E-expansion

FIG. 10-173 Slip-typ e expansion joint. (Ftom Kellogg, Design of Piping Systems, Wiley, New York, 1965.)... 

As the wave evolves from that point, the losses connected with rearward expansion decrease. If a charge of a small diam is considered, then lateral expansion depends on the path over which the wave has traveled. The increase in cross section of a cylinder, i.e., expansion in the lateral direction, leads to a reduction in pressure and to a decrease in the deton velocity in comparison with detonation propagating in a constant cross-section cylinder. The decrease in deton vel causes, in turn, the diminution of shock amplitude wave and impairs the conditions under which the reaction can proceed. The loss caused by lateral expansion is known as lateral loss. Propagation of detonation is possible only if this loss is not smaller than a certain limit, which is characteristic for each expl... 

We discuss here the basic ideas of the renormalization group, using the discrete chain model. This is not the most elegant or powerful approach, and in Part Til of this book we will present a much more efficient scheme. However, the present approach is conceptually the simplest, and it allows us to explain all the relevant features dilatation symmetry and scaling, fixed points and universality, crossover. Furthermore, technical aspects like the e-expansion also come up. We are then prepared to discuss the Qualitative concept of scaling in its general form and to work out some consequences. 

Having constructed renormalized perturbation theory, in a further step we have to determine the renormalized parameters u, tir as functions of / e, n, / r. As it stands, Eq. (11.1) cannot be used for that purpose, since the Taylor expansion of the renormalization factors suffers from ail the drawbacks of the unrenormalized cluster expansion. A priori, in strict e-expansion it orders according to powers of u ln / n). Now the renormalized perturbation... 

These results show that the minimal subtraction scheme eliminates ambiguities inherent in an extrapolation of e-expansion results to physical dimension d = 3. For the flow equations no extrapolation is necessary. Furthermore they are strictly independent of the parameter bu(z). 

If we want to evaluate the results in strict e-expansion we need in addition... 

Aiming at the e-expansion we still have to analyze the integral. The analysis is simple. We write... 

In the construction of the RGf dimension d = 4 plays a special role as upper critical dimension of the thebry. This for instance shows up in the estimate of the nonuniversal corrections to the theorem of renormalizability, or in the feature that the nontrivial fixed point u merges with the Gaussian fixed point for d — 4. It naturally leads to the e-expansion. However, the RG mapping constructed in minimal subtraction only trivially depends on e. Also results of renormalized perturbation theory do not necessarily ask for further expansion in e. Equation (12.25) gives an example. We should thus consider the practical implications of the -expansion in some more detail. 

Consider the e-expansion (12.27) of the renormalized end-to-end distribution. It contains the constant b and thus depends on our renormalization scheme. This dependence can be eliminated by replacing the chain length, which is a microscopic parameter, by the end-to-end distance... 

This quantity will be analyzed in detail in Sect. 15.4. The e-expansion result reads... 

This calculation illustrates a general feature we may write down scaling laws for normalized quantities in terms of scaled momenta qRg (or qf e, equivalently), scaled concentrations cpRand ip replacing the coupling. Such relations involve only physically observable macroscopic quantities. They must have a uniquely defined -expansion, where ip = 0(e) acts as an expansion parameter. The result is necessarily independent of any conventions of the renormalization scheme. Not even the form of the RG flow equations matters. Furthermore, in establishing such results, we never have to invoke a condition like hr = 1. These are the great virtues of consistent e-expansion. 

Strict e-expansion cannot yield this result. It expands the exponent according to 1 jv 2—g/4-h0(e2) (cf. Eq. (8,34)) and yields a power series in ln(q2i 2) ... 

To summarize, strict e-expansion a priori seems to yield unambiguous results. Closer inspection, however, reveals that in low order calculations considerable ambiguity is hidden in the definition of the physical observables used as variables or chosen to calculate. What is worse, the e-expansion does not incorporate relevant physical ideas predicting the behavior outside the small momentum range or beyond the dilute limit. In particular, it does not give a reasonable form for crossover scaling functions. On the other hand, it can be used to calculate well-defined critical ratios, which are a function of dimensionality only, Even then, however, the precise definition of the ratio matters,... 

On a deeper level we observe that the e-expansion does not properly respect the structure of the theory. As discussed before (see Sects. 8.3 or 10.2, in particular), we should first use the RG to map the system to an uncritical manifold, which for dilute systems is determined by Nr = 0(1), for instance. In a second step the scaling functions on the uncritical manifold can be calculated by renormalized perturbation theory. These two steps are well separated conceptually. The exponents, for instance, are properties of the RG, independent of specific scaling function. Strict e-expansion mixes these two steps since it simultaneously expands exponents and scaling functions. The expected scaling structure then has to be put in by hand at the end of the calculation. 

Avoiding the e-expansion, we can work directly with renormalized perturbation theory in powers of u, evaluated in d = 3 dimensions. We still have to decide what quantities to expand, and it clearly will make some difference whether we for instance use the result (12.25) for Pr(p, ft/, /) or rather expand In Pr(P- nR, /). Having chosen some basic set of quantities, however, we consistently evaluate observables of interest without further expansion. We for instance calculate ii2, perturbatively, constructing ijj from Eq. (12.33) without further manipulation. Clearly then some trivial change in the definition of %fj has no deeper consequences. 

A most accurate form of the renormalization group mapping has been constructed in field theory, using results of the e-expansion carried to order e5 and additional information on the asymptotically high orders of the expansion. We will present the results as specialized to polymer theory in Sect, 13.1. It is this form of the mapping which we will use in all further analysis. 

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