Skip to main content
SpringerLink
Log in

Random surfaces in statistical mechanics: Roughening, rounding, wetting,...

  • Articles
  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

In this paper we study several problems in statistical mechanics involving systems of fluctuating extended objects, such as interacting steps and domain walls. We reconsider the roughening transition and relate it to the free energy of a gas of steps and to the rounding of facets in the equilibrium shape of crystals, defined via the Wulff construction. Using an idealized description due to Fisher and Fisher we analyze the commensurate-incommensurate transition in uniaxial systems in terms of a gas of interacting domain walls. We also study the interactions between a domain wall and a rigid wall and between two interfaces, a problem which is central for the understanding of wetting. Among our results are a quantitative analysis of entropic repulsion between extended objects and a calculation of deviations from mean-field theory in the commensurate-incommensurate transition in dimension 2 ⩽d ⩽ 3.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. G. Gallavotti, A. Martin-Löf, and S. Miracle-Solé, in Statistical mechanics and mathematical problems, Battelle Recontres 1971 (Lecture Notes in Physics No. 20, Springer, Berlin, 1973).

    Google Scholar 

  2. R. Brandenberger and C. E. Wayne,J. Stat. Phys. 27:425 (1982).

    Google Scholar 

  3. R. L. Dobrushin,Theory Prob. Appl. 17:582 (1972);18:253 (1973).

    Google Scholar 

  4. J. Fröhlich and T. Spencer,Commun. Math. Phys. 81:527 (1981).

    Google Scholar 

  5. H. N. V. Temperley,Proc. Cambridge Philos. Soc. 48:683 (1952);Phys. Rev. 103:1 (1956).

    Google Scholar 

  6. M. Göpfert and G. Mack,Commun. Math. Phys. 82:545 (1982).

    Google Scholar 

  7. J. Bricmont, J. R. Fontaine, and J. L. Lebowitz,J. Stat. Phys. 29:193 (1982).

    Google Scholar 

  8. H.-J. Brascamp, E. H. Lieb, and J. L. Lebowitz,Bull. Int. Stat. Inst. 46:Paper no. 62 (1975).

  9. R. K. P. Zia, Interfacial problems in statistical physics, inStatistical and Particle Physics, K. C. Bowler and A. J. McKane, eds. (SUSSP, Edinburgh, 1984).

    Google Scholar 

  10. C. Rotman and M. Wortis,Phys. Rep. 103:59 (1984).

    Google Scholar 

  11. G. Wulff,Z. Krist. Mineral. 34:449 (1901); C. Herring, in Structure and properties of solid surfaces, R. Gomer and G. S. Smith, eds. (University of Chicago Press, Chicago, 1955).

    Google Scholar 

  12. M. Fisher and D. Fisher,Phys. Rev. B 25:3192 (1982).

    Google Scholar 

  13. O. McBryan and T. Spencer,Commun. Math. Phys. 53:299 (1977).

    Google Scholar 

  14. E. Brézin, The wetting transition, inApplications of Field Theory to Statistical Mechanics, L. Garrido, ed. (Lecture Notes in Physics No. 216, Springer, Berlin, 1985).

    Google Scholar 

  15. W. Selke, Interfacial adsorption in multi-state models, inStatic Critical Phenomena in Inhomogeneous Systems, A. Pekalski and J. Sznajd, eds. (Lecture Notes in Physics No. 206, Springer, Berlin, 1984).

    Google Scholar 

  16. M. Fisher,J. Stat. Phys. 34:667 (1984).

    Google Scholar 

  17. K. Binder, Critical behaviour at surfaces, inPhase Transitions and Critical Phenomena, Vol. 8, C. Domb and J. L. Lebowitz; eds. (Academic Press, New York, 1983).

    Google Scholar 

  18. D. B. Abraham,Phys. Rev. Lett. 44:1165 (1980); D. B. Abraham and J. De Coninck,J. Phys. A16:L333 (1983).

    Google Scholar 

  19. T. W. Burckhardt,J. Phys. A14:L63 (1981); J. T. Chalker,J. Phys. A14:1431 (1981); H. J. Hilhorst and J. M. J. van Leeuwen,Physica 107A:319 (1981); M. Vallade, J. Lazjierowicz,J. Phys. (Paris) 42:1505 (1981).

    Google Scholar 

  20. J. Bricmont, A. El Mellouki, and J. Fröhlich, in preparation.

  21. S. A. Pirogov and Ya. G. Sinaï,Theor. Math. Phys. 25:1185 (1975);26:39 (1976). For reviews see Ya. G. Sinaï,Theory of Phase Transitions; Rigorous Results (Pergamon Press, New York, 1982); J. Slawny, in Phase transitions and critical phenomena, Vol. 10, C. Domb and J. L. Lebowitz, eds. (Academic Press, New York, 1986) (in press).

    Google Scholar 

  22. H. van Beijeren,Commun. Math. Phys. 40:1 (1975).

    Google Scholar 

  23. G. Gallavotti,Commun. Math. Phys. 27:103 (1972).

    Google Scholar 

  24. D. B. Abraham and P. Reed,Commun. Math. Phys. 49:35 (1976).

    Google Scholar 

  25. T. H. Berlin and M. Kac,Phys. Rev. 86:821 (1952).

    Google Scholar 

  26. C. Rotman and M. Wortis,Phys. Rev. B 24:6274 (1981); J. E. Avron, H. van Beijeren, L. S. Schulman, and R. P. K. Zia,J. Phys. A15:L81 (1982).

    Google Scholar 

  27. A. D. Sokal, unpublished (1980); for a related result seeJ. Stat. Phys. 28:431 (1982).

  28. J. Slawny,J. Stat. Phys. 20:711 (1979).

    Google Scholar 

  29. M. Zahradnick,Commun. Math. Phys. 93:559 (1984).

    Google Scholar 

  30. D. Preiss, unpublished.

  31. R. B. Griffiths, in Phase transitions and critical phenomena, Vol. 1, C. Domb and M. S. Green, eds. (Academic Press, New York, 1972).

    Google Scholar 

  32. E. Nelson, Probability theory and Euclidean quantum field theory, inConstructive Quantum Field Theory, G. Velo and A. S. Wightman, eds. (Lecture Notes in Physics No. 25, Springer,. Berlin, 1973).

    Google Scholar 

  33. H. van Beijeren and G. S. Sylvester,J. Funct. Anal. 28:145 (1978).

    Google Scholar 

  34. J. Fröhlich and Y. M. Park,Commun. Math. Phys. 59:235 (1978).

    Google Scholar 

  35. M. J. De Oliveira, R. B. Griffiths,Surf. Sci. 71:687 (1978).

    Google Scholar 

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

    Google Scholar 

  37. R. Pandit, M. Schick, and M. Wortis,Phys. Rev. B 26:5112 (1982), and references therein.

    Google Scholar 

  38. M. P. Nightingale, W. F. Saam, and M. Schick,Phys. Rev. B 30:3830 (1984).

    Google Scholar 

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

    Google Scholar 

  40. C. Fortuin, P. Kasteleyn, and J. Ginibre,Commun. Math. Phys. 22:89 (1971); see also Ref. 60.

    Google Scholar 

  41. J. Bricmont and J. Slawny, in preparation.

  42. D. B. Abraham, private communication.

  43. A. Messager and S. Miracle-Solé,J. Stat. Phys. 17:245 (1977).

    Google Scholar 

  44. A. G. Basuev,Theor. Math. Phys. 58:171 (1984); see also F. F. Galeb,Trans. Moscow Math. Soc. 44:111 (1982).

    Google Scholar 

  45. R. B. Griffiths, C. A. Hurst, and S. Sherman,J. Math. Phys. 11:790 (1970).

    Google Scholar 

  46. M. Blume,Phys. Rev. 141:517 (1960); H. W. Capel,Physica 32:966 (1966).

    Google Scholar 

  47. W. Selke, D. A. Huse, and D. M. Kroll,J. Phys. A17:3019 (1984).

    Google Scholar 

  48. J. Bricmont and J. L. Lebowitz,J. Stat. Phys. 34:651 (1984).

    Google Scholar 

  49. J. L. Lebowitz and A. Martin-Löf,Commun. Math. Phys. 25:276 (1972).

    Google Scholar 

  50. R. S. Ellis, J. L. Monroe, and C. Newman,Commun. Math. Phys. 46:167 (1976).

    Google Scholar 

  51. H. Widom,Trans. Am. Math. Soc. 88:491 (1958); S. V. Parter,Trans. Am. Math. Soc. 100:263 (1961).

    Google Scholar 

  52. J. Fröhlich, R. B. Israel, E. H. Lieb, and B. Simon,Commun. Math. Phys. 62:1 (1978), and references therein.

    Google Scholar 

  53. B. Simon,Commun. Math. Phys. 77:111 (1980); E. H. Lieb,Commun. Math. Phys. 77:127 (1980).

    Google Scholar 

  54. D. Brydges, J. Fröhlich, and T. Spencer,Commun. Math. Phys. 83:123 (1982).

    Google Scholar 

  55. J. Glimm and A. Jaffe,Quantum Physics (Springer, Berlin, 1981).

    Google Scholar 

  56. J. Fröhlich and T. Spencer, inNew Developments in Quantum Field Theory, M. Lévy and P. Mitter, eds. (Plenum, New York, 1977).

    Google Scholar 

  57. J. Fröhlich, B. Simon, and T. Spencer,Commun. Math. Phys. 50:527 (1976).

    Google Scholar 

  58. R. Schor,Commun. Math. Phys. 59:219 (1978).

    Google Scholar 

  59. H. J. Brascamp and E. H. Lieb,J. Funct. Anal. 22:366 (1976).

    Google Scholar 

  60. G. Battle and L. Rosen,J. Stat. Phys. 22:128 (1980).

    Google Scholar 

  61. E. Seiler, Gauge theories as a problem of constructive quantum field theory and statistical mechanics (Lecture Notes in Physics No. 159, Springer, Berlin, 1982).

    Google Scholar 

  62. J. Bricmont and J. R. Fontaine,Commun. Math. Phys. 86:337 (1982).

    Google Scholar 

  63. G. Gallavotti and S. Miracle-Solé,Phys. Rev. B 5:2555 (1972).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut de Physique Théorique, Université Catholique de Louvain, Louvain-La-Neuve, Belgium

    J. Bricmont & A. El Mellouki

  2. Theoretical Physics, CH-8093, ETH-Hönggerberg, Zürich, Switzerland

    J. Fröhlich

Authors
  1. J. Bricmont
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. El Mellouki
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. Fröhlich
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

On leave from École Normale Supérieure, Rabat, Morocco.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bricmont, J., El Mellouki, A. & Fröhlich, J. Random surfaces in statistical mechanics: Roughening, rounding, wetting,.... J Stat Phys 42, 743–798 (1986). https://doi.org/10.1007/BF01010444

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01010444

Key words

Search

Navigation