A $d$-dimensional model of the canonical ensemble of open strings

We propose a $d$-dimensional model of the canonical ensemble of open self-avoiding strings. We consider the model of a solitary open string in the $d$-dimensional Euclidean space $\mathbb{R}^d$, $2\le d<4$, where the string configuration is described by the arc length $L$ and the distance $R$ between string ends. The distribution of the spatial size of the string is determined only by its internal physical state and interaction with the ambient medium. We establish an 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 using the renormalization group method to investigate the asymptotic behavior of this density in the case where $R\to\infty$ and $L\to\infty$. We consider the model of an ensemble of $M$ open strings with the mean string length over the ensemble given by $\bar L$, and we use the Darwin–Fowler method to obtain the most probable distribution of strings over their lengths in the limit as $M\to\infty$. Averaging the probability density $W(R,L)$ over the canonical ensemble eventually gives the sought density $\langle W(R,\bar L)\rangle$.