Phenomenological renormalization

In general, percolation is one of the principal tools to analyze disordered media. It has been used extensively to study, for example, random electrical networks, diffusion in disordered media, or phase transitions. Percolation models usually require approximate solution methods such as Monte Carlo simulations, series expansions, and phenomenological renormalization [16]. While some exact results are known (for the Bethe lattice, for instance), they are very rare because of the complexity of the problem. Monte Carlo simulations are very versatile but lack the accuracy of the other methods. The above solution methods were employed in determining the critical exponents given in the following section. 

The special significance of this result was first realized by Nightingale [70], who showed how it could be reinterpreted as a renormalization group transformation of the infinite system. The phenomenological renormalization (PR) equation for finite systems of sizes L and L is given by... 

The finite-size scaling theory combined with transfer matrix calculations had been, since the development of the phenomenological renormalization in 1976 by Nightingale [70], one of the most powerful tools to study critical phenomena in two-dimensional lattice models. For these models the partition function and all the physical quantities of the system (free energy, correlation length, response functions, etc) can be written as a function of the eigenvalues of the transfer matrix [71]. In particular, the free energy takes the form... 

The second approach is the phenomenological renormalization (PR) [24,70] method, where the sequence of the pseudocritical values of X can be calculated by knowing the first and the second lowest eigenvalues of the matrix for two different orders, N and N1. The critical Xc can be obtained by searching for the fixed point of the phenomenological renormalization equation for a finite-size system [70],... 

Phenomenological renormalization of a polymer chain the strip method... 

In section 2, we briefly outline density functional theory and the various approximations employed including the local density approximation (LDA), and discuss generalizations of LDA to incorporate spin (LSDA), orbital, and relativistic effects. We also discuss phenomenological renormalized band schemes based on the slave-boson or Kondo phase-shift methods. 

It should be emphasized that in the present microscopic context /3 is a constant independent of temperatmre and has to be compared to the T = 0 value of the phenomenological Griineisen parameter i2 T), which is obtained directly from thermal expansion experiments as discussed in the following section. It is possible that an improved theory which includes vertex corrections and fluctuations beyond the mean-field model would lead to a strongly renormalized and temperature-dependent Griineisen parameter. 

Bethe lattice treatment of phenomenological relaxation problem mentioned above has also some limitations. It predicts a transition temperature higher than that of a bravais lattice. Also, predicting the critical exponents is not reliable. Therefore, one must consider the relaxation problem on the real lattices using more reliable equilibrium theories to get a much clear relaxation picture. In particular, renormalization group theory of relaxational sound dynamics and dynamic response would be of importance in future. 

The theory of renormalization group transformations of the conformational space of polymer chains contains the phenomenological parameters f and ( which are not natural, experimentally produced ones. However, the results of theory are approximated, to a large extent, with the variable 5 which is written through experimentally measured quantities as... 

Our presentation of the scaling relationships has been strkdy phenomenological. A more fundamental apinoach, based on renormalization group ideas, can be found in various a anced texts. ... 

Like the exact QDT counterpart [cf. Eq. (4.6)], the POP-CS-QDT preserves both the reduced Gaussian dynamics and the effective local field pictinre for the DBO system. Its TZg [Eq. (4.11a)] has the same dissipation superoperator terms as those in ]Zf [Eq. (4.6b)]. The first and the last terms in the right-hand-side of Eq. (4.11a) for TZg or Eq. (4.6b) for are mainly responsible for the energy renormalization (or self-energy) contribution [38] and their dynamics implications are often neglected in phenomenological quantum master equations such as the optical Bloch-Redfield theory [36]. Note that the bath response function relates to the spectral density as [cf. Eq. (2.8)]... 

Besides these stochastic interpretations, deterministic interpretations are presently developed GRAY [12], KUMPINSKY and EPSTEIN [13], propose systemic approaches, commonly used in chemical engineering several ideal reactors are coupled by conservative flows with expandable coefficients, so that by-passes or dead zones may be taken into account. NICOLIS and FRISCH 14] use a quasi-Semenov equation in the limit of large diffusion coefficients and obtain a renormalization of k , DEWEL et al. [15] use a phenomenological theory of turbulent mixing to study surface effects produced by the feed of the reactor. 

Several theoretical attempts have been made recently to give a more complete description of the transport coefficients of mixtures in the critical region. Folk Moser (1993) performed dynamic renormalization-group calculations for binary mixtures near plait points and obtained nonasymptotic expressions for the kinetic coefficients. Kiselev Kulikov (1994) derived phenomenological crossover functions for the transport coefficients by factorizing the Kubo formulas for the transport coefficients a, p and y. This approach is referred to as the decoupled-mode approximation (Ferrell 1970). Their calculations yield for the thermal conductivity the expected finite enhancement in the asymptotic critical region and also a smooth crossover to the background far away from... 

The five primary steps of the derivation are (1) extension of the renormalization group derivation of the hydrodynamic scaling model for Ds to treat the zero-shear viscosity t], (2) description of the experimental phenomenology of r] c),... 

Unfortunately, a full description of the crossover from tricritical behavior to nonmean-field critical behavior is a difficult theoretical problem [48]. Here, we shall not dwell on recent developments based on the renormalization group approach, since this is outside the scope of the present review, but we only mention the phenomenological extension of the crossover scaling description, Eqs. (24)-(27), to incorporate the Ising behavior [3, 30,49]. There one starts from the observation that the variable appearing in the Ginzburg criterion, r Gi cx is simply proportional to the... 

Renormalization in this context means an attempt to find a physically plausible ansatz for the unknown memory matrix given by Eq. 106. In principle one could postulate a power law with an exponent being a fitting parameter to experimental data after having derived expressions of observables on this basis. However, in the frame of the renormalized Rouse models (RRM) a somewhat less formal and less phenomenological approach is possible. 

It remains to future developments to find a physical, more elementary and less heuristic basis for the renormalization ansatz, or at least some physical arguments and principles making this sort of ansatz more and more plausible. At present the justification of this sort of modeling can only be based on a phenomenological argument, namely the success of describing experimental findings. 

---