The canonical ensemble of open self-avoiding strings

Statistical models of a single open string avoiding self-intersections in the $ d $-dimensional Euclidean space $\mathbb{R}^{d}$, $ 2\leq d < 4$, and the ensemble of strings are considered. The presentation of these models is based on the Darwin–Fowler method, used in statistical mechanics to derive the canonical ensemble. The configuration of the string in space $\mathbb{R}^{d}$ is described by its contour length $ L $ and the spatial distance $ R $ between its ends. We establish an integral equation for a transformed probability density $ W(R,L) $ of the distance $ R $ similar to the known Dyson equation, which is invariant under the continuous group of renormalization transformations. This allows us using the renormalization group method to investigate the asymptotic behavior of this density in the case where $ R\rightarrow \infty $ and $ L \rightarrow \infty $. For the model of an ensemble of $ M $ open strings with the mean string contour length over the ensemble given by $\bar{L} $, we obtain the most probable distribution of strings over their lengths in the limit as $ M\rightarrow \infty $. Averaging the probability density $ W(R,L) $ over the canonical ensemble eventually gives the sought density $ \langle W(R,L) \rangle $.