The idea of renormalization

The idea of renormalizing the CCSD(T) method via the MMCC formalism, as described above, can be easily extended to the CCSD(TQ) case. The resulting CR-CCSD(TQ) approaches are examples of the MMCC(2,4) scheme, defined by Eq. (44), in which we improve the results of the CCSD calculations by adding the non-iterative corrections Sq(2, 4), defined in terms of moments 9Jt fc(2) and 0Jll lcd(2), to the CCSD energies. Two variants of the CR-CCSD(TQ) method, labeled by the extra letters a and b , are particularly useful. The CR-CCSD(TQ),a and CR-CCSD(TQ),b energies are calculated in the following manner [11-14,24,33,34] ... 

Feigenbaum went on to develop a beautiful theory that explained why a and 5 are universal (Feigenbaum 1979). He borrowed the idea of renormalization from statistical physics, and thereby found an analogy between a, 5 and the universal exponents observed in experiments on second-order phase transitions in magnets, fluids, and other physical systems (Ma 1976). In Section 10.7, we give a brief look at this renormalization theory. 

When renormalizing the functions in the diffusion equation, Oono and Freed (l )81b) used the ideas of renormalization schemes for tfie dynamic properties of substance in the critical region. Analysis of this situation, as well as intuitive scaling (sec section 4.5), has led to an equality between the static and dynamic indices... 

Even non-theoretical readers are urged to peruse the introductory part of Chapter 20 where the idea of renormalization is explained without any detailed mathematical calculations. Nonetheless we feel it important to make some comments here on this question. 

From one point of view, (109) can be interpreted as a manifestation of the noncanonical nature of the microscopic equation of motion and supports the idea of dissipative effects on the microscopic level (for time scale t < t/). From another point of view (109) can be related to the coarse graining of the phase volume minimum cells. The concept of fractional evolution is due to the action of the averaging operator [45]. Each application of the averaging operator is equivalent to a loss of information regarding the short time mobility and is closely associated with the renormalization approach ideas [239]. 

As shown below, an attempt is made to solve this problem using the ideas of the renormalization group transformation method and the theory of fractals, which is also called the geometry of chaos. 

The notion of an effective Hamiltonian can be defined almost rigorously from the perspective of the renormalization group. One intriguing idea that has emerged from work in the critical phenomena arena is that in addition to the importance of renormalization transformations near the critical point, there may be merit to imitating the systematic way in which degrees of freedom are eliminated even when there is no critical point in question. 

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... 

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. 

In the following we elaborate the renormalization group scheme for the theory of SAWs on the percolation cluster as described by the effective Hamiltonian developed in section 2.4. Extending the ideas of Meir and Harris [22] in this respect we refer to this as the MH-model. The motivation for this model is to calculate the average of a logarithm - as usual for a quenched average-... 

The idea of universality goes back to the law of corresponding states, by which all liquids and gases have the same state equations accurate up to the renormalization of length and energy scales. 

I hc fundamental idea of renormalization theory is in establishing relationships between physical quantities. For instance, if = 0, then g = 0, but if z oo, g has its finite limit 5. Besides, for d > 4, g = 0. Hence, for small positive quantities e = 4 — d, g can be expanded into a series in powers of s. The e expansion of the critical indices can be obtained identically. 

However, only in renormalization group methods did the ideas of step-by-step scaling transformations find their rigorous analytical and beautiful realization. One of such procedures was put forward by de (Jennes and described in his book. 

A theory of finite-order walks has been developed which permits the derivation, from data for such walks, of the coefficient y for self-avoiding walks ((r ) N. In three dimensions, y= 1.203, in agreement with other calculations, but the two-dimensional value (1.469) seemed a little low. The idea of thermal blobs , within which ideal statistics apply, derives from the renormalization-group theory of chain statistics. However, expansion within a short segment of a long Monte Carlo chain is partly caused by interactions of the atoms of the segment with the rest of... 

This regrouping of the fugacity expansion allows for the easy incorporation of steric effects. Now, unlike Andersen who tamed the arbitrarily large bonds through the introduction of a renormalized F, Wertheim uses the idea of multiple densities, splitting the total density of the fluid as... 

The diffusion in a velocity field with a wide velocity spectrum supposedly describing turbulence is considered in the spirit of cascade-renormalization ideas. For the latter case of isotropic turbulence, we construct an ordinary differential equation for the turbulent diffusion coefficient. 

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