Effective coupling function

According to Eq. (5.52), the function f t g,fo) is an invariant of a renormalization procedure, in the course of which the generalized time is divided into tjx steps, each of length, x, the bare coupHng parameter is replaced by its effective counterpart and the initial value of the function at t = 1 is replaced with the function evaluated at the new scale length x. In the limit g( = 0 the conservation equation (5.52) must reduce to the corresponding equation (5.6), appropriate to the ideal system. Therefore, the effective coupling function must satisfy the condition g t 0) = 0, Vt. 

If the conservation equation (5.52) is to be satisfied, the as yet unspecified effective coupling function must be a functional of the object function /(t g(,/o). A self-consistent procedure for establishing this functional relationship is presented later in this section. Once this has been done, one can derive coupled functional equations of evolution for the excess part Sf t g) and the effective coupling function g(t, g). From these equations, estimates of (5/(t g) can be constructed that are applicable to greater ranges of group parameter and bare coupling parameter values than is possible with a standard perturbation series such as... 

If one then is able to find an effective coupling function g(t g) such that the excess part of / satisfies the nonlinear functional equation... 

The effective coupling function thus far remains essentially unspecified. However, it is obvious that this function plays a central role in the RG theory and, as we now shall demonstrate, the proper choice of this function is not arbitrary, as sometimes has been suggested [14], but a natural consequence of the basic conservation equation (5.52). To obtain the functional equation of evolution for g(t g) we rewrite Eq. (5.57) in the multiplicative form... 

Similarly, the evolution equation for the effective coupling function can be written in the form... 

The formula (5.71) allows one to compute the value of excess function Sf t = r" fif), provided that the value of this function is known for a time equal to the scale t and that one is able to determine g(z g), the value of the effective coupling function for this same scale. To obtain the latter, we have only to solve the equation [(5.71) for n = 2]... 

While the functional equations (5.61) and (5.65), are very well suited to the above mentioned, iterative method for determining the (generalized) time evolution of the dynamic system f(t, g,fo), most presentations of the RG method have instead used an approach based on differential equations. These differential equations can be written directly as equations of evolution for the object function Sf t g) and the related effective coupling function g(t g) [3-5,16] or as a pair of partial differential equations known as the Callan-Symanzik equations [3-5,17]. These three forms of the RG theory are essentially equivalent. However, we personally favor the functional equation approach, not only from a computational point of view but because it provides better insight into the workings of the postulates of the self-similarity based RG technique. 

The pair of Lie equations (5.84) and (5.87) govern the evolution of the dynamical system 6f t g). As we previously have emphasized, the effective coupling function must be determined self-consistently, as a function of Sf. This critical step in the procedure is accomplished by expressing the generator P g) as the following functional of the object function df ... 

The formulation of the GPRG method that has been presented here differs from earlier presentations of the old renormalization group method [3-5], not only by virtue of the manner in which the effective coupling function is introduced [see the derivation of Eq. (5.65)] but by the role this function plays in enforcing the validity of the self-similarity assumption [see the derivation of Eq. (5.57)]. The intimate relationship between the effective coupling function and the object function, as illustrated by Eqs. (5.75) and (5.89), is an essential part of the theory that is unique to our approach. The remainder of this section is devoted to two examples that are illustrative of how the method works in practice. 

The effective coupling function g K g) then can be determined from the equation [(5.101) with JV = K ... 

This solution then is used to generate the effective coupling function, according to the prescription [cf. (5.102) or (5.103)]... 

Figure 5.2. Effective coupling function for a 5-spin block plotted versus the bare coupling parameter g = J/k T.

It now will be shown that Eq. (5.109) requires the effective coupling function to be an invariant of the scaling transformation and thus a solution of Eq. (5.65). We begin by rewriting (5.109) in the form... 

The proof is now complete since this can be so only if the effective coupling function satisfies Eq. (5.65). We leave it to the reader to verify that this same requirement of invariance follows from (5.112) as well. 

The objects g) appearing here are the nested iterates of the effective coupling function defined previously by Eqs. (5.72)-(5.74). 

The effective coupling function specific to this scale is found by solving... 

The equation of evolution for the effective coupling function again is given by (5.86). However, the infinitesimal generator P g) now is related to the excess function by the formula... 

The Callan-Symanzik equation for the effective coupling function is the same as before, namely, Eq. (5.95). 

The effective coupling function for the Kuhn segment is obtained by solving the equation... 

The asymptotic N -> oo) behavior of the swelhng factor is determined by the fixed-point spectrum of the effective coupling function, defined as the roots g K) of the nonlinear equation... 

Figure 5.5. The GPRG effective coupling function g(4 g) for a 4-bond randomly jointed polymeric macrosegment plotted versus the bare coupling parameter g.

If there is only one stable and nontrivial fixed point g of the effective coupling function g(K g), it then follows directly from Eq. (5.167) that... 

We recently performed calculations of the partition function for a randomly jointed chain with hard-sphere excluded-volume interactions [21], namely, the same model for which the swelling factor was calculated in Section IV.A. The effective coupling function for the two-bond K = 2) Kuhn segment is displayed in Figure 5.8. It is apparent that the spectrum of fixed points becomes quasicontinuous for g > 2. (An examination of the roots of the equation 3Q K" g) — [3Q(K = 0 confirms this... 

Here, g K g) denotes an effective coupling function that satisfies the scaling invariance relationship (5.159). Finally, the evolution of the compressibility factor is governed by the GPRG functional equation... 

If the effective coupling function a t a) exhibits a stable fixed point a (r) for any given value r of the scale, its asymptotic form will be given by the familiar scaling relation... 

The curves of Figure 5.14 show how the values of the fixed point a (r) and the anomalous dimension /1(t) depend on the choice of scale. Both a (r) and A(t) decrease monotonically as the value of the scale increases. To determine the proper threshold value t, in excess of which the propagator amplitude is self-similar for all values of the group parameter t, we must acquire independent information about the expected asymptotic behavior of the effective coupling function such as, for example, the correct limiting value (at t oo) of the coupling function. One possibility is that the onset of self-similarity will become apparent if one selects a sufficiently large value of the scale. Implementation of this notion requires an extrapolation of our calculated results to infinite values of t, a procedure that leads to the results (t oo) 0.95 and X(z co) 0.12. 

Here, g t g) is an effective coupling function that must be so chosen that f t g, /o) is indeed an invariant of the transformation... 

This last relationship is satisfied only if the effective coupling function is a solution of the functional equation... 

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