Skip to main content
SpringerLink
Log in

Walks, walls, wetting, and melting

Journal of Statistical Physics volume 34, pages 667–729 (1984)Cite this article

Abstract

New results concerning the statistics of, in particular,p random walkers on a line whose paths do not cross are reported, extended, and interpreted. A general mechanism yielding phase transitions in one-dimensional or linear systems is recalled and applied to various wetting and melting phenomena in (d=2)-dimensional systems, including fluid films and p×1 commensurate adsorbed phases, in which interfaces and domain walls can be modelled by noncrossing walks. The heuristic concept of an effective force between a walk and a rigid wall, and hence between interfaces and walls and between interfaces, is expounded and applied to wetting in an external field, to the behavior of the two-point correlations of a two-dimensional Ising model belowT c and in a field, and to the character of commensurate-incommensurate transitions ford=2 (recapturing recent results by various workers). Applications of random walk ideas to three-dimensional problems are illustrated in connection with melting in a lipid membrane model.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

  1. W. Feller,An Introduction to Probability Theory and its Applications, Vol. 1 (John Wiley & Sons, New York, 1950).

    Google Scholar 

  2. D. A. Huse and M. E. Fisher,Phys. Rev. B 29:239 (1984).

    Google Scholar 

  3. D. A. Huse, A. M. Szpilka, and M. E. Fisher,Physica 121A:363 (1983).

    Google Scholar 

  4. P.-G. de Gennes,J. Chem. Phys. 48:2257 (1968).

    Google Scholar 

  5. F. J. Dyson,Commun. Math. Phys. 21:269 (1971).

    Google Scholar 

  6. J. Fröhlich and T. Spencer,Commun. Math. Phys. 84:87 (1982).

    Google Scholar 

  7. M. E. Fisher,Physics 3:255 (1967).

    Google Scholar 

  8. M. E. Fisher and B. U. Felderhof,Ann. Phys. (N.Y.) 58:176, 217, 268, 281 (1970).

    Google Scholar 

  9. H. N. V. Temperley,Phys. Rev. 103:1 (1956).

    Google Scholar 

  10. M. E. Fisher,Rept. Prog. Phys. 30:615 (1967).

    Google Scholar 

  11. M. E. Fisher,Proc. Nobel Symp. 24,Collective Properties of Physical Systems, B. Lundqvist and S. Lundqvist, eds. (Academic Press, New York, 1974), p. 16.

    Google Scholar 

  12. J. W. Cahn,J. Chem. Phys. 66:3667 (1977).

    Google Scholar 

  13. C. Ebner and W. F. Saam,Phys. Rev. Lett. 38:1486 (1977).

    Google Scholar 

  14. D. B. Abraham,Phys. Rev. Lett. 44:1165 (1980).

    Google Scholar 

  15. J. D. Weeks, inOrdering in Strongly Fluctuating Condensed Matter Systems, T. Riste, ed. (Plenum, New York, 1980), p. 293.

    Google Scholar 

  16. S. T. Chui and J. D. Weeks,Phys. Rev. B 23:2438 (1981).

    Google Scholar 

  17. T. W. Burkhardt,J. Phys. A 14:L63 (1981).

    Google Scholar 

  18. J. M. J. van Leeuwen and H. J. Hilhorst,Physica 107A:319 (1981).

    Google Scholar 

  19. H. N. V. Temperley,Proc. Cambridge Philos. Soc. 48:683 (1952).

    Google Scholar 

  20. D. A. Huse and M. E. Fisher,Phys. Rev. Lett. 49:793 (1982).

    Google Scholar 

  21. M. R. Moldover and J. W. Cahn,Science 207:1073 (1980).

    Google Scholar 

  22. O'D. Kwon, D. Beaglehole, W. W. Webb, B. Widom, J. W. Schmidt, J. W. Cahn, M. R. Moldover and B. Stephenson,Phys. Rev. Lett. 48:185 (1982).

    Google Scholar 

  23. D. W. Pohl and W. I. Goldburg,Phys. Rev. Lett. 48:1111 (1982).

    Google Scholar 

  24. J. W. Schmidt and M. R. Moldover,J. Chem. Phys. 79:379 (1983).

    Google Scholar 

  25. J. S. Rowlinson and B. Widom,Molecular Theory of Capillarity (Clarendon Press, Oxford, 1982).

    Google Scholar 

  26. D. Poland and H. A. Scheraga,Theory of Helix-Coil Transitions in Biopolymers (Academic Press, New York, 1970).

    Google Scholar 

  27. C. L. Stevens and G. Felsenfeld,Biopolymers 2:293 (1964).

    Google Scholar 

  28. M. E. Fisher,J. Chem. Phys. 45:1469 (1966).

    Google Scholar 

  29. J. G. Dash and J. Ruvalds, eds.,Phase Transitions in Surface Films (Plenum Press, New York, 1980).

    Google Scholar 

  30. R. Imbihl, R. J. Behm, K. Christmann, G. Ertl, and T. Matsushima,Surf. Sci. 117:257 (1982).

    Google Scholar 

  31. M. Bretz,Phys. Rev. Lett. 38:501 (1977).

    Google Scholar 

  32. D. E. Moncton, P. W. Stephens, R. J. Birgeneau, P. M. Horn, and G. S. Brown,Phys. Rev. Lett. 46:1533 (1981);49:1679 (1982).

    Google Scholar 

  33. S. Alexander,Phys. Lett. 54A:353 (1975).

    Google Scholar 

  34. E. Domany, M. Schick, J. S. Walker, and R. B. Griffiths,Phys. Rev. B 18:2209 (1978).

    Google Scholar 

  35. R. J. Baxter,J. Stat. Phys. 26:427 (1981).

    Google Scholar 

  36. R. J. Baxter and P. A. Pearce,J. Phys. A 15:897 (1982);16:2239 (1983).

    Google Scholar 

  37. M. Stone,Phys. Rev. D 23:1862 (1981).

    Google Scholar 

  38. S. Ostlund,Phys. Rev. B 24:398 (1981);23:2235 (1981).

    Google Scholar 

  39. D. A. Huse,Phys. Rev. B 24:5180 (1981).

    Google Scholar 

  40. V. L. Pokrovsky and A. L. Talapov,Zh. Eksp. Teor. Fiz. 78:269 (1980) [Sov. Phys. JETP 51:134(1980)].

    Google Scholar 

  41. J. Villain and M. Gordon,J. Phys. C 13:3117 (1980); F. D. M. Haldane and J. Villain,J. Phys. (Paris) 42:1673 (1981).

    Google Scholar 

  42. S. N. Coppersmith, D. S. Fisher, B. I. Halperin, P. A. Lee, and W. F. Brinkman,Phys. Rev. Lett. 46:549 (1981);Phys. Rev. B 25:349 (1982).

    Google Scholar 

  43. M. E. Fisher and D. S. Fisher,Phys. Rev. 25:3192 (1982).

    Google Scholar 

  44. T. Nattermann,J. Phys. (Paris) 43:631 (1982);J. Phys. C 16:4125 (1983).

    Google Scholar 

  45. R. Pandit and M. Wortis,Phys. Rev. B 25:3226 (1982).

    Google Scholar 

  46. H. Nakanishi and M. E. Fisher,Phys. Rev. Lett. 49:1565 (1982).

    Google Scholar 

  47. D. B. Abraham and E. R. Smith,Phys. Rev. B 26:1480 (1982).

    Google Scholar 

  48. W. J. Camp and M. E. Fisher,Phys. Rev. Lett. 26:73 (1971);Phys. Rev. B 5:946 (1972).

    Google Scholar 

  49. T. T. Wu,Phys. Rev. 149:380 (1966).

    Google Scholar 

  50. L. P. Kadanoff,Nuovo Cimento 44B:276 (1966).

    Google Scholar 

  51. R. Hartwig and M. E. Fisher,Adv. Chem. Phys. 15:333 (1969).

    Google Scholar 

  52. B. M. McCoy and T. T. Wu,The Two-dimensional Ising Model (Harvard University Press, Cambridge, Massachusetts, 1973).

    Google Scholar 

  53. H. B. Tarko and M. E. Fisher,Phys. Rev. B 11:1217 (1975).

    Google Scholar 

  54. W. J. Camp,Phys. Rev. B 6:960 (1972).

    Google Scholar 

  55. M. E. Fisher and W. J. Camp,Phys. Rev. Lett. 26:565 (1971).

    Google Scholar 

  56. B. M. McCoy and T. T. Wu,Phys. Rev. D 18:1259, 1243, 1253 (1978).

    Google Scholar 

  57. D. B. Abraham,Phys. Rev. Lett. 50:291 (1983).

    Google Scholar 

  58. M. E. Fisher and A. E. Ferdinand,Phys. Rev. Lett. 19:169 (1967).

    Google Scholar 

  59. D. B. Abraham, G. Gallavotti, and A. Martin-Löf,Physica 65:73 (1973).

    Google Scholar 

  60. P. Bak,Rept. Prog. Phys. 45:587 (1982).

    Google Scholar 

  61. P. Bak and V. J. Emery,Phys. Rev. Lett. 36:978 (1976).

    Google Scholar 

  62. H. J. Schulz,Phys. Rev. B 22:5274 (1980);Phys. Rev. Lett. 46:1685 (1981).

    Google Scholar 

  63. J. Villain and P. Bak,J. Phys. (Paris) 42:657 (1981).

    Google Scholar 

  64. P. W. Kasteleyn,Physica 27:1209 (1961).

    Google Scholar 

  65. H. N. V. Temperley and M. E. Fisher,Philos. Mag. 6:1061 (1961).

    Google Scholar 

  66. M. E. Fisher,Phys. Rev. 124:1664 (1961).

    Google Scholar 

  67. P. W. Kasteleyn,J. Math. Phys. 4:287 (1964).

    Google Scholar 

  68. F. J. Wegner,Phys. Rev. B 5:4529 (1972).

    Google Scholar 

  69. K. G. Wilson and J. Kogut,Phys. Rept. 12C:75 (1974).

    Google Scholar 

  70. M. E. Fisher,Rev. Mod. Phys. 46:597 (1974).

    Google Scholar 

  71. K. G. Wilson,Rev. Mod. Phys. 47:773 (1975);55:583 (1983).

    Google Scholar 

  72. H. J. Schulz, B. I. Halperin, and C. L. Henley,Phys. Rev. B 26:3797 (1982).

    Google Scholar 

  73. B. Durhuus, J. Fröhlich, and T. Jonsson (preprint, 1983).

  74. T. Izuyama and Y. Akutsu,J. Phys. Soc. Japan 51:50, 730 (1982).

    Google Scholar 

  75. J. F. Nagle,Ann. Rev. Phys. Chem. 31:157 (1980);Proc. R. Soc. London Ser. A 337:569 (1974).

    Google Scholar 

  76. S. M. Bhattacharjee, J. F. Nagle, D. A. Huse, and M. E. Fisher,J. Stat. Phys. 32:361 (1983).

    Google Scholar 

  77. M. E. Fisher, inCritical Phenomena, Proceedings of the 1970 Enrico Fermi International School of Physics, Course No. 51, Varenna, M. S. Green, ed. (Academic Press, New York, 1971), p. 1.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Baker Laboratory, Cornell University, 14853, Ithaca, New York

    Michael E. Fisher

Authors
  1. Michael E. Fisher
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Fisher, M.E. Walks, walls, wetting, and melting. J Stat Phys 34, 667–729 (1984). https://doi.org/10.1007/BF01009436

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01009436

Key words

search

Navigation