Skip to main content
SpringerLink
Log in

Exact Solution of the Percus–Yevick Approximation for Hard Spheres …and Beyond

  • Chapter
  • First Online:
A Concise Course on the Theory of Classical Liquids

Part of the book series: Lecture Notes in Physics ((LNP,volume 923))

  • 1866 Accesses

Abstract

One of the milestones of the statistical-mechanical theory of liquids in equilibrium was the exact analytical solution of the Percus–Yevick integral equation for three-dimensional hard spheres by, independently, Wertheim and Thiele in 1963. The solution was later extended to mixtures, hard hyperspheres in odd dimensions, and sticky hard spheres. This chapter presents some of those solutions within the framework of the rational-function approximation, which lends itself to generalizations beyond the Percus–Yevick approximation.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics (Wiley, New York, 1974)

    MATH  Google Scholar 

  2. M.S. Wertheim, Phys. Rev. Lett. 10, 321 (1963)

    Article  ADS  MathSciNet  Google Scholar 

  3. E. Thiele, J. Chem. Phys. 39, 474 (1963)

    Article  ADS  Google Scholar 

  4. M.S. Wertheim, J. Math. Phys. 5, 643 (1964)

    Article  ADS  MathSciNet  Google Scholar 

  5. J.L. Lebowitz, Phys. Rev. 133, A895 (1964)

    Article  ADS  MathSciNet  Google Scholar 

  6. J.S. Høye, L. Blum, J. Stat. Phys. 16, 399 (1977)

    Article  ADS  Google Scholar 

  7. C. Freasier, D.J. Isbister, Mol. Phys. 42, 927 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  8. E. Leutheusser, Physica A 127, 667 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  9. R.J. Baxter, J. Chem. Phys. 49, 2770 (1968)

    Article  ADS  Google Scholar 

  10. J.W. Perram, E.R. Smith, Chem. Phys. Lett. 35, 138 (1975)

    Article  ADS  Google Scholar 

  11. B. Barboy, Chem. Phys. 11, 357 (1975)

    Article  ADS  Google Scholar 

  12. B. Barboy, R. Tenne, Chem. Phys. 38, 369 (1979)

    Article  ADS  Google Scholar 

  13. S.B. Yuste, A. Santos, Phys. Rev. A 43, 5418 (1991)

    Article  ADS  Google Scholar 

  14. S.B. Yuste, M. López de Haro, A. Santos, Phys. Rev. E 53, 4820 (1996)

    Article  ADS  Google Scholar 

  15. M. López de Haro, S.B. Yuste, A. Santos, in Theory and Simulation of Hard-Sphere Fluids and Related Systems, ed. by A. Mulero. Lecture Notes in Physics, vol. 753 (Springer, Berlin, 2008), pp. 183–245

    Google Scholar 

  16. J.R. Solana, Perturbation Theories for the Thermodynamic Properties of Fluids and Solids (CRC Press, Boca Raton, 2013)

    Book  MATH  Google Scholar 

  17. R.D. Rohrmann, A. Santos, Phys. Rev. E 76, 051202 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  18. L. Carlitz, Duke Math. J. 24, 151 (1957)

    Article  MathSciNet  Google Scholar 

  19. J. Abate, W. Whitt, Queueing Syst. 10, 5 (1992)

    Article  MathSciNet  Google Scholar 

  20. A. Santos, Radial distribution function for hard spheres. Wolfram Demonstrations Project (2013), http://demonstrations.wolfram.com/RadialDistributionFunctionForHardSpheres/

  21. H. Reiss, H.L. Frisch, J.L. Lebowitz, J. Chem. Phys. 31, 369 (1959)

    Article  ADS  MathSciNet  Google Scholar 

  22. E. Helfand, H.L. Frisch, J.L. Lebowitz, J. Chem. Phys. 34, 1037 (1961)

    Article  ADS  Google Scholar 

  23. J.L. Lebowitz, E. Helfand, E. Praestgaard, J. Chem. Phys. 43, 774 (1965)

    Article  ADS  Google Scholar 

  24. A. Santos, Phys. Rev. Lett. 109, 120601 (2012)

    Article  ADS  Google Scholar 

  25. K.W. Kratky, J. Stat. Phys. 27, 533 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  26. S. Labík, J. Kolafa, A. Malijevský, Phys. Rev. E 71, 021105 (2005)

    Article  ADS  Google Scholar 

  27. N. Clisby, B.M. McCoy, J. Stat. Phys. 122, 15 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  28. R.J. Wheatley, Phys. Rev. Lett. 110, 200601 (2013)

    Article  ADS  Google Scholar 

  29. C. Zhang, B.M. Pettitt, Mol. Phys. 112, 1427 (2014)

    Article  ADS  Google Scholar 

  30. A.J. Schultz, D.A. Kofke, Phys. Rev. E 90, 023301 (2014)

    Article  ADS  Google Scholar 

  31. J. Kolafa, S. Labík, A. Malijevský, Phys. Chem. Chem. Phys. 6, 2335 (2004). See also http://www.vscht.cz/fch/software/hsmd/

    Article  Google Scholar 

  32. S.B. Yuste, A. Santos, J. Stat. Phys. 72, 703 (1993)

    Article  ADS  Google Scholar 

  33. S.B. Yuste, A. Santos, Phys. Rev. E 48, 4599 (1993)

    Article  ADS  Google Scholar 

  34. S.B. Yuste, A. Santos, M. López de Haro, J. Chem. Phys. 108, 3683 (1998)

    Article  ADS  Google Scholar 

  35. A. Santos, S.B. Yuste, M. López de Haro, J. Chem. Phys. 109, 6814 (1998)

    Article  ADS  Google Scholar 

  36. A. Malijevský, S.B. Yuste, A. Santos, J. Chem. Phys. 125, 074507 (2006)

    Article  ADS  Google Scholar 

  37. A. Santos, R.D. Rohrmann, Phys. Rev. E 87, 052138 (2013)

    Article  ADS  Google Scholar 

  38. M. Mandell, H. Reiss, J. Stat. Phys. 13, 113 (1975)

    Article  ADS  Google Scholar 

  39. Y. Rosenfeld, J. Chem. Phys. 89, 4272 (1988)

    Article  ADS  Google Scholar 

  40. M. Heying, D. Corti, J. Phys. Chem. B 108, 19756 (2004)

    Article  Google Scholar 

  41. T. Boublík, J. Chem. Phys. 53, 471 (1970)

    Article  ADS  Google Scholar 

  42. E.W. Grundke, D. Henderson, Mol. Phys. 24, 269 (1972)

    Article  ADS  Google Scholar 

  43. L.L. Lee, D. Levesque, Mol. Phys. 26, 1351 (1973)

    Article  ADS  Google Scholar 

  44. G.A. Mansoori, N.F. Carnahan, K.E. Starling, J.T.W. Leland, J. Chem. Phys. 54, 1523 (1971)

    Article  ADS  Google Scholar 

  45. M. Barošová, A. Malijevský, S. Labík, W.R. Smith, Mol. Phys. 87, 423 (1996)

    Article  ADS  Google Scholar 

  46. M.A. Miller, D. Frenkel, Phys. Rev. Lett. 90, 135702 (2003)

    Article  ADS  Google Scholar 

  47. M.A. Miller, D. Frenkel, J. Chem. Phys. 121, 535 (2004)

    Article  ADS  Google Scholar 

  48. J. Largo, M.A. Miller, F. Sciortino, J. Chem. Phys. 128, 134513 (2008)

    Article  ADS  Google Scholar 

  49. M.A. Miller, D. Frenkel, J. Phys. Condens. Matter 16, S4901 (2004)

    Article  ADS  Google Scholar 

  50. A.J. Post, E.D. Glandt, J. Chem. Phys. 84, 4585 (1986)

    Article  ADS  Google Scholar 

  51. N.A. Seaton, E.D. Glandt, J. Chem. Phys. 84, 4595 (1987)

    Article  ADS  Google Scholar 

  52. N.A. Seaton, E.D. Glandt, J. Chem. Phys. 86, 4668 (1987)

    Article  ADS  Google Scholar 

  53. N.A. Seaton, E.D. Glandt, J. Chem. Phys. 87, 1785 (1987)

    Article  ADS  Google Scholar 

  54. R.D. Rohrmann, A. Santos, Phys. Rev. E 89, 042121 (2014)

    Article  ADS  Google Scholar 

  55. R.O. Watts, D. Henderson, R.J. Baxter, Adv. Chem. Phys. 21, 421 (1971)

    Google Scholar 

  56. M.G. Noro, D. Frenkel, J. Chem. Phys. 113, 2941 (2000)

    Article  ADS  Google Scholar 

  57. H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, Oxford, 1971)

    Google Scholar 

  58. L.E. Reichl, A Modern Course in Statistical Physics, 2nd edn. (Wiley, New York, 1998)

    MATH  Google Scholar 

  59. S. Fishman, M.E. Fisher, Physica A 108, 1 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  60. A. Malijevský, A. Malijevský, S.B. Yuste, A. Santos, M. López de Haro, Phys. Rev. E 66, 061203 (2002)

    Article  ADS  Google Scholar 

  61. A. Malijevský, S.B. Yuste, A. Santos, M. López de Haro, Phys. Rev. E 75, 061201 (2007)

    Article  ADS  Google Scholar 

  62. S.B. Yuste, A. Santos, M. López de Haro, J. Chem. Phys. 128, 134507 (2008). Erratum: 140, 179901 (2014)

    Google Scholar 

  63. R.D. Rohrmann, A. Santos, Phys. Rev. E 83, 011201 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  64. R.D. Rohrmann, A. Santos, Phys. Rev. E 84, 041203 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  65. A. Malijevský, A. Santos, J. Chem. Phys. 124, 074508 (2006)

    Article  ADS  Google Scholar 

  66. A. Malijevský, S.B. Yuste, A. Santos, Phys. Rev. E 76, 021504 (2007)

    Article  ADS  Google Scholar 

  67. R. Fantoni, A. Giacometti, A. Malijevský, A. Santos, J. Chem. Phys. 131, 124106 (2009)

    Article  ADS  Google Scholar 

  68. S.B. Yuste, A. Santos, J. Chem. Phys. 101, 2355 (1994)

    Article  ADS  Google Scholar 

  69. L. Acedo, A. Santos, J. Chem. Phys. 115, 2805 (2001)

    Article  ADS  Google Scholar 

  70. J. Largo, J.R. Solana, S.B. Yuste, A. Santos, J. Chem. Phys. 122, 084510 (2005)

    Article  ADS  Google Scholar 

  71. S.B. Yuste, A. Santos, M. López de Haro, Mol. Phys. 109, 987 (2011)

    Article  ADS  Google Scholar 

  72. A. Santos, S.B. Yuste, M. López de Haro, Condens. Matter Phys. 15, 23602 (2012)

    Article  Google Scholar 

  73. A. Santos, S.B. Yuste, M. López de Haro, M. Bárcenas, P. Orea, J. Chem. Phys. 139, 074503 (2013)

    Article  ADS  Google Scholar 

  74. R. Fantoni, A. Santos, Phys. Rev. E 84, 041201 (2011)

    Article  ADS  Google Scholar 

  75. R. Fantoni, A. Santos, Phys. Rev. E 87, 042102 (2013)

    Article  ADS  Google Scholar 

  76. R. Fantoni, A. Santos, J. Chem. Phys. 140, 244513 (2014)

    Article  ADS  Google Scholar 

  77. M.A.G. Maestre, R. Fantoni, A. Giacometti, A. Santos, J. Chem. Phys. 138, 094904 (2013)

    Article  ADS  Google Scholar 

  78. M. Robles, M. López de Haro, J. Chem. Phys. 107, 4648 (1997)

    Article  ADS  Google Scholar 

  79. A. Díez, J. Largo, J.R. Solana, J. Chem. Phys. 125, 074509 (2006)

    Article  ADS  Google Scholar 

  80. S.B. Yuste, A. Santos, M. López de Haro, Mol. Phys. 98, 439 (2000)

    Article  ADS  Google Scholar 

  81. E. Waisman, Mol. Phys. 25, 45 (1973)

    Article  ADS  Google Scholar 

  82. D. Henderson, L. Blum, Mol. Phys. 32, 1627 (1976)

    Article  ADS  Google Scholar 

  83. J. Largo, J.R. Solana, L. Acedo, A. Santos, Mol. Phys. 101, 2981 (2003)

    Article  ADS  Google Scholar 

  84. A. Lang, G. Kahl, C.N. Likos, H. Löwen, M. Watzlawek, J. Phys. Condens. Matter 11, 10143 (1999)

    Article  ADS  Google Scholar 

  85. S. Zhou, J.R. Solana, J. Chem. Phys. 131, 204503 (2009)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Física and Instituto de Computación Científica Avanzada (ICCAEx), Universidad de Extremadura, Badajoz, Spain

    Andrés Santos

Authors
  1. Andrés Santos
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Santos, A. (2016). Exact Solution of the Percus–Yevick Approximation for Hard Spheres …and Beyond. In: A Concise Course on the Theory of Classical Liquids. Lecture Notes in Physics, vol 923. Springer, Cham. https://doi.org/10.1007/978-3-319-29668-5_7

Download citation

Publish with us

Policies and ethics

Search

Navigation