Equivalence of discrete or continuous chains

This chapter is concerned with important but technical aspects of the actual calculations. A reader more interested in the results may go on directly to the next chapter, where the explicit form of the RG mapping is presented. 

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In Chap. 7 we have shown that the bare discrete chain or continuous chain models are naively equivalent only close to the 0-point. We thus might wonder whether the equivalence of the two models, shown above to one loop order, can hold generally. We thus have to show that starting from these different bare theories we nevertheless can construct identical renormalized theories. We consider the renormalized continuous chain limit (RCL), used in the theorem of renormalizability. 

In this chapter we first show that the continuous chain model is renor-malizable by taking the naive continuous chain limit of the theorem of renor-malizability. We then argue that we can construct renormalization schemes for the continuous or the discrete chain models, equivalent in the sense that they yield the same renormalized theory (Sect. 12.1). In Sect. 12.2 we estab-... 

The expression for the total energy (eqn [72] or eqn [73]) allows us to write down the Eangevin equations of motion for the chain and slip-link coordinates in the form of eqn [13] (or equivalently of eqn [14]). In order to do so for the slip-links positions Xj, we must make a choice between a discrete and a continuous description. One possibility is to assume that Xj is an integer between 0 and N, that is, each time step each slip-link must be on one particular monomer. The natural dynamics then are of Monte-Carlo type we can generate a... 

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