys. 63, 1099
  80. Raveché, H. J., and C. A. Stuart, 1976, J. Math. Phys. 17, 1949
  81. Reichl, L. E., 1980, A Modern Course in Statistical Physics (University of Texas, Austin)
  82. Ruelle, D., 1969, Statistical Mechanics: Rigorous Results (Benjamin, New York)
  83. Rybicki, G. R., 1971, Astrophys. Space Sci. 14, 56
  84. Salzberg, A. S., 1965, J. Math. Phys. 6, 158
  85. Schwartz, L., 1966, Théorie des distributions (Hermann, Paris)
  86. Siegert, A. J. F., and E. Meeron, 1966, J. Math. Phys. 7, 741
  87. Spohn, H., 1980, Rev. Mod. Phys. 53, 569
  88. Stell, G., 1964, in The Equilibrium Theory of Classical Fluids, edited by H. L. Frisch and J. Lebowitz (Benjamin, New York)
  89. Stodólkiewicz, J. S., 1963, Acta Astron. 13, 30
  90. Suzuki, T., and K. Nagasaki, 1988, Trans. Am. Math. Soc. 309, 591
  91. Tricomi, F. G., 1957, Integral Equations (Interscience, New York)
  92. Vainberg, M. M., 1973, Variational Method and Method of Monotone Operators (Wiley, New York)
  93. Van den Berg, M., T. C. Dorlas, J. T. Lewis, and J. V. Pulé, 1990, Commun. Math. Phys. 127, 41
  94. Vlasov, A. A., 1938, Sov. Phys. JETP 8, 1
  95. Weeks, J. D., S. A. Rice, and J. J. Kojak, 1970, J. Chem. Phys. 52, 2416

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Rev. Mod. Phys. 63, 129– Published 1 January, 1991

    © 1991 American Physical Society

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