Journal of Mathematical Sciences

, Volume 237, Issue 3, pp 337–352 | Cite as

The Canonical Ensemble of Open Self-Avoiding Strings

  • V. I. AlkhimovEmail author


Statistical models of a single open string avoiding self-intersections in the d-dimensional Euclidean space ℝd, 2 ≤ 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 ℝd 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 → ∞ and L → ∞. For the model of an ensemble of M open strings with the mean string contour length over the ensemble given by \( \overline{L} \), we obtain the most probable distribution of strings over their lengths in the limit as M → ∞. Averaging the probability density W(R, L) over the canonical ensemble eventually gives the sought density 〈W(R, L)〉.


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Applied Mathematics, Faculty of Information TechnologyMoscow State University of Psychology and EducationMoscowRussia

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