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 distribution P_N (θ) and the even moments of winding angle 〈θ_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(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 function exp[-θ²/ln N] in terms of a variable θ/√ln N and that 〈θ_N^2k〉 ∝ (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. P_N(θ) is then described much better by a stretched exponential function exp[-|θ|^α/ln N] and 〈θ_N^2k〉 ∝ (ln N)^2k/α with α = 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.