Dynamic renormalization group theory

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. 

On the other hand, the renormalization-group theory of dynamic critical phenomena predicts that asymptotically diverge as (Hohenberg Halperin 1977). The two predictions can be reconciled if one considers equation (6.31) as a first approximation to the power law (6.23), with z = 8/15jr = 0.054 as a first-order estimate for the exponent z. Until recently, the theoretical estimates for the exponent z appeared to be consistently lower than the experimental values z 0.065 (Bhattacharjee Ferrell 1983b). But this problem is now resolved, and the most recent theoretical estimate is z = 0.063 (Hao 1991). 

Non-mean field corrections can be treated by renormalization group theory,which is not discussed here. In order to leave the tree approximation we turn to percolation description and model the microgels as finite clusters. The percolation theory itself is not essential for this description, but only the fractal character of the clusters on their scale of extension. Polymeric fractals have been discussed already in Section 8.2.6 and we use the properties here as well. Indeed most of the results may be applied here. In Ref. 123 the dynamics of the sol phase is discussed extensively, but we do not want to go in these details here. 

Tulig and Tirrell [13] have recently summarized advances in polymer diffusion theory relevant to diffusion-controlled termination. This summary will be repeated here. Significant advances in the understanding of static and dynamic properties of flexible macromolecules in goocSJ, solvents have been made with scaling concepts, renormalization group techniques and ideas of polymer chain reptation after de Gennes [29-3 ] ... 

We shall use the dynamical scaling theory to describe the hydrodynamic properties of polymer solutions, focusing mainly on the expected universal behavior. We use a Flory approximation for the power law behavior, which turns out to be a much easier approach and allows a simple understanding of the important physical features often masked by a heavier formalism. For comparison with experiment we shall sometimes quote more detailed results obtained by renormalization group calculations. We will also discuss briefly the deviations from universal behavior related to crossover effects. 

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