Renormalization group theory first-order

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... 

The nematic to smectic A phase transition has attracted a great deal of theoretical and experimental interest because it is the simplest example of a phase transition characterized by the development of translational order [88]. Experiments indicate that the transition can be first order or, more usually, continuous, depending on the range of stability of the nematic phase. In addition, the critical behaviour that results from a continuous transition is fascinating and allows a test of predictions of the renormalization group theory in an accessible experimental system. In fact, this transition is analogous to the transition from a normal conductor to a superconductor [89], but is more readily studied in the liquid crystal system. 

Recently, Oono [119] and Oono and Kohmoto [120] applied the renormalization group theory to polymer hydrodynamics of the Kirkwood-Riseman scheme. They computed and Pe to first order in e, where e = 4 — d with d being the dimensionality of space, and obtained in three dimensions... 

Lee et al. [50] calculated the characteristic frequency by renormalization group theory to first order in e (= 4 — d). However, the agreement of their formula with experiment was not very encouraging. 

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). 

Let us now embed the renormalization group, Constructed in Chap. 8, iftto this general framework. As mentioned above, relation (8.5) shows that the RG we are searching for must be a nonlinear representation of the group of dilatations in the space of parameters. , n,/ e). These are the microscopic parameters of the model, and the representation shall leave macroscopic observables invariant. Furthermore we want the representation to show a nontrivial fixed point. In Sect. 8.2 we have constructed such a representation based on first order perturbation theory. The invariance constraint is obeyed within deviations of order 1+e 2, no = n(A = 1). Equations (8.38), (8.42) give the parameter flow under this nonlinear representation in the standard form (10.28),... 

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