Renormalization method

This equation describes not only the crystal growth, but with an additional noise term it also describes the evolution of the surface width and is called the Edward-Wilkinson model. An even better treatment has been performed by renormalization methods and other techniques [44,51-53]. 

According to equation 15, eigenvalues of the superoperator Hamiltonian matrix, H, are poles (electron binding energies) of the electron propagator. Several renormalized methods can be defined in terms of approximate H matrices. The... 

At this point all the vibronic eigenstates can be easily evaluated numerically by means of the recursion method or alternatively the renormalization method. 

These are iterative procedures that allow the calculation and the Green s function matrix elements without explicit diagonalization of the Hamiltonian [18]. In the present case the Hamiltonian is factorized into double chains, and the renormalization method can be conveniently and efficiently applied, since its implementation simply requires the handling and the inversion of small matrices of rank two. For more details and elaboration of the renormalization procedures, see, for example, Ref. [23]. 

This PWE renormalization method was also called noncovariant contrary to the covariant approach described above where the covariant procedure in 4-dimensional momentum space was used to separate out and cancel the divergent terms. In principle, the noncovariant procedure should not lead to any differences provided that both bound -term and counterterm are described in the same way. Such a difference may arise only if the counterterm, unlike the bound -term is written in covariant form [12]. 

The difficulty with low Z values can be avoided in another version of the noncovariant renormalization method, developed in [18,19]. Within this approach an explicit expression for the matrix element ( 7o- bou( a) ) is used ... 

The numerical results reviewed above were obtained for infinite lattices. How do the various quantities of interest behave near the percolation threshold in a large but finite lattice This problem has been studied by renormalization methods, which are essentially equivalent to finite-size scaling. For finite lattices the percolation transition is smeared out over a range of p, and one must expect a similar trend in other functions, including the conductivity. Computer simulations by the Monte Carlo method have been carried out for bond percolation on a three-dimensional simple cubic lattice by Kirkpatrick (1979). Five such experimental curves are shown in Fig. 40, each of which corresponds to a cube of size b, containing bonds. In Fig. 40 the vertical axis gives the fraction p of such samples that percolate (i.e., have opposite faces con-... 

For many of the commonly used renormalized methods, such as 2ph-TDA, NR2, and ADC(3), the operator space spans the h, p, 2hp, and 2ph subspaces [7,22]. Reference states are built from Hartree-Fock determinantal wavefunctions plus perturbative corrections. The resulting expressions for various blocks of the superoperator Hamiltonian matrix may be evaluated through a given order in the fluctuation potential. 

Both classes of self-energy approximations yield useful data for C60. Nondiagonal, renormalized methods reveal the presence of correlation states in photoelectron spectra. 

The difference between and the full renormalized potential is a well-behaved function that is evaluated numerically. The interest in the renormalization procedure is now mainly a theoretical one as formal results regarding screening and other thermodynamic parameters can be obtained this way. Results applicable to both pure one-component fluids or mixtures can be obtained. The numerical solution of integral equations, such as the SSOZ and CSL equations, for sites with charge interactions should no longer use the renormalization method but rather the method we are about to describe. 

Almost the whole of the above method is valid for charged sites once c, and 7a, are replaced by c°, and 7", in the renormalized method and by c , and 7 , in Ng s method. The principal change arises in the derivative dcxJdyx -For the PY and HNC closures, respectively... 

The homopolymer arms in miktoarm stars are predicted, by the renormalization method, to expand monotonically, reaching a limit which is dictated by the increase in the size of the other kind of homopolymer arms. The whole molecule seems to increase in size monotonically with molecular weight and functionality. The same behavior is predicted for any part of the molecules consisting of two or more arms of different homopolymers. This work was expanded with the study of A2B and A3B miktoarm copolymers. The dimensionless ratio 6q, which expresses quantitatively the effect of heterointeractions between unlike units on the conformational properties of copolymers, was calculated by an intrinsic viscosity analysis. It is defined as... 

In fact, by using renormalization methods, it was possible to calculate15 the first two terms of the expansion of K in powers of e = 4 — d. Using this result and the exact result Kd=1 = 1/2, we can represent K by the approximate expression (see Chapter 13, Section 1.4.2)... 

However, very interesting results were obtained by applying analytic renormalization methods to perturbation series. These techniques initiated by Wilson himself, were developed by many physicists, and, among them, Brezin, Le Guillou, and Zinn-Justin played an important role. 

The iterative renormalization method on lattices was extended to the case of polymers by H. Hilhorst in 1976.8 The technique used by Hilhorst relies on the polymer-magnetic system correspondence for n - 0, as it is described in Chapter 11, Section 3.2. Hilhorst introduced spins located on the sites of a cyclic triangular lattice a spin lattice site M. The components take the values — n1/2, 0 or n1/2 and have to fulfil the condition that only one of these components is different from zero. The Hamiltonian of the system is given by the sum... 

Analytic renormalization methods were applied to the Landau-Ginzburg method (around 1970) but, originally for a space dimension close to 4, dimension 4 being marginal. However, it appeared later that analytical methods could also be used directly in dimension 3, as G. Parisi suggested in 1973 (Cargese Summer School). 

Interpreting the fixed point is not obvious, if one adopts the point of view and the methods originally developed by K. Wilson. On the contrary, the interpretation is transparent, if one uses the most recent renormalization methods. Then the expansion parameter is a number defining the intensity of a physical quantity this quantity itself is expressed by using the characteristic length of the system as the length unit. Therefore, if, as we believe, there exists a limiting critical system, the expansion parameter which is a physical observable must have a finite limit. 

However, the principles and the techniques of renormalization theory are not directly related to the existence of fields. They apply whenever one deals with a critical system, i.e. whenever one has to describe large-scale phenomena which depend only globally on the chemical microstructure. Thus, because an ensemble of long polymers in a solution constitutes a critical system, renormalization principles and renormalization techniques must be directly applicable to their study. Actually, this idea appeared quite naturally. It led to the decimation method which has been described previously and which lacks efficiency. However, the same idea can be applied in a much better way. This direct renormalization method (des Cloizeaux 1980)37,38 consists in adapting to polymers methods which had been successful in field theory.39 In other words, the aim is to bypass the Laplace de Gennes transformation (see Chapter 11). This method applies to semi-dilute solutions as well as to dilute solutions. 

Entropy can also be calculated by starting from the continuous model and, in this case, we expect a similar result. However, we must note that in the continuous case, the entropy of the system is really infinite and that a finite entropy can be obtained only after performing a subtractive renormalization. Actually, this question has been tackled several times, either indirectly with the help of a zero component field theory,3,4 or by using the direct renormalization method.2 We shall now describe in detail the latter approach. 

Until now, the exponents 0, have been calculated (in powers of e) only from field theory however, it is also possible to use the direct renormalization method. Thus, as an example, we shall use this method to explicitly calculate the exponents 0, to first-order in e. For this purpose we shall use formula (13.1.140) which connects the 0, to the [Pg.580]

A systematic study of demixtion curves was undertaken as early as 1942 by Flory both from experimental and theoretical points of view. In particular, he showed that the dissymmetry of the demixtion curves is large for high molecular masses. Nevertheless, the top of the demixtion curve can be considered as an ordinary critical point. The critical opalescence associated with it was studied by P. Debye and collaborators in 1962, but correct calculations of critical exponents and of critical properties could not be made before 1972 or so, and had to wait for the renormalization methods discovered by K. Wilson. 

In the following, we consider only monodisperse systems and we study tricritical systems by using the direct renormalization method. It is also possible to proceed indirectly by introducing a tricritical field theory and the correspondence which exists between field theory and polymer theory. This approach... 

Here, we use the direct renormalization method which relies on a slightly different conceptual approach. Nevertheless the points of view are equivalent and the results obtained through field theory can be directly recovered, (as we shall now show) by using more recent results obtained (1985) by Duplantier.15... 

Thus, to calculate the swelling of an isolated chain and the osmotic pressure of a set of chains in the vicinity of the Flory point, a determination of the partition functions 3 (E, — H S) and 3 N x S) by dimensional renormalization is sufficient. The calculations will be performed to first orders in x and y for a space dimension d = 3 — e (0 < s < 1), and we note that the purely repulsive terms have been already calculated in Chapter 10. The partition functions will be represented by series in terms of x and y. Finally, in order to study the behaviour of long polymers, we shall treat these series by using the direct renormalization method. 

In held theory, the dimensional regularization can also be used but, in general, one prefers other renormalization methods which always give finite results for all values of d. These methods are equivalent to a direct introduction of 91 s then, the renormalization constants depend on the masses, but this is not a drawback, since in field theory, the masses are considered as constant. 

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