Fisher renormalization

Table 1. Comparison of the critical exponents of NIPA gel with some known Ising systems. The number in the parentheses indicates the error of the corresponding exponent. The row next to the last is obtained directly by using an = —0.05. The last row is the Fisher renormalized results...

Clearly, the simultaneous presence of crossover from Ising to mean-field criticality, a transition from two-component behavior of solutions to one-component behavior, and the possible presence of Fisher renormalization renders any analysis difficult. 

According to the classical Wilson-Fisher renormalization process [12], the parameters of the Hamiltonian (Eq. 20) will evolve under an anisotropic rescaling of the wavevectors of the form and... 

The Ising nature of the SmAi-SmA2 transition is confirmed [78, 96, 97] with Fisher-renormalized exponents [98]. 

The rare case of a crossover to the Fisher renormalized Ising behavior was found in the two PB(1,4)/PS and PB(1,2)/PS blends near Tq (see Figs. 22-23), but these two cases look quite different. Similar observations were made in asymmetric (polymer blend-solvent systems. A large effect was... 

M.E. Fisher, Renormalization group theory its basis and formulation in statistical physics. Rev. 

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

GLD Cited by N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Addison-Wesley, Reading, Mass., 1992. BDFN Adapted from tables by J. J. Binney, N. J. Dowrick, A. J. Fisher and M. E. J. Newman, The Theory of Critical Phenomena, An Introduction to the Renormalization Group, Clarendon Press, Oxford, 1992. 

Aharony, A. and Fisher, M.E. Critical behavior of magnets with dipolar interactions. Renormalization group near four dimensions. Phys. Rev. B, 1973, 8, p. 3323-3341. 

Rzoska, S. J., Chrapec, J., and Ziolo J. (1987) Fisher s renormalization for the nonlinear dielectric effect from isothermal measurements, Phys. Rev. A 36, 2885-2889... 

After these caveats, fig. 17 shows qualitatively the dimensionality dependence of the order parameter exponent /5, the response function exponent y, and correlation length exponent v. Although only integer dimensionalities d = 1,2, 3 are of physical interest (lattices with dimensionalities d = 4,5, 6 etc. can be studied by computer simulation, see e.g. Binder, 1981a, 1985), in the renormalization group framework it has turned out useful to continue d from integer values to the real axis, in order to derive expansions for critical exponents in terms of variables = du — d or e1 = d — dg, respectively (Fisher, 1974 Domb and Green, 1976 Amit, 1984). As an example, we quote the results for r) and v (Wilson and Fisher, 1972)... 

The magnitude of the effect of an impurity depends upon its amount and its nature. Fisher and Scesney have made detailed calculations for a soluble model (a mobile-electron ferromagnet), utilizing a dilution parameter X which is approximately Tg - r° /T where T is the critical temperature of the system with the impurity while 7g is that of the pure system. They find that appreciable values of X are required in order to observe a renormalization effect. For example, they find as a useful approximation for the observed jS deter-minedt by a log-log fit of experimental data in the range of t from 10 to 10 ... 

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). 

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

On developing this concept, Oono and Freed obtained unambiguous correspondence between the renormalization procedure of the conformational space of a polymer chain and the characteristic values in critical phenomena within Wilson-Fisher .s theoretical field appro lch (some differences do, certainly, exist and will be discussed at the end of this subsection). 

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. 

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