Primitive steps, definition

Eq. (18.17) is simply the definition of the Doi-Edwards damping function with the independent-alignment approximation (Chapters 8 and 12). The damping functions of Doi and Edwards with and without the independent-alignment approximation both explain well the experimental results of a well-entangled nearly monodisperse polymer system (Fig. 9.2). Note that here the unit vector u° represents the orientation of a Fraenkel segment as opposed to the orientation associated with an entanglement strand or primitive step in the Doi-Edwards theory. 

A definitive step in extending the CM is to include the relaxation processes occurring before the a-relaxation and to establish the existence of the primitive... 

These g t) data do not favor the reptation model, but are not definitive enough to rule out the possibility of reptation motion. Thus, Kolinski et al. [57] stepped further by calculating the longitudinal and lateral displacements of the primitive path used in the Doi-Edwards theory. Such calculations should give us direct information on local chain motions. 

Establishing the complexity of the method is quite intricate, unless some assumptions are made. Thus, assume that properties are non-recursive, and that all primitives used in T r) are deterministic, and let c be the number of disjuncts in LA r), p the number of properties of P r), and t the number of recursive atoms in LA(r). There are p proofs to be made. Each proof-tree has only two choice-points, namely the DCI resolution of the head of the initial goal, where there are c possibilities, and the DCI resolution of the t recursive atoms, where there are p - 1 possibilities, all other proof steps being deterministic. Each proof tree thus has size 0(cp ). Hence, the time complexity of the method is 0(cp ). This assumes that there is a fixed maximum number of atoms for the definitions of the used primitives. Indeed, if that number is a function of c, t, or p, then this complexity analysis doesn t hold. 

In the following section we calculate these long-time properties from the simplest assumptions about the short-time dynamics of unentangled loops. These assumptions are based on the definition of unentangled loops given below, that provides a unique assignment of the loops to the steps of the primitive path. 

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