Use of Renormalization

There are basically four motivations for using renormalization techniques in the theory of fluids, namely, (1) a desire to remove divergences in individual graphs, (2) a desire to exploit extensive cancellation among terms in the series, (3) a desire to group terms in some particular way, such as to collect together all terms that are of the same order in some hopefully small parameter, and (4) a desire to generate tractable approximations for properties of the fluid. 

The second motivation, a desire to exploit extensive cancellation among terms in the cluster series, is similar in spirit to the first, where we exploit cancellation of divergences. If the new bond that is introduced in the reduction is much smaller and weaker than the individual graphs in the definition of that bond, the resulting series that contains the new bond will have fewer and smaller terms than the original series. 

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The comparison of experimental and theoretical MWD curves has shown that these curves for PDMDAAC, synthesized at c=4.0 and 5.0 mol/l, are simulated directly at the following parameters of aPj(N) curve D=l.65 (a 0.182), o =0.25 and b=. Such simulation for c=A.O mol/l is adduced in Fig. 94 (curve 3), from which a good conformity of theory and experiment may be observed. For MWD curves at Co=1.0 and 2.5 mol/l such direct simulation proves to be unsatisfactory, since for them the position of a maximum on MWD curve corresponds to iV =175 and 375, but for the theoretical curves such low values of N are not attained (Figs. 91-93). Nevertheless, to achieve agreement between theory and the experiment by the use of renormalization is possible for these curves, that is attributed to the automodality of PJwhite noise of intensity as follows [210] ... 

Since polyaiylates are formed from two different monomers (diane and di-chloroanhydride of l,l-dichloro-2,2-di(n-carboxiphenyl) ethylene), then as particle mass for them half of the molecnlar weight of PAr hnk was accepted and N calcnlation was performed according to the indicated weight. Comparison of Figs. 29-31, on the one hand, and Fig. 32, on the other hand, shows, that direct simulation of MWD curves by P,(N) curves cannot be obtained successfully, since MWD curves have too low Nmax values. Nevertheless, to achieve agreement between the theory and the experiment by use of renormalization is possible for these curves. This is due to the automodality of P (N) distribution on the variable... 

In good solvents, the mean force is of the repulsive type when the two polymer segments come to a close distance and the excluded volume is positive this tends to swell the polymer coil which deviates from the ideal chain behavior described previously by Eq. (1). Once the excluded volume effect is introduced into the model of a real polymer chain, an exact calculation becomes impossible and various schemes of simplification have been proposed. The excluded volume effect, first discussed by Kuhn [25], was calculated by Flory [24] and further refined by many different authors over the years [27]. The rigorous treatment, however, was only recently achieved, with the application of renormalization group theory. The renormalization group techniques have been developed to solve many-body problems in physics and chemistry. De Gennes was the first to point out that the same approach could be used to calculate the MW dependence of global properties... 

The lattice gas model of Bell et al. [33] neither gave any detailed mechanism of the orientational ordering nor separated the contributions of the headgroup and the acyl chain. Lavis et al. [34] discussed Ref. 33 critically and concluded that the sharp kink point in the isotherm at transition was an artifact of the mean field approximation used. An improved correspondence to experimental data was claimed by the use of the real-space renormalization group method [35]. The same authors returned to the problem [35] and concluded that in addition to the orientation of the molecules, chain melting had to be included in a model which could interpret the phase transitions. 

Our group has made extensive use of the RNG k-e model (Nijemeisland and Dixon, 2004), which is derived from the instantaneous Navier-Stokes equations using the Renormalization Group method (Yakhot and Orszag, 1986) as opposed to the standard k-e model, which is based on Reynolds averaging. The... 

By using these renormalized conditional probabilities, the probability of any amino acid X being observed in a new sequence at profile position, k, can be estimated given that amino acids set Tk has been previously observed at that position in the defining set as a probability density mixture ... 

Using the self-consistently obtained solutions of Eq. 19, the calculated chemical shift (Jiso = (To + a(T) is calculated and compared to the experimental data in Fig. 4. Even though the experimentally observed transition is broader than the calculated one, the agreement between theory and experiment is good. As the discontinuity in the lattice-related mode is small at Tc, where Tc corresponds to a = 0, the chemical shift does not show a discontinuity at Tc within numerical accuracy. It is important to note here that the S-shape in the cf T) data is a direct consequence of using the renormalized frequencies as defined in Eq. 19. 

In this connection there is an important question concerning the infinite selfenergy of a point charge in classical as well as in quantum field theory. The latter uses a renormalization process to solve the problem, namely, by subtracting two infinities to end up with a finite result. Despite the success of such a procedure, a more physically satisfactory way is needed [80]. Possibly the present theory may provide such an alternative, by tackling the divergence problem in a more surveyable manner. The finite result of a difference between two infinities due to renormalization theory would then be replaced by a finite result obtained from the product of an infinity and a zero, as demonstrated by the present analysis. 

The associated crossover will be calked concentration crossover . The neighborhood of the 0-liuiit will be addressed as 0-region1, etc. The renormalization group will be found to suggest the use of some modified variables, which however does not change the essential contents of the limits. 

In this part,we first explain in general terms the construction of renormalized perturbation theory. We show how the RG results from the arbitrariness of r and establish the general scaling form (Chap. 11). We then turn to the specific technique of minimal subtraction and show how to calculate the scaling functions (Chap. 12). The RG mapping, used in the sequel, is presented and discussed in Chap. 13. We finally (Chap. 14) illustrate the theory with an evaluation of the tree approximation. 

The theorem of renormalizability can be read in two ways. With the renormalized theory taken to be fixed, it implies the existence of a one-parameter class, parameterized by , of bare theories, all equivalent to the given renormalized theory and thus equivalent to each other. This aspect is related to universality a whole class of microscopic models yields the same scaling functions. In the next chapter we will use this aspect to get rid of the technical complications of the discrete chain model. We can however also interpret the theorem as establishing the existence of a one-parameter class of renormalized theories, all equivalent to a given bare theory. This class is parameterized by the length scale r or the scaling parameter... 

The only, but essential, use we make here of Eq. (11.19) is to extract the properties of the functions W u) etc. Note that by construction these functions are independent of (L, M), of momenta, and of chain lengths. Writing down Eq. (11.19) for three different choices of (L,M) we therefore have a system of three equations linear in W, 1/V, t/, which allows us to express these functions in terms of renormalized cumulants, Three important results immediately follow. 

In composite Liouville space, the expectation values of renormalized observables, calculated by using the renormalized a matrix, are expressed by ... 

Here, again, the sum runs over the atoms of the molecule and i/, p are their orbital quantum numbers, respectively. Obviously, the obtained eigen-levels and orbitals are different from those of the free molecule because the interaction with the leads is taken into account in Eq. (8) through H. The correspondence to the free molecule levels and orbitals can be found by projecting TJ( )1 ( onto the orbitals of the free molecule. This way, the terms HOMO-derived or LUMO-derived levels can be used for the corresponding groups of renormalized molecular levels. 

As seen above, most of the recent SCIETs are involved with the function y (r). The use of this renormalized indirect correlation function in the diagrams expansions has a simple and practical justification The h(r) diagrams for the bridge function contain an exp[co(r) — factor in the integrand expressions,... 

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