Abstract
The well-established relation between Potts models withv spin values and random-cluster models (with intracluster bonding favored over intercluster bonding by a factorv) is explored, but with the random-cluster model replaced by a much generalized polymer model, implying a corresponding generalization of the Potts model. The analysis is carried out in terms a given defined functionR(ρ), an entropy/free-energy density for the polymer model in the casev=1, expressed as a function of the density ρ of units. The aim of the analysis is to determine the analogR v (ρ) ofR(ρ) for general nonnegativev in terms ofR(ρ), and thence to determine the critical value of density ρvg at which gelation occurs. This critical value is independent ofv up to a valuev P, the Potts-critical value. What is principally required ofR(ρ) is that it should show a certain given concave/convex behavior, although differentiability and another regularizing condition are required for complete conclusions. Under these conditions the unique evaluation ofR v (ρ) in terms ofR(ρ) is given in a form known to hold for integralv but not previously extended. The analysis is carried out in terms of the Legendre transforms of these functions, in terms of which the phenomena of criticality (gelation) and Potts criticality appear very transparently andv P is easily determined. The value ofv P is 2 under mild conditions onR. Special interest attaches to the functionR 0(ρ), which is shown to be the greatest concave minorant ofR(ρ). The naturalness of the approach is demonstrated by explicit treatment of the first-shell model.
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References
M. Aizenmann, J. T. Chayes, L. Chayes, and C. M. Newman, Discontinuity of the magnetisation in one-dimensional 1/|x−y|2 Ising and Potts models,J. Stat. Phys. 50:1–40 (1988).
H. W. J. Blöte, M. P. Nightingale, and B. Derrida, Critical exponents of two-dimensional Potts and bond percolation models,J. Phys. A 14:L45-L49 (1981).
P. J. Flory,Principles of Polymer Chemistry (Cornell University Press, Ithaca, New York, 1953).
I. J. Good, Cascade theory and the molecular weight averages of the sol fraction,Proc. R. Soc. A 272:54–59 (1963).
P. W. Kasteleyn and C. M. Fortuin, Phase transitions in lattice systems with random local properties,J. Phys. Soc. Jpn. 26(Suppl.):11–14 (1969).
T. C. Lubensky and J. Isaacson, Field theory for the statistics of branched polymers, gelation and vulcanisation,Phys. Rev. Lett. 41:829–832 (1978).
D. Stauffer, A. Coniglio, and M. Adam, Gelation and critical phenomena, inAdvances in Polymer Science (Springer, 1982).
W. H. Stockmayer, Theory of molecular size distribution and gel formation in branched chain polymers,J. Chem. Phys. 11:45–55 (1943).
W. H. Stockmayer, Theory of molecular size distribution and gel formation in branched polymers. II. General cross-linking,J. Chem. Phys. 12:125–131 (1944).
P. Whittle, Statistical processes of aggregation and polymerisation,Proc. Camb. Phil. Soc. 61:475–495 (1965).
P. Whittle, The equilibrium statistics of a clustering process in the uncondensed phase,Proc. R. Soc. A 285:501–519 (1965).
P. Whittle, Polymerisation processes with intrapolymer bonding, Parts I, II and III,Adv. Appl. Prob. 12:94–115, 116–134, 135–153 (1980).
P. Whittle, A direct derivation of the equilibrium distribution for a polymerisation process,Teoriya Veroyatnostei 26:360–361 (1981).
P. Whittle,Systems in Stochastic Equilibrium (Wiley, New York, 1986).
F. Y. Wu, The Potts model,Rev. Mod. Phys. 54:235–268 (1982).
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Whittle, P. Polymer models and generalized Potts-Kasteleyn models. J Stat Phys 75, 1063–1092 (1994). https://doi.org/10.1007/BF02186757
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DOI: https://doi.org/10.1007/BF02186757