Renormalization approach

Not all the results of scaling theory are equally well supported by the more fundamental renormalization approach. In Sect. 9,3 we discuss which features should not be taken too literally. We furthermore should note that scaling theory in general ignores such complications as polydispersity effects. Ln the sequel we therefore explicitly consider a monodisperse system. 

Dryga A, Warshel A (2010) Renormalizing SMD the renormalization approach and its use in long time simulations and accelerated PMF calculations of macromolecules. J Phys Chem B 114(39) 12720-12728... 

Eq. (2) presents the basis for the covariant renormalization approach. The explicit expressions are known for E Ten(E), X u 6 in momentum space. For obtaining these expressions the standard Feynman approach [11,12] or dimensional regularization [13] can be used. They are free from ultraviolet divergencies but acquire infrared divergencies after the renormalization. However, these infrared divergencies, contained in X 1) and cancel due to the Ward identity X -1) = —A1 1 and the use of the Dirac equation for the atomic electron in the reference state a) ... 

In order to write down explicitely the matrix element ( a70X n ) the bound-state wave functions are expanded in terms of free-electron wave functions. The result is shown in Fig. 5. This renormalization approach can be called direct . The next step gave the name to the approach described this is the partial wave expansion (PWE). Both terms on the right-hand side of Eq. (12) are expanded in partial waves. Then each term of this expansion both for x ou and Xfree is finite but the sum over partial waves is divergent. Combining both expansions one can write... 

Fig. 5. The graphical representation of the direct renormalization approach. The triangle with the letter n inside means the expansion of the wave function for the bound electron state n in terms of free electron wave functions...

This PWE was used in [18] to obtain the numerical results. For the numerical implementation the B-spline approximation [21] was chosen that represents actually the refined version of the space discretization approach. In Table 1 the convergence of the PWE approach with the multicommutator expansion is presented for the lowest-order SE correction for the ground state of hydrogenlike ions with Z = 10. The minimal set of parameters for the numerical spline calcuations was chosen to be the number of grid points N = 20, the number of splines k = 9. This minimal set allowed to keep a controlled inaccuracy below 10%. What is most important for the further generalization of the PWE approach to the second-order SESE calculation is that with Zmax = 3 the inaccuracy is already below 10% (see Table 1). The same picture holds with even higher accuracy for larger Z values. The direct renormalization approach is not necessarily connected with the PWE. In [19] this approach in the form of the multicommutator expansion (Eq. (16)) was employed in combination with the Taylor expansion in powers of (Ea — En>)r 12 The numerical procedure with the use of B-splines and 3 terms of Taylor series yielded an accuracy comparable with the PWE-expansion with Zmax = 3. 

An investigation devoted to the evaluation of SESE a) irreducible contribution was accomplished by S. Mallampalli and J. Sapirstein [22]. Using the same covariant renormalization approach with the potential expansion and employing... 

The work [26] was devoted to the application of the PWE renormalization approach to the evaluation of the SESE a) irreducible contribution. In this work the multicommutator expansion version of the PWE [18] and the numerical B-spline approach was used. The results disagree strongly with Mallampalli and Sapirstein calculations for low and intermediate Z values but agree with [23,24,25],... 

Fig. 7. The graphical representation of the direct PWE renormalization approach. The double and ordinary solid lines with the cross denote the quadratic denominators in the bound and free electron propagators. The other notations axe the same as in Figure 5. The graphs a)-c) correspond to Eqs. (18)-(20) and the additional counterterm correspond to Eq. (21), respectively...

Eq. (17) is an analogy of the first-order diagonal (n = a) equation (11). In Fig. 7 an analogy of Fig. 5, i.e., the graphical representation of the PWE renormalization approach to A 2 ren is depicted. Below we give the explicit expressions for all contributions from the graphs with double electron lines in Fig. 7, i.e., bound electron terms ... 

The role of undulation on the equilibrium of lipid bilayers was also examined by Lipowsky and Leibler,18 who used a nonlinear functional renormalization group approach, and by Sornette,14 who employed a linear functional renormalization approach. It was theoretically predicted that a critical unbinding transition (corresponding to a transition from a finite to an infinite swelling) can occur by varying either the temperature or the Hamaker constant. However, the renormalization group procedures do not offer quantitative information about the systems, when they are not in the close vicinity of this critical point. 

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]. 

Eq. (2) presents the basis for the covariant renormalization approach. The explicit expressions are known for in momentum space. For... 

Renormalization approach to intermittency functional version) Show that if the renormalization procedure in Exercise 10.7.8 is done exactly, we are led to the functional equation... 

In the partial-wave renormalization approach, which we consider in this paper, we have shown that these correction terms vanish for the first-order selfenergy and also for Coulomb screened self-energy. However, in the last case we obtain non-zero correction terms for the different subgroups of diagrams,... 

The pure problem can be solved easily by a Real space renormalization approach where one needs only the renormalization of the Boltzmann factor y = exp(u/T). Let yniZn, and En be the renormalized weight, partition function and energy at the nth generation. By decimating the diamonds the recursion relations are given by... 

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