Renormalization group fixed points

We discuss here the basic ideas of the renormalization group, using the discrete chain model. This is not the most elegant or powerful approach, and in Part Til of this book we will present a much more efficient scheme. However, the present approach is conceptually the simplest, and it allows us to explain all the relevant features dilatation symmetry and scaling, fixed points and universality, crossover. Furthermore, technical aspects like the e-expansion also come up. We are then prepared to discuss the Qualitative concept of scaling in its general form and to work out some consequences. 

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),... 

This assumption is fulfilled for all realizations of the renormalization group of interest here. Furthermore, u = 0 is a trivial fixed point, since the expansion of W (u) starts as... 

In the limits of large / a trajectory of renormalization group transformation terminates in a fixed point. The dimensionless correlation length 2/ can be used as a measure of the remoteness of a percolation system from a critical point... 

The existence of such a Hamiltonian was discussed (in 1979) by Griffiths.6 The successive transformati ons define the renormalization group of the system. The Hamiltonian er is the fixed point of these transformations. This Hamiltonian Jt critical properties of the system. 

We see certain similarities of disorder or randomness becoming marginally relevant at some dimension dc = 2 for the random medium problem, but dc = l for the RANI problem. A new fixed point emerges above this critical dimension. For the random medium problem, this implies the existence of a new phase and a disorder induced phase transition, but for the random interaction problem it defines a new type of critical behaviour. These results based on the exact RG analysis [36,37] were later on also recovered from a dynamic renormalization group study [38]. 

The renormalization group flows of the variables A and B in this case is shown in Fig. 14. There is the trivial fixed point A = B = 0, which is reached if a < Xc u). If a > Xc u), the fixed point A = B = oo is reached. The two-dimensional space of possible initial conditions (A B >) is divided into the basins of attraction of these two fixed points. The common boundary of these basins is a line. This line is an invariant sub-manifold of the renormalization flows (i.e. points starting on the line remain on the line). On this line we have three fixed points ... 

The line where the three phases meet is also an invariant manifold for the renormalization group flows. We find three fixed points on this line ... 

This chapter will compare experimental measurements of and Dp with c, P, and M dependences predicted by various models. Most models can be grouped into two major classes, differing in the functional forms that they predict for D c), namely a scaling (power) law and a stretched-exponential law. Both forms follow [32] from the same theoretical renormalization-group treatment, depending on the location of the supporting fixed point. The analysis below will examine whether either form describes experiment. Note that the forms are incompatible. Data described by a power law cannot be described with a stretched exponential, and vice versa, other than as a tangential approximant. 

To analyze the universal properties of the friction coefficient /, one should write the basic renormalization group equation and choose fixed points. Solving this equation together with dimensional analysis (Oono el al., 1981) (see section 5.2) yields the seeding form of /. 

At the fixed points, the renormalization group equation 17 reduces to... 

Edwards continuous chain with its corresponding Hamiltonian IIq is obviously the most suitable, in every respect, model of a polymer chain. However, this model involves plenty of fine details of the conformational structure, which actually have no influence on the experimentally measured quantities, eg. the mean-srjuare end-to-end distance. The theoreticians (Freed, des Cloizeaux, Oono, Ohta, Duplanticr, Schaffer, et el.) have found such renormalization group procedures of the source Hamiltonian Ho to drive it to the fixed point Hamiltonian //, which allow access, by the conventional methods of statistical physics, to characteristic quantities close to their experimental values. 

These properties all indicate that the solutionlike-meltlike transition is mathematical rather than physical in nature. Furthermore, the nature of the positive-function renormalization group is such that a fixed point at the origin naturally yields exponential behavior, while a fixed point away from the origin leads to power-law behavior, precisely as observed for r (c)(9). 

Time and frequency do not enter the above calculations. However, the solutionlike-meltlike transition suggested a structure for fixed points of the Altenberger-Dahler renormalization group. An ansatz extending the structure from a single concentration variable to a two-variable concentration-time plane indicated a possible form for the complex viscosity(14). Chapter 13 successfully compares the ansatz predictions with experiment. This two-parameter temporal scaling approach has since been applied successfully to describe viscoelastic functions of linear polymers and soft-sphere melts(15), of star polymers(16), and of hard-sphere colloids(17). 

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