Renormalization transformation group

It has to be specially noted that the set of S, for s real numbers is an Abelian group while that of >C, for s non-negative numbers is a commutative semigroup. S, and K, commute therefore, the sot of R, for s non-negative numbers is also a commutative semigroup called a renormalization transformation group (renormalization group). ... 

The notion of an effective Hamiltonian can be defined almost rigorously from the perspective of the renormalization group. One intriguing idea that has emerged from work in the critical phenomena arena is that in addition to the importance of renormalization transformations near the critical point, there may be merit to imitating the systematic way in which degrees of freedom are eliminated even when there is no critical point in question. 

The renormalization transformation in the problem of polymer chain conformations in the Kadanoff-Wilson fashion forms, in essence, a semigroup. A version of such transformations based on the true group (also called the renormalization group) was applied by Alkhimov (1991, 1994). This method provides an asymptotical solution of the exact equation for the eiid-to-eiid distance probability distribution of a self-avoiding trajectories. The following formula has been obtained for the critical index i/ in d-dimension space ... 

Figure 2.68 Grid model of a porous medium (left) and renormalization group transformation replacing a cluster of grid cells by a unit cell of larger scale (right).

The most direct influence on the current work is the recent canonical diago-nalisation theory of White [22]. This, in turn, is an independent redevelopment of the flow-renormalization group (flow-RG) of Wegner [23] and Glazek and Wilson [24]. As pointed out by Freed [25], canonical transformations are themselves a kind of renormalization, and our current theory may be viewed also from a renormalization group perspective. 

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... 

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... 

Under renormalization group transformation the correlation length E, decreases so that... 

To illustrate the basics of renormalization group transformation we shall consider a triangular lattice, which is featured prominently in the literature. The initial cell is connecting if all three knots are made or two of them are made, and one is vacant. In this case the probability, for the cell to be connecting, is [1]... 

As shown below, an attempt is made to solve this problem using the ideas of the renormalization group transformation method and the theory of fractals, which is also called the geometry of chaos. 

After k steps of renormalization group transformation, the density function becomes... 

---