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Two Hard Spheres in a Spherical Pore: Exact Analytic Results in Two and Three Dimensions

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Abstract

The partition function and the one- and two-body distribution functions are evaluated for two hard spheres with different sizes constrained into a spherical pore. The equivalent problem for hard disks is addressed too. We establish a relation valid for any dimension between these partition functions, second virial coefficient for inhomogeneous systems in a spherical pore, and third virial coefficients for polydisperse hard spheres mixtures. Using the established relation we were able to evaluate the cluster integral b 2(V) related with the second virial coefficient for the Hard Disc system into a circular pore. Finally, we analyse the behaviour of the obtained expressions near the maximum density.

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References

  1. Kratky, K.W.: Phys. A 87, 584 (1977)

    Article  Google Scholar 

  2. Janse van Rensburg, E.J.: J. Phys. A 26, 4805 (1993)

    Article  ADS  Google Scholar 

  3. Labík, S., Kolafa, J., Malijevsky, A.: Phys. Rev. E 71, 021105 (2005)

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  MATH  ADS  Google Scholar 

  5. Henderson, D.: Mol. Phys. 30, 971 (1975)

    Article  ADS  Google Scholar 

  6. Barker, J.A., Henderson, D.: Rev. Mod. Phys. 48, 587 (1976)

    Article  ADS  MathSciNet  Google Scholar 

  7. Luding, S.: Phys. Rev. E 63, 042201 (2001)

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Thiele, E.: J. Chem. Phys. 39, 474 (1964)

    Article  ADS  Google Scholar 

  10. Carnahan, N.F., Starling, K.E.: J. Chem. Phys. 51, 635 (1969)

    Article  ADS  Google Scholar 

  11. Wang, X.Z.: Phys. Rev. E 66, 031203 (2002)

    Article  ADS  Google Scholar 

  12. Miandehy, M., Modarress, H.: J. Chem. Phys. 119, 2716 (2003)

    Article  ADS  Google Scholar 

  13. Rusanov, A.I.: J. Chem. Phys. 121, 1873 (2004)

    Article  ADS  Google Scholar 

  14. Eisenberg, E., Baram, A.: Phys. Rev. E. 73, 025104(R) (2006)

    Article  ADS  Google Scholar 

  15. Tonks, L.: Phys. Rev. 50, 955 (1936)

    Article  ADS  MATH  Google Scholar 

  16. van Laar, J.J.: Proc. Roy. Acad. Sci. Amsterdam 1, 273 (1899)

    Google Scholar 

  17. Boltzmann, L.: Proc. Roy. Acad. Sci. Amsterdam 1, 398 (1899)

    Google Scholar 

  18. Boltzmann, L.: Nederlandse Akad. Wetensch. 7, 477 (1899)

    Google Scholar 

  19. Rowlinson, J.S.: Mol. Phys. 7, 593 (1964)

    Article  ADS  MathSciNet  Google Scholar 

  20. Hemmer, P.C.: J. Chem. Phys. 42, 1116 (1964)

    Article  ADS  MathSciNet  Google Scholar 

  21. van Laar, J.J.: Amsterdam Akad. Versl. 7, 350 (1899)

    Google Scholar 

  22. Nijboer, B.R.A., Van Hove, L.: Phys. Rev. 85, 777 (1952)

    Article  ADS  MATH  Google Scholar 

  23. Ree, F.H., Hoover, W.G.: J. Chem. Phys. 40, 939 (1964); 46, 4181 (1967)

    Article  ADS  MathSciNet  Google Scholar 

  24. Bates, M.A., Frenkel, D.: Phys. Rev. E 61, 5223 (2000)

    Article  ADS  Google Scholar 

  25. Wu, G.-W., Sadus, R.J.: J. Chem. Phys. 120, 11686 (2004)

    Article  ADS  Google Scholar 

  26. Zheng, Z., Hu, G., Zhang, J.: Phys. Rev. E 53, 3246 (1996)

    Article  ADS  Google Scholar 

  27. Nèmeth, Z.T., Lowen, H.: J. Phys. Condens. Matter 10, 6189 (1998)

    Article  ADS  Google Scholar 

  28. Nèmeth, Z.T., Lowen, H.: Phys. Rev. E 59, 6824 (1999)

    Article  ADS  Google Scholar 

  29. Hu, G., Zheng, Z., Yang, L., Kang, W.: Phys. Rev. E. 64, 045102(R) (2001)

    ADS  Google Scholar 

  30. Kegel, W.K.: Phys. Rev. E 63, 037104 (2001)

    Article  ADS  Google Scholar 

  31. Awazu, A.: Phys. Rev. E 63, 032102 (2001)

    Article  ADS  Google Scholar 

  32. Rosenfeld, Y., Schmidt, M., Lowen, H., Tarazona, P.: Phys. Rev. E 55, 4245 (1997)

    Article  ADS  Google Scholar 

  33. Tarazona, P.: Physica A 306, 243 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. Rosenfeld, Y., Tarazona, P.: Mol. Phys. 95, 141 (1998)

    Article  Google Scholar 

  35. Hansen, J.-P., McDonald, I.R.: Theory of Simple Liquids, 2nd ed. Academic Press, New York (1990)

    MATH  Google Scholar 

  36. Ree, F.H., Hoover, W.G.: J. Chem. Phys. 41, 1635 (1964)

    Article  ADS  MathSciNet  Google Scholar 

  37. Ree, F.H., Keeler, R.N., McCarthy, S.L.: J. Chem. Phys. 44, 3407 (1966)

    Article  ADS  Google Scholar 

  38. Clisby, N., McCoy, B.M.: J. Stat. Phys. 114, 1343 (2004); arXiv:cond-mat/0303098v2, and references therein

    Article  MathSciNet  MATH  ADS  Google Scholar 

  39. Lyberg, I.: J. Stat. Phys. 119, 747 (2005); arXiv:cond-mat/0410080v2, and references therein

    Article  MathSciNet  MATH  ADS  Google Scholar 

  40. Rowlinson, J.S., McQuarrie, D.A.: Mol. Phys. 61, 525 (1987)

    Article  ADS  Google Scholar 

  41. Hirschfelder, J.O., Curtis, C.F., Bird, R.B.: Molecular Theory of Gases and Liquids. Willey, New York (1954), p. 161

    MATH  Google Scholar 

  42. Blaak, R.: Mol. Phys. 95, 695 (1998)

    Article  ADS  Google Scholar 

  43. Saija, F., Fiumara, G., Giaquinta, P.V.: Mol. Phys. 87, 991 (1996); 92, 1089 (1997); 89, 1181 (1996)

    Article  ADS  Google Scholar 

  44. Wheatley, R.J., Saija, F., Giaquinta, P.V.: Mol. Phys. 94, 877 (1998)

    Article  ADS  Google Scholar 

  45. Wheatley, R.J.: Mol. Phys. 93, 675 (1998); 93, 965 (1998); 96, 1805 (1999)

    Article  ADS  Google Scholar 

  46. McQuarrie, D.A., Rowlinson, J.S.: Mol. Phys. 60, 977 (1987)

    Article  ADS  Google Scholar 

  47. Munakata, T., Hu, G.: Phys. Rev. E 65, 066104 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  48. Uranagase, M., Munakata, T.: Phys. Rev. E 74, 066101 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  49. Suh, S.-H., Lee, J.-W., Moon, H., MacElroy, J.M.D.: Korean J. Chem. Eng. 21, 504 (2004)

    Article  Google Scholar 

  50. Suh, S.-H., Kim, S.-C.: Phys. Rev. E 69, 026111 (2004)

    Article  ADS  Google Scholar 

  51. Cao, Z., Li, H., Munakata, T., He, D., Hu, G.: Phys. A 334, 187 (2004)

    Article  MathSciNet  Google Scholar 

  52. Leff, H.S., Coopersmith, M.H.: J. Math. Phys. 8, 306 (1967)

    Article  ADS  Google Scholar 

  53. Percus, J.K.: J. Stat. Phys. 15, 505 (1976)

    Article  ADS  MathSciNet  Google Scholar 

  54. Burgos, E., Bonadeo, H.: J. Chem. Phys. 88, 1163 (1988)

    Article  ADS  Google Scholar 

  55. Hill, T.L.: Statistical Mechanics. Dover, New York (1956)

    MATH  Google Scholar 

  56. Bellemans, A.: Physica 28, 493 (1962); 29, 548 (1963)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  57. Stecki, J., Sokolowski, S.: Mol. Phys. 39, 343 (1980)

    Article  ADS  Google Scholar 

  58. Stecki, J., Sokolowski, S.: Phys. Rev. A 18, 2361 (1978)

    Article  ADS  Google Scholar 

  59. Sokolowski, S., Stecki, J.: Acta Phys. Pol. A 55, 619 (1979)

    MathSciNet  Google Scholar 

  60. Sokolowski, S., Stecki, J.: Mol. Phys. 35, 1483 (1978)

    Article  ADS  Google Scholar 

  61. Tejero, C.F., Ripoll, M.S., Pérez, A.: Phys. Rev. E 52, 3632 (1995)

    Article  ADS  Google Scholar 

  62. Hoover, H.G., Salsburg, Z.W.: J. Chem. Phys. 54, 2129 (1971)

    Article  ADS  Google Scholar 

  63. Salsburg, Z.W., Wood, W.W.: J. Chem. Phys. 37, 798 (1962)

    Article  ADS  Google Scholar 

  64. Wood, W.W.: J. Chem. Phys. 20, 1334 (1952)

    Article  ADS  Google Scholar 

  65. Kim, S.-C., Munakata, T.: J. Phys. Korean Soc. 43, 997 (2003)

    Google Scholar 

  66. González, A., White, J.A., Román, F.L.: J. Chem. Phys. 109, 3637 (1998)

    Article  ADS  Google Scholar 

Download references

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  1. Consejo Nacional de Investigaciones Científicas y Técnicas y Departamento de Física, Comisión Nacional de Energía Atómica, Av. Gral. Paz 1499 (RA-1650) San Martín, Buenos Aires, Argentina

    Ignacio Urrutia

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  1. Ignacio Urrutia
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Correspondence to Ignacio Urrutia.

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Urrutia, I. Two Hard Spheres in a Spherical Pore: Exact Analytic Results in Two and Three Dimensions. J Stat Phys 131, 597–611 (2008). https://doi.org/10.1007/s10955-008-9513-3

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  • DOI: https://doi.org/10.1007/s10955-008-9513-3

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