Renormalization

In this section we give an intuitive introduction to Feigenbaum s (1979) renormalization theory for period-doubling. For nice expositions at a higher mathematical level than that presented here, see Feigenbaum (1980), Collet and Eckmann (1980), Schuster (1989), Drazin (1992), and Cvitanovic (1989b). 

Feigenbaum phrased his analysis in terms of the superstable cycles, so let s get some practice with them. 

the map has a superstable 2-cycle. Let p and g denote the points of the cycle. Superstability requires that the multiplier A = (-2p)(-2g) = 0, so the point X = 0 must be one of the points in the 2-cycle. Then the period-2 condition / (0,7 , ) = 0 implies 7 ,-(7 ,) =0. Hence 7 , =1 (since the other root gives a fixed point, not a 2-cycle).  

The renormalization theory is based on the self-similarity of the figtree—the twigs look like the earlier branches, except they are scaled down in both the x and r directions. This structure reflects the endless repetition of the same dynamical processes a 2 -cycle is born, then becomes superstable, and then loses stability in a period-doubling bifurcation. 

Now we need to convert these qualitative observations into formulas. A helpful first step is to translate the origin of. r to. r, by redefining x as x - x ,. This rede- 

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T.M. Habashy, E.Y. Chow, and D.G. Dudley, Profile inversion using the renormalized source-type integral equation approach, 1990, IEEE Trans. Antennas Propagat., 38,... 

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Jayaram, B., Beveridge, D. L. A simple method to estimate free energy from molecular simulation Renormalization on the unit interval. J. Phys. Chem. 94 (1990) 7288-7293... 

Such renormalization can be obtained in the framework of the small polaron theory [3]. Scoq is the energy gain of exciton localization. Let us note that the condition (20) and, therefore, Eq.(26) is correct for S 5/wo and arbitrary B/ujq for the lowest energy of the exciton polaron. So Eq.(26) can be used to evaluate the energy of a self-trapped exciton when the energy of the vibrational or lattice relaxation is much larger then the exciton bandwidth. 

A slightly improved form of this equation is the renormalized Davidson correction, which is also called the Brueckner correction ... 

The contracted basis set created from the procedure above is listed in Figure 28.3. Note that the contraction coefficients are not normalized. This is not usually a problem since nearly all software packages will renormalize the coefficients automatically. The atom calculation rerun with contracted orbitals is expected to run much faster and have a slightly higher energy. 

However, as given by group renormalization theory (45), the values of the universal exponents depend on the (thermodynamic) dimensionality of the system. For four dimensions (as required by the phase rule for the existence of tricritical points), the exponents have classical values. This means the values are multiples of 1/2. The dimensions of the volume of tietriangles are (31)... 

The solution of the spin-boson problem with arbitrary coupling has been discussed in detail by Leggett et al. [1987]. The displacement of the equilibrium positions of the bath oscillators in the transition results in the effective renormalization of the tunneling matrix element by the bath overlap integral... 

The situation simplifies when V Q) is a parabola, since the mean position of the particle now behaves as a classical coordinate. For the parabolic barrier (1.5) the total system consisting of particle and bath is represented by a multidimensional harmonic potential, and all one should do is diagonalize it. On doing so, one finds a single unstable mode with imaginary frequency iA and a spectrum of normal modes orthogonal to this coordinate. The quantity A is the renormalized parabolic barrier frequency which replaces in a. multidimensional theory. In order to calculate... 

In the derivation of (5.38) we have extracted the <5-function term from the phonon Green s function which, in turn, renormalized the bare potential V to the adiabatic one An expression similar to (5.37) can be obtained for an arbitrary bath whenever the coupling is suflSciently weak and the functional Z[Q(t)] can be expanded into the series... 

The first-order term in this expansion renormalizes the potential V Q) while the bilinear term is analogous to the last term in (5.38). This is the linear-response theory for the bath. In fact, it shows... 

Substitution of this for the golden-rule expression (1.14) together with the renormalized tunneling matrix element from (5.60) gives (5.64), after thermally averaging over the initial energies E-,. In the biased case the expression for the forward rate constant is... 

Sethna [1981] considered two limiting cases. The calculation of action in the fast flip approximation (a>j CO ) proceeds by utilizing the expansion exp ( — cu,-1t ) 1 — cu t. After substituting the first term, i.e. the unity, in (5.72) we get precisely the quantity which yields the Franck-Condon factor in the rate constant. The next term cancels the adiabatic renormalization and changes KM)... 

In the opposite case of slow flip limit, cojp co, the exponential kernel can be approximated by the delta function, exp( —cUj t ) ii 2S(r)/coj, thus renormalizing the kinetic energy and, consequently, multiplying the particle s effective mass by the factor M = 1 + X The rate constant equals the tunneling probability in the adiabatic barrier I d(Q) with the renormalized mass M, ... 

As discussed before, the mass renormalization is a reflection of the fact that the particle traces a distance longer than 2Qq in the total multidimensional coordinate space. 

Here and V( are the terms of initial and final states, the diabatic coupling. We have explicitly added the counterterm YCjQ /lcoj in order to cancel the adiabatic renormalization caused by vibrations. We shall consider the particular case of two harmonic diabatic terms,... 

Turbulence modeling capability (range of models). Eddy viscosity k-1, k-e, and Reynolds stress. k-e and Algebraic stress. Reynolds stress and renormalization group theory (RNG) V. 4.2 k-e. low Reynolds No.. Algebraic stress. Reynolds stress and Reynolds flux. k- Mixing length (user subroutine) and k-e. 

An appropriate model of the Reynolds stress tensor is vital for an accurate prediction of the fluid flow in cyclones, and this also affects the particle flow simulations. This is because the highly rotating fluid flow produces a. strong nonisotropy in the turbulent structure that causes some of the most popular turbulence models, such as the standard k-e turbulence model, to produce inaccurate predictions of the fluid flow. The Reynolds stress models (RSMs) perform much better, but one of the major drawbacks of these methods is their very complex formulation, which often makes it difficult to both implement the method and obtain convergence. The renormalization group (RNG) turbulence model has been employed by some researchers for the fluid flow in cyclones, and some reasonably good predictions have been obtained for the fluid flow. 

It is well known that it is difficult to solve numerically integral equations for models with Coulomb interaction [69,70]. One needs to develop a renormalization scheme for the long-range terms of ion-ion correlations. Here we must do that for ROZ equations. 

The correlation functions of the partly quenched system satisfy a set of replica Ornstein-Zernike equations (21)-(23). Each of them is a 2 x 2 matrix equation for the model in question. As in previous studies of ionic systems (see, e.g.. Refs. 69, 70), we denote the long-range terms of the pair correlation functions in ROZ equations by qij. Here we apply a linearized theory and assume that the long-range terms of the direct correlation functions are equal to the Coulomb potentials which are given by Eqs. (53)-(55). This assumption represents the mean spherical approximation for the model in question. Most importantly, (r) = 0 as mentioned before, the particles from different replicas do not interact. However, q]f r) 7 0 these functions describe screening effects of the ion-ion interactions between ions from different replicas mediated by the presence of charged obstacles, i.e., via the matrix. The functions q j (r) need to be obtained to apply them for proper renormalization of the ROZ equations for systems made of nonpoint ions. 

L. Schafer, T. A. Witten. Renormalization field theory of polymer solutions. I. Scaling laws. J Chem Phys 66 2121-2130, 1977 A. Knoll, L. Schafer, T. A. Witten. The thermodynamic scaling function of polymer solution. J Physique 42 161-m, 1981. 

D. J. Amit. Field Theory, the Renormalization Group and Critical Phenomena. Singapore World Scientific Publishing, 1984. 

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]. 

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