Abstract
A field-theoretic representation is presented to count the number of configurations of a single self-avoiding walk on a hypercubic lattice ind dimensions with periodic boundary conditions. We evaluate the connectivity constant as a function of the fractionf of sites occupied by the polymer chain. The meanfield approximation is exact in the limit of infinite dimensions, and corrections to it in powers ofd −1 can be systematically evaluated. The connectivity constant and the site entropy calculated throughout second order compare well with known results in two and three dimensions. We also find that the entropy per site develops a maximum atf≳1−(2d)−1. Ford=2 (d=3), this maximum occurs atf~0.80 (f~0.86) and its value is about 50% (30%) higher than the entropy per site of a Hamiltonian walk (f=1).
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References
P. J. Flory,Principles of Polymer Chemistry (Cornell University, Ithaca, 1953);Statistical Mechanics of Chain Molecules (Interscience, New York, 1969); H. Yamakawa,Modern Theory of Polymer Solutions (Harper and Row, New York, 1971).
P. G. de Gennes,Scaling Concepts in Polymer Physics (Cornell University Press, New York, 1979); M. Doi and S. F. Edwards,Theory of Polymer Dynamics (Clarendon Press, Oxford, 1986); K. F. Freed,Renormalization Group Theory of Macromolecules (Wiley-Interscience, New York, 1987).
M. G. Watts,J. Phys. A. 8:61 (1975), and references therein. The Connectivity Constant for Hypercubic Lattices in four, five and six Dimensions was Calculated by M. E. Fisher and D. S. Gaunt,Phys. Rev. 133:224 (1964).
For a recent review on Monte Carlo simulation of lattice models of polymers, see A. Baumgartner, inApplications of the Monte Carlo Method in Statistical Physics, K. Binder, ed. (Springer, Berlin, New York, 1984), p. 145.
K. Kremer and K. Binder, to be published.
For a nice description of properties of Hamiltonian walks and a complete list of references, see B. Duplantier and F. David,J. Stat. Phys., in press.
B. Duplantier,J. Stat. Phys. 49:411 (1987).
P. D. Gujrati and M. Goldstein,J. Chem. Phys. 74:2596 (1981).
P. W. Kasteleyn,Physica 29:1329 (1962).
E. H. Lieb,Phys. Rev. Lett. 18:692 (1967).
T. G. Schmaltz, G. E. Hite, and D. J. Klein,J. Phys. A 17:445 (1984).
H. Orland, C. Itzykson, and C. de Dominicis,J. Phys. Lett. (Paris)46:353 (1985).
B. Duplantier and H. Saleur,Nucl. Phys. B. 290[FS20]:291 (1987).
K. F. Freed,J. Phys. A 18:871 (1985).
M. G. Bawendi, K. F. Freed, and U. Mohanty,J. Chem. Phys. 84:7036 (1986).
M. G. Bawendi and K. F. Freed,J. Chem. Phys. 86:3720 (1987).
M. G. Bawendi and K. F. Freed,J. Chem. Phys. 85:3007 (1986).
M. G. Bawendi, K. F. Freed, and U. Mohanty,J. Chem. Phys. 87:5534 (1987); M. G. Bawendi and K. F. Freed,J. Chem. Phys., in press; K. F. Freed and A. I. Pesci,J. Chem. Phys. Lett. 87:7342 (1987); A. I. Pesci and K. F. Freed,J. Chem. Phys., in press.
A. M. Nemirovsky, M. G. Bawendi, and K. F. Freed,J. Chem. Phys. 87:7272 (1987).
C. Itzykson and J. B. Zuber,Quantum Field Theory (McGraw-Hill, New York, 1980).
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Nemirovsky, A.M., Coutinho-Filho, M.D. From dilute to dense self-avoiding walks on hypercubic lattices. J Stat Phys 53, 1139–1153 (1988). https://doi.org/10.1007/BF01023861
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DOI: https://doi.org/10.1007/BF01023861