Strong Correlations and Equation of State of Dense Gases

  • Werner EbelingEmail author
  • Vladimir E. Fortov
  • Vladimir Filinov
Part of the Springer Series in Plasma Science and Technology book series (SSPST)


We summarize the physics and the statistical theory of strongly coupled gases. Starting from the binary correlation functions, we develop systematic expansions with respect to the density, the so-called virial expansions, and also fugacity expansions. Further, we discuss solutions of integral equations (PY and HNC) for dense systems. Quantum effects are treated by the methods of Slater and Beth-Uhlenbeck. Finally, we discuss strongly correlated Fermi-Dirac and Bose-Einstein gases, and Yukawa fluids.


  1. A.A. Abrikosov, L. Gorkov, I.E. Dzyaloshinskyi, Methods of quantum field theory in statistical physics, Moscow 1962 (Engl, Pergamon London, 1965)Google Scholar
  2. J.A. Barker, D. Henderson, Perturbation theory and EOS for fluids. J. Chem. Phys. 47, 2856 (1967)ADSCrossRefGoogle Scholar
  3. N. Bessis, G. Bessis, G. Corbel, B. Dakhel, Bound state energies of the exponentially screened potentials. J. Chem. Phys. 63, 3744–3749 (1975)ADSCrossRefGoogle Scholar
  4. S.T. Belyaev, Energy spectrum of a non-ideal Bose gas. JETP 7, 299 (1958)zbMATHGoogle Scholar
  5. E. Beth, G.E. Uhlenbeck, The quantum theory of the non-ideal gas II. Behaviour at low temperatures. Physica 4, 915–924 (1937)zbMATHGoogle Scholar
  6. V.B. Bobrov, S.A. Trigger, Criterion of superfluidity, excitations and heat capacity singularity in superfluid helium. Progr. Theor. Exp. Phys. 04301 (2013)Google Scholar
  7. V.B. Bobrov, A.G. Zagorodny, S.A. Trigger, Coulomb interaction and Bose-Einstein condensate (Russ.). Low Temp. Phys. 41, 1154–1163 (2015)Google Scholar
  8. N.N. Bogoliubov, On the theory of superfluidity. Akad. Nauk USSR, Phys. Ser. 11, 1947 (1947)Google Scholar
  9. N.N. Bogoliubov, N.N. Bogoliubov Jr., Introduction to Quantum Statistical Mechanics, New York Gordon and Breach 1992Google Scholar
  10. N.N. Bogoliubov, Collected papers, vol. 112, Fizmatlit Moscow (2005–2009)Google Scholar
  11. K.A. Brueckner, K. Sawada, Bose-Einstein gas with repulsive interactions. Phys. Rev. 106, 1117 (1957)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. E.D.M. Costa, N.H.T. Lemes, M.O. Alves, R.C.O. Sebastiao, J.P. Braga, Quantum second virial coefficient calculation for the 4-He dimer on a recent potential. J. Braz. Chem. Soc. 24, 363–368 (2013)CrossRefGoogle Scholar
  13. C.A. Croxston, Liquid State Physics, A statistical mechanical introduction (Cambridge University Press, Cambridge, 2009)Google Scholar
  14. K.B. Davis, M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn, W. Ketterle, Bose-Einstein condensation in a gas of sodium atoms. Phys. Rev. Lett. 75, 3969 (1995)ADSCrossRefGoogle Scholar
  15. H.E. DeWitt, Thermodynamic functions of a partially degenerate, fully ionized gas. J. Math. Phys. 2, 27 (1961)CrossRefGoogle Scholar
  16. H.E. De Witt, Statistical mechanics of high-temperature quantum plasmas beyond the ring approximation. J. Math. Phys. 7, 616–626 (1966)ADSMathSciNetCrossRefGoogle Scholar
  17. R. Dutt, U. Mukherjee, Improved approximation for the bound \(s\)-states of the static screened Coulomb potential. Z. Phys. A. Atoms Nucl. 302, 199–201 (1981)ADSCrossRefGoogle Scholar
  18. W. Ebeling, Statistical derivation of the mass action law or interacting gases and plasmas. Physica 73, 573–584 (1974)ADSCrossRefGoogle Scholar
  19. W. Ebeling, H.J. Hoffmann, G. Kelbg, Quantum statistics of high-temperature plasmas in thermodynamic equilibrium. Contr. Plasma Phys. 7, 233 (1967)Google Scholar
  20. W. Ebeling, W.D. Kraeft, D. Kremp, Theory of Bound States and Ionisation Equilibrium in Plasmas and Solids (Akademie, Berlin, 1976); extended Russ. translation Mir, Moscow, 1979Google Scholar
  21. W. Ebeling, W.D. Kraeft, G. Röpke, On the quantum statistics of bound states in Rutherford’s model. Ann. Physik (Berlin) 524, 311–326 (2012)ADSCrossRefzbMATHGoogle Scholar
  22. W. Ebeling, W.D. Kraeft, G. Röpke, Bound states in Coulomb systems. Contr. Plasma Phys. 52, 7–16 (2012)ADSCrossRefzbMATHGoogle Scholar
  23. R.P. Feynman, Atomic theory of the two-fluid model of liquid helium. Phys. Rev. 94, 262 (1954)ADSCrossRefzbMATHGoogle Scholar
  24. R.P. Feynman, Statistical Mechanics (Benjamin, Reading Mass, 1972)Google Scholar
  25. R.P. Feynman, A.R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965)zbMATHGoogle Scholar
  26. M. Fisher, D.M. Zuckerman, Exact thermodynamic formulation of chemical association. J. Chem. Phys. 70, 100 (1998)Google Scholar
  27. V.E. Fortov, A.V. Ivlev, S.A. Khrapak, A.G. Khrapak, G.E. Morfill, Complex (dusty) plasmas: current status, open issues, perspectives. Phys. Rep. 421, 1–103 (2005)ADSMathSciNetCrossRefGoogle Scholar
  28. H.L. Friedman, Ionic Solution Theory (Interscience, New York, London, 1962)Google Scholar
  29. H.L. Friedman, W. Ebeling, Theory of interacting and reacting particles. Rostocker Physikalische Manuskripte 4, 33–48 (1979)Google Scholar
  30. V.M. Galitskii, Energy spectrum of a Fermi gas. JETP 7, 104 (1958)Google Scholar
  31. V.M. Galitskii, A.B. Migdal, S.T. Belyaev, Applications of the quantum-field theory methods to the many-body problems. JETP 7, 96 (1958)MathSciNetGoogle Scholar
  32. M. Gellman, K.A. Brueckner, Phys. Rev. 106, 364 (1957)ADSMathSciNetCrossRefGoogle Scholar
  33. D. Greenberger, K. Hentschel, F. Weinert (eds.), Compendium of Quantum Physics Concepts, Experiments, History and Philosophy (Springer, Berlin, 2009)zbMATHGoogle Scholar
  34. D. Henderson, Analytic methods for the Percus-Yevick hard sphere correlation functions. Cond. Matter Phys. 12, 127–135 (2009)CrossRefGoogle Scholar
  35. J.O. Hirschfelder, C.F. Curtiss, R.B. Bird, Molecular Theory of Gases and Liquids (Wiley, New York, 1954)zbMATHGoogle Scholar
  36. T.L. Hill, Statistical Mechanics (McGraw Hill, New York, 1956). Moskva 1960Google Scholar
  37. K. Huang, Introduction to Statistical Physics (Taylor and Francis, Abingdon, 2001)zbMATHGoogle Scholar
  38. K. Huang, Statistical Mechanics (Wiley, New York, 1987)zbMATHGoogle Scholar
  39. G. Kelbg, H.J. Hoffmann, Quantenstatistik realer Gase und Plasmen. Ann. Physik 14, 310–318 (1964)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. K. Kilimann, W. Ebeling, Energy gap and line shifts for H-like ions in dense plasmas. Z. Naturforsch. 45a, 613–617 (1990)Google Scholar
  41. J.E. Kilpatrick, W.E. Keller, E.F. Hammel, N. Metropolis, Second virial coefficient of He 3 and He 4. Phys. Rev. 94, 1103 (1954)ADSCrossRefzbMATHGoogle Scholar
  42. Y. Kano, N. Mishima, Ann. Phys. N.Y. 51, 203 (1969)Google Scholar
  43. W.D. Kraeft, W. Ebeling, D. Kremp, Phys. Lett. 29A, 466 (1969)ADSCrossRefGoogle Scholar
  44. W.D. Kraeft, D. Kremp, W. Ebeling, G. Röpke, Quantum Statistics of Charged Particle Systems (Berlin and Pergamon Press, New York, Akademie, 1986)CrossRefGoogle Scholar
  45. D. Kremp, W.D. Kraeft, Analyticity of the second virial coefficient as a function of the interaction parameter and compensation between bound and scattering states. Phys. Lett. A 38, 167–168 (1972)ADSCrossRefGoogle Scholar
  46. D. Kremp, M. Schlanges, W.D. Kraeft, Quantum Statistics of Nonideal Plasmas (Springer, 2005)Google Scholar
  47. L.D. Landau, E.M. Lifshits, Quantum mechanics: Non-relativistic theory, Nauka Moska 1976 (In Engl. Pergamon Press, Oxford, 1977)Google Scholar
  48. L.D. Landau, E.M. Lifshits, Statistical Physics (Part I). Nauka Moska (1976), in German Berlin 1979, in Engl. Butterworth–Heinemann 1980Google Scholar
  49. T.K. Langin, T. Strickler, N. Maksimovic, P. McQuillen, T. Pohl, D. Vrinceanu, T.C. Killian, Demonstrating universal scaling for dynamics of Yukawa OCP. Phys. Rev. E 93, 023201 (2016)ADSCrossRefGoogle Scholar
  50. H. Lehmann, W. Ebeling, Wave function properties and shifted energy levels of bound states in a plasma. Z. Naturforsch. 46a, 583–589 (1991)Google Scholar
  51. F.G. Lether, Analytical expansions and numerical approximations of Fermi-Dirac integrals. J. Sci. Computing 15, 479 (2000)Google Scholar
  52. E. Montroll, J. Ward, Quantum statistics of interacting particles. Phys. Fluids 1, 55–72 (1958)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. T. Morita, Equation of state of high temperature plasma. Prog. Theor. Phys. (Kyoto) 22, 757–774 (1959)ADSCrossRefzbMATHGoogle Scholar
  54. N.F. Mott, Phil. Mag. 6, 287 (1961)ADSCrossRefGoogle Scholar
  55. T. Ott, M. Bonitz, L.G. Stanton, M.S. Murillo, Coupling strength in Coulomb and Yukawa one-component plasmas. Phys. Plasma 21, 113704 (2014)CrossRefGoogle Scholar
  56. C.J. Pethick, H. Smith, Bose-Einstein Condensation in Dilute Gases (Cambridge University Press, Cambridge, 2001)CrossRefGoogle Scholar
  57. D. Pines, P. Nozieres, The theory of quantum liquids, Benjamin, New York, Amsterdam 1966 (Russ. Transl. Mir, Moskva, 1967)Google Scholar
  58. L.P. Pitaevskii, S. Stringari, Bose-Einstein Condensation (Clarendon Press, Oxford, 2003)zbMATHGoogle Scholar
  59. J.K. Percus, G.J. Yevick, Analysis of classical statistical mechanics by means of collective coordinates. Phys. Rev. 110, 1103 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  60. F. Perrot, M.W.C. Dharma-Wardana, Phys. Rev A 30, 2619 (1984)ADSCrossRefGoogle Scholar
  61. D. Kremp, W.D. Kraeft, W. Ebeling, Quantum statistics of the second virial coefficients and scattering theory. Physica 51, 164–164 (1971)CrossRefGoogle Scholar
  62. D. Pines, P. Nozieres, The Theory of Quantum Liquids (Benjamin, New York, 1966); Russ. Translation Mir, Moskva (1967)Google Scholar
  63. F.J. Rogers, H.C. Graboske, D.J. Harwood, Bound eigenstates of the static screened Coulomb potential. Phys. Rev. A 1, 1577–1586 (1970)ADSCrossRefGoogle Scholar
  64. M. Ross, Physics of dense fluids, Proc. Conf. Advances in High Pressure Studies, Corfu 1986, (Preprint UCRL-94997)Google Scholar
  65. M. Ross, Phys. Rev. B 54, 9589 (1996); 58, 669 (1998)Google Scholar
  66. J.C. Slater, Introduction to Chemical Physics (Mc Graw Hill, New York, 1939)Google Scholar
  67. M. Steinberg, J. Ortner, W. Ebeling, Phys. Rev. E 58, 3806 (1998)ADSCrossRefGoogle Scholar
  68. S.A. Trigger, W. Ebeling, G.J.F. van Heijst, D. Litinski, Analysis of linear and nonlinear conductivity of plasma systems by Fokker–Planck equations, arXiv:1412.4282 (2014), Plasma Physics (2015)
  69. S.A. Trigger, A.G. Zagorodny, Brownian motion of grains and negative friction in dusty plasmas, Conden. Matter Phys. 7(3(39)), 629–638 (2004)Google Scholar
  70. G.E. Uhlenbeck, E. Beth, The quantum theory of the non-ideal gas I. Deviations from the classical theory. Physica 3, 729–745 (1936)zbMATHGoogle Scholar
  71. A.A. Vedenov, A.I. Larkin, Equation of state of plasmas. Sov. Phys. JETP 9, 806–821 (1959)MathSciNetzbMATHGoogle Scholar
  72. A. Wasserman, T.J. Buckholtz, H.E. De Witt, Evaluation of some Fermi-Dirac integrals. J. Math. Phys. 11, 427 (1970)ADSMathSciNetCrossRefGoogle Scholar
  73. R. Zimmermann, Many-particle theory of of highly excited semiconductors, Teubner Leipzig (1987)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Werner Ebeling
    • 1
    Email author
  • Vladimir E. Fortov
    • 2
  • Vladimir Filinov
    • 3
  1. 1.Institut für PhysikHumboldt Universität BerlinBerlinGermany
  2. 2.Russian Academy of SciencesMoscowRussia
  3. 3.Joint Institute for High TemperaturesRussian Academy of SciencesMoscowRussia

Personalised recommendations