Scaling, renormalization group theories

Fisher M 1983 Scaling, universality and renormalization group theory Critical Phenomena (Lecture Notes in Physics vol 186) (Berlin Springer)... 

The region around (S2) 2q s 1 is extremely sensitive to structure and, thus, cannot be described by some laws of universality similar to those of the scaling or renormalization group theory. Moreover, this region contains the most valuable information on any particular system. 

The correlation length is also found to scale in a power-law fashion, and it becomes very large at the transition temperature. One of the most significant results of renormalization group theory is to show that the behavior of the correlation length in the critical region is the basis of the power-law singularities observed in the other thermodynamic properties. 

The key aspects of the modern understanding of phase transitions and the development of renormalization group theory can be summarized as follows. First was the observation of power-law behavior and the realization that critical exponents were, to some extent, universal for all kinds of phase transitions. Then it became clear that theories that only treated the average value of the order parameter failed to account for the observed exponents. The recognition that power-law behavior could arise from functions that were homogeneous in the thermodynamic variables and the scale-invariant behavior of such functions... 

A. Carpinteri et al Cohesive crack model description of ductile to brittle size-scale transition dimensional analysis vs. renormalization group theory. Eng. Fract. Mech. 70(14) 1809-1839 (2003)... 

D. Scaling and Renormalization Group Theories 1. Self-Similarity... 

M. Bishop, J. H. R. Clarke, and J. J. Freire, /. Cbem. Phys., 98, 3452 (1993). Computer Simulation Study of the Validity of Scaling and Renormalization Group Theories for Two-Dimensional Star Polymers. 

Renormalization group theory (see, e.g., [35]) lies at the heart of this theory, justifying the use of scaling laws in the asymptotic limit, i.e., for infinitely long polymer chains and for dilute solutions. For semidilute solutions, however, this criterion is not so crucial because the polymer chains are overlapping and many properties, e.g., osmotic pressure, are independent of the chain length. 

This is the so-called Flory-Fisher scaling law (De Gennes 1979). The critical exponent v = 1 in (4.21) at the dimensionality d = 1 v = 3/4 at d = 2 v = 3/5 at d = 3 and v = 1/2 at d = 4. These critical exponents are consistent with that of self-avoiding walks obtained above from the computer simulations. The scaling law for the ideal chain model occurs only in 4D space of SAWs. In 3D space, the renormalization group theory yields the critical exponent as v = 0.588 0.001, which is in good consistency with the computer simulation results (Le Guillou and Zinn-Justin 1977). 

The scaling argument is developed quite generally in statistical mechanics and indeed historically was the source of the renormalization group theory. Suppose therefore that a similar property exists... 

Fisher, M.E. (1988) Scaling, universality, and renormalization group theory, in F. J.W. Hahne (ed.). Critical Phenomena Lecture Notes in Physics Springer, Berlin, 186, 1-139. 

Figure 9 Dimensionless apparent molecular weight Mapp times the concentration c of giant surfactant micelles of hexadecyl trimethyl ammonium bromide in brine, obtained by means of light scattering measurements at a temperature of 306 K. Circles experimental data of Buhler and coworkers [50]. Line renormalization group theory for self-assembled flexible chains fitted to the data [51,52], Also indicated by the dashed lines are the predictions of scaling theory in the dilute and semidilute regimes [1,53], which agree well with results of Monte Carlo simulation [52],...

As in the blob model, the scaling theory does not provide an estimate of the numerical coefficient missing in Eq. 4.19. The theory tells only that /n(x) x " when X 1. If needed, we can find the exact relationship in experiments or computer simulations with the help of the scaling prediction. Often, the renormalization group theory can provide the missing coefficient. 

The scaling function obtained in the renormalization group theory by Ohta and Oono has the same power relationship at p p as that of Eq. 4.19. They proposed an interpolation formula ... 

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