Entropy Extension in the Gap

All we require of the extension of Scomm.dis ( ) in the gap is that the resulting entropy be continuous and concave. 

In most computations, the analytic form of the communal entropy Scomm,dis( ) will be known over the range E Eq, so the extension is unique. Otherwise, the extended portion of the entropy can be arbitrary provided it does maintain continuity and concavity. This arbitrariness is irrelevant as its values over the range [ o Tr] will not determine the physics of the system. This will become apparent in Section 10.12. 

Making a distinction between ordered and disordered microstates has been a time-honored practice in theoretical physics. In the context of polymers, this was carried out by Flory [51] in his study of polymer melting, which was later followed by Gibbs and Di Marzio [18] in their highly celebrated work on glass transition in polymers. 

Flory considered a simple model of semiflexible polymers in which each gauche bond has a penalty e 0 each over a trans bond for which the energy is zero. Let g denote the density of gauche bonds, so the energy is simply E = Nge, where N — oo is the number of lattice sites. Let us consider a single polymer whose monomers cover all the lattice sites of a square lattice. In the crystalline state, the polymer bonds are all trans, so g = 0. All bonds are parallel, which can be either in the horizontal or in the vertical directions on a lattice, oriented so that its lattice bonds are either horizontal or vertical. In the disordered state, there are equal number of bonds in both orientations. Let Hh and ny respectively denote the density of horizontal and vertical 

6) We treat nonstationary metastable state as one of the partial equilibriums and follow Landau and Lifshitz ([16], p. 27) to define its entropy according to (10.2). This entropy will continue to increase with time as different parts of the system move toward the same stationary state. This explains why FG lies below DAK. Using the entropy function given by FG, we can also calculate the corresponding free energy, which is shown in the inset by FG. 

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