Abstract
We consider the statistical mechanics of a fluctuating string (1D solid-on-solid model) ofN columns with a contact energy term displaying a critical wetting transition. For this model we derive a contour integral representation for the finite-size partition function. From this representation we derive a polynomial representation and obtain the Lee-Yang zeros forN ≲, 100. Through the asymptotic evaluation of the contour integral we evaluate the zeros for higherN. This asymptotic evaluation displays a Stokes phenomenon providing a different viewpoint of the mechanism by which a phase transition can arise, supplementing the picture of Lee and Yang. We also reproduce and extend somewhat the results of Smith for the finite-size scaling limit of the partition function.
Similar content being viewed by others
References
C. N. Yang and T. D. Lee,Phys. Rev. 87:404 (1952); T. D. Lee and C. N. Yang,Phys. Rev. 87:410 (1952).
C. J. Thompson,Mathematical Statistical Mechanics (Princeton University Press, Princeton, New Jersey, 1972), p. 85; G. E. Uhlenbeck and G. W. Ford,Lectures in Statistical Mechanics (American Mathematical Society, Providence, Rhode Island, 1963).
R. B. Pearson,Phys. Rev. B 26:6285 (1982); B. Bonnier and Y. Leroyer,Phys. Rev. B 44:9700 (1991); S. Katsura, Y. Abe, and M. Yamamoto,J. Phys. Soc. Jpn. 30:347 (1971).
M. E. Fisher, inLectures in Theoretical Physics, Vol. VII(c) (University of Colorado Press, Boulder, Colorado, 1965), p. 1.
T. Asano,J. Phys. Soc. Jpn. 29:350 (1970); M. Suzuki and M. E. Fisher,J. Math. Phys. 12:235 (1971).
G. G. Stokes,Trans. Comb. Phil. Soc. 9:379 (1847);10:106 (1864) [Reprinted inMathematical and Physical Papers by the Late Sir George Gabriel Stokes (Cambridge University Press, Cambridge, 1904), Vol. II, p. 329; Vol. IV, p. 77].
C. M. Bender and S. A. Orszag,Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1984).
F. W. J. Olver,Asymptotics and Special Functions (Academic Press, London, 1974).
M. V. Berry,Proc. R. Soc. A 422:7 (1989);Publ. Math. Inst. Hautes Études Sci. 68:211 (1989); F. W. J. Olver,SIAM J. Math. Anal. 22:1460 (1991).
C. Itzykson, J. B. Zuber, and R. B. Pearson,Nucl. Phys. B 220:415 (1983).
E. R. Smith,J. Stat. Phys. 60:529 (1990).
M. L. Glasser, V. Privman, and L. S. Schulman,J. Stat. Phys. 45:451 (1986); M. L. Glasser, V. Privman, and L. S. Schulman,Phys. Rev. B 35:1841 (1987).
G. Forgacs, R. Lipowsky, and Th. M. Nieuwenhuizen, inPhase Transitions and Critical Phenomena, Vol. 14, Section 4, C. Domb and J. L. Lebowitz, eds. (Academic Press, London, 1988).
K. M. Watson,Phys. Rev. 103:489 (1956); B. J. Baumgartl,Z. Phys. 198:148 (1967); W. G. Gibson,Phys. Rev. A 6:2469 (1972).
A. Erdeyli, W. Magnus, F. Oberhettinger, and F. G. Tricomi,Higher Transcendental Functions, Vol. 1 (McGraw-Hill, New York, 1953).
S. Wolfram,Mathematica: A System for Doing Mathematics by Computer, 2nd ed. (Addison-Wesley, Redwood City, California, 1991).
M. Abramowitz and I. A. Stegun,Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1968).
M. E. Fisher and M. N. Barber,Phys. Rev. Lett. 28:1516 (1972).
R. G. Newton,Scattering Theory of Waves and Particles, 2nd ed. (Springer-Verlag, New York, 1982).
J. Stephenson and R. Couzens,Physica A 129:201 (1984); W. van Saarloos and D. A. Kurtze,J. Phys. A 17:1301 (1984).
C. Itzykson and J. M. Luck, inProceedings of the Brasov International Summer School 1983 (Birkhäuser, Boston, 1985).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Pisani, C., Smith, E.R. Lee-Yang zeros and stokes phenomenon in a model with a wetting transition. J Stat Phys 72, 51–78 (1993). https://doi.org/10.1007/BF01048040
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01048040