Polymer Model and Classification of Configurations

We relate the high-temperature expansion of a one-dimensional w-component axis spin model to a polymer problem, as has been discussed elsewhere by us [50,107, 

We set V = 0 for a site occupied by a solvent. In the following, we only speak of bonds (p 1) between monomers, as there are no bonds attached to a solvent. If there is no bond between two neighboring monomers, we treat them as disconnected. There are no interactions between the solvents, and between a solvent and a monomer. As we will see, the model exhibits a first-order transition at T = Tu- We 

A bond of multiplicity pj gives rise to loops, whose number is given by h = Pj —1. Let Vi denote the valence at a site. The number of chemical bonds Band the number of loops I are given by 

We see that the activity of each loop is m, and the activity of each chemical bond is K. The multiplicity W B,L) determines the polymer system entropy. We can treat (—B) as analogous to the energy E so that we can consider the entropy as a function of 

The following will be established. At high temperatures, we have microstates in which each component is finite in size hence, the multiplicity of each bond is even. This condition then uniquely defines disordered microstates. There are two distinct percolating components = N). One ofthem has all bond multiplicities even. We will treat this to represent a disordered microstate for the obvious reason. The other percolating component has all bond multiplicities odd. This is obviously a component with a different symmetry, and we will see below that it represents the ordered state. The possible odd multiplicities of its bonds uniquely determine ordered microstates. The sets of disordered and ordered microstates are obviously disjoint, which ensures that there will be no stable nuclei in the metastable state if it occurs, as discussed in Section 10.1.6, and distinguishes our approach from the PL approach. Because of distinct symmetries of the two phases, our model genuinely represents a melting transition, and not a liquid-gas transition in which the symmetry remains the same in both equilibrium phases. 

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