Winding angle distribution of self-avoiding walks in two dimensions

摘要

Winding angle problem of two-dimensional self-avoiding walks (SAWs) on a square lattice is studied intensively by the scanning Monte Carlo simulation at high, theta (Theta), and low-temperatures. The winding angle distribution P-N(theta) and the even moments of winding angle (theta(N)(2k)) are calculated for lengths of SAWs up to N = 300 and compared with the analytical prediction. At the infinite temperature (good solvent regime of linear polymers), P-N(theta) is well described by either a Gaussian function or a stretched exponential function which is close to Gaussian, so, it is not incompatible with an analytical prediction that it is a Gaussian function exp[-theta(2) / In N] in terms of a variable theta/root InN and that (theta(N)(2k)) proportional to (InN)(k). However, the results for SAWs at Theta and low-temperatures (Theta and bad solvent regime of linear polymers) significantly deviate from this analytical prediction. P-N(theta) is then described much better by a stretched exponential function exp[-heta(alpha)/ln N] and (theta(N)(2k)) proportional to (In N)(2k/alpha) with or = 1.54 and 1.51 which is far from being a Gaussian. We provide a consistent numerical evidence that the winding angle distribution for SAWs at the finite temperatures may not be a Gaussian function but a nontrivial distribution, possibly a stretched exponential function. [References: 14]