Position-space renormalisation group for isolated polymer chains?

We develop a cell position-space renormalisation group (PSRG) with which we study the scaling properties of isolated polymer chains. We model a chain by a self-avoiding walk constrained to a lattice. For rescaling factors b S 6, we calculate recursion relations analytically on the square lattice with several different choices for the PSRG weight function. We also calculate implicit cell-to-cell transformations in which a cell of size b is rescaled to a cell of size b'. The results of these PSRGS improve both as b increases and as b/b'+ 1. In addition, we construct a true infinitesimal PSRG transformation, which appears to become exact as the dimensionality d approaches 1; we obtain the closed-form expression for the correlation length exponent, U = (d - l)/(d In d). The Flory formula deviates from this already at first order in (d - 1). We also develop a constant-fugacity Monte Carlo method which enables us to simu- late-in an unbiased way within the grand canonical ensemble-chains of up to lo3 bonds. With this method, we extend the PSRG to larger cells (b s 150) on the square lattice. Our numerical method provides high statistical accuracy for all cell sizes. However, in the range of b we study, the asymptotic behaviour of our results appears to depend on the choice of weight function. One weight function provides smooth behaviour as a function of b, and with it we extrapolate to find U = 0.756* 0.004. Further work is required to resolve the apparent anomalies in the results based on the other weight functions.

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