Renormalization-group Techniques

There are a few other solution schemes. There is [78] novel work paying close attention to sets of states incorporating explicit resonating VB character. Another simple approach (which many do not acknowledge as a many-body scheme) entails 

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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... 

In this chapter, we will focus on the entanglement behavior in QPT for the two-dimensional array of quantum dots, which provide a suitable arena for implementation of quantum computation [88, 89, 103]. For this purpose, the real-space renormalization group technique [91] will be utilized and developed for the finite-size analysis of entanglement. The model that we will be using is the Hubbard model [83],... 

Two methods appear to be very powerful for the study of critical phenomena field theory as a description of many-body systems, and cell methods grouping together sets of neighboring sites and describing them by an effective Hamiltonian. Both methods are based on the old idea that the relevant scale of critical phenomena is much larger than the interatomic distance and this leads to the notion of scale invariance and to the statistical applications of the renormalization group technique.93... 

The values of the exponents quoted in Table XII have been estimated numericcilly by renormalization group techniques. Intuitively, there should be a close relationship between conductivity and percolation probability, and one would guess that their critical exponents should be identical. This is not true. Dead ends contribute to the mass of the infinite network described by the percolation probability, but not to the electric current it carries. Figure 39 shows the different growth of the percolation probability and the conductivity. It is convenient to set the conductivity equal to unity at = 1, as in Fig. 39. We note, in passing, that diffusivity is proportional to conductivity, in agreement with Einstein s result in statistical physics that diffusivity is proportional to mobility. 

For turbulent flows, by means of dynamical renormalization group techniques, Yakhot [40] proposed... 

Unfortunately, theoretical understanding of polyelectrolytes is less developed than the understanding of the properties of neutral polymers. Some reasons are that the presence of long-range interactions renders the application of renormalization group techniques and scaling ideas much more difficult than in the neutral case. The reason is that many new length scales... 

R N. The exponent v = 0.588 has been calculated using renormalization group techniques [9, 10], enumeration techniques for short chain lengths and Monte Carlo simulations [13]. 

The method which held the promise of overcoming the difficulty of exploding dimensionalities is the renormalization-group technique in which one systematically throws out the degrees of freedom of a many-body system. While this technique found dramatic success in the Kondo problem [62], its straightforward extension to intereicting lattice models was quite inaccurate [63]. 

We now briefly indicate how the real-space renormalization group technique of the previous section can be extended to other fractals. 

The fractal dimension was calculated by various renormalization group techniques and by computer simulaticms [19,20]. Here, we describe the Hory approximation which, although being wrong [1-6], gives the fractal dimension within a very good accuracy for all dimensions. In this approximation one assumes that the free energy can be written as... 

Let us stress that this is the fractal dimension of the polymers in the reaction bath. We assume that all polymers that constitute the sol have this same fractal dimension. This was calculated by renormalization group techniques and computer simulations [31,32,33,34]. We will give a simple Flory derivation [35] that is close to the former results for all space dimensions. The polydispersity exponent t can be shown to be related to the fractal dimension. 

Figure 2.44. Critical indices ft and 7 a,s parameters of plotting the order parameter dimension n U.S the space dimension

Aharony (1975) and Pelcovits et al. (1978) have investigated the nature of the phase transition for an anisotropic amorphous magnet with an isotropic distribution of easy axis directions using renormalization group techniques. The latter... 

Tulig and Tirrell [13] have recently summarized advances in polymer diffusion theory relevant to diffusion-controlled termination. This summary will be repeated here. Significant advances in the understanding of static and dynamic properties of flexible macromolecules in goocSJ, solvents have been made with scaling concepts, renormalization group techniques and ideas of polymer chain reptation after de Gennes [29-3 ] ... 

More recent theoretical studies have been carried out using renormalization group techniques both for diblock and triblock copolymers, and good quantitative agreement is claimed between the... 

An informative quantity is the static and dynamic scattering function. From this one finds the size, different averages of molecular weight or, in the case of dynamics, the diffusion constant, via the first cumulant. For linear Gaussian chains all these quantities have been calculated extensively, including non-mean field corrections to the size, by renormalization group techniques. ... 

In Figure 4 the experimental results are also compared with the theoretical expression derived by Alessandrini and Carignano [70] by renormalization group techniques. Accordingly,... 

Douglas and Freed introduced renormalization group techniques into the TP theory [32,49]. Accordingly,... 

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