Shape of a Self-Avoiding Walk or Polymer Chain

Abstract

If p_n(r) is the probability density of self‐avoiding walks of m steps which reach the point r it is proved rigorously that the generating function $$\mathrm{P(\theta ,}\mathbf{r}\mathrm{)=}{\displaystyle \sum _{\text{n=1}}^{\infty }}{p}_n(\mathbf{r})\exp \mathrm{(-n\theta )}$$ decays exponentially with r. This result is used to derive restrictions on the form of the distribution p_n(r). In particular it is argued that if, for large n the distribution in d dimensions approaches a limiting shape, R_n^−dF(r/R_n), where the scaling length R_n, which measures the mean end‐to‐end distance, varies as r₀n^v, then the shape factor has the form $$\mathrm{F(y)=A(y)}\exp {\mathrm{(-y}}^{\delta }\mathrm{),}$$ where δ=1/(1—v) and where A(y) does not vary exponentially fast for large y. Accepting the values v₂=¾ and v₃=⅗ for d=2 and 3, as suggested originally by Flory and since supported by numerical and theoretical calculations, yields δ₂=4 and δ₃=2½ so that F(y) has the form conjectured recently by Domb et al. on the basis of the numerical analysis of finite lattice chains.