Winding Angle Distribution Of Self-Avoiding Walks In Two Dimensions

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 (Θ), and low-temperatures. The winding angle distributionPN(θ)and the even moments of winding angle$\langle \theta_N^{2k} \rangle$are calculated for lengths of SAWs up toN = 300and compared with the analytical prediction. At the infinite temperature (good solvent regime of linear polymers),PN(θ)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 functionexp[-θ2/lnN]in terms of a variable$\theta /\sqrt{\ln N}$and that$\langle \theta_N^{2k} \rangle\propto (\ln N)^k$. However, the results for SAWs at Θ and low-temperatures (Θ and bad solvent regime of linear polymers) significantly deviate from this analytical prediction.PN(θ)is then described much better by a stretched exponential functionexp[-|θ|α/ln N]and$\langle \theta_N^{2k} \rangle\propto (\ln N)^{2k/\alpha}$withα = 1.54and 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.