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 TMF, 2014, Volume 180, Number 1, Pages 125–144 (Mi tmf8633)

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

V. I. Alkhimov

Moscow City University for Psychology and Pedagogy, Moscow, Russia

Abstract: 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 $\langle W(R,\bar L)\rangle$.

Keywords: $d$-dimensional model, open string model, master equation, renormalization group, asymptotic distribution, canonical ensemble

DOI: https://doi.org/10.4213/tmf8633

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English version:
Theoretical and Mathematical Physics, 2014, 180:1, 862–879

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Revised: 18.03.2014

Citation: V. I. Alkhimov, “A $d$-dimensional model of the canonical ensemble of open strings”, TMF, 180:1 (2014), 125–144; Theoret. and Math. Phys., 180:1 (2014), 862–879

Citation in format AMSBIB
\Bibitem{Alk14}
\by V.~I.~Alkhimov
\paper A~$d$-dimensional model of the~canonical ensemble of open strings
\jour TMF
\yr 2014
\vol 180
\issue 1
\pages 125--144
\mathnet{http://mi.mathnet.ru/tmf8633}
\crossref{https://doi.org/10.4213/tmf8633}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3344499}
\elib{http://elibrary.ru/item.asp?id=21826702}
\transl
\jour Theoret. and Math. Phys.
\yr 2014
\vol 180
\issue 1
\pages 862--879
\crossref{https://doi.org/10.1007/s11232-014-0185-7}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84905656926}

• http://mi.mathnet.ru/eng/tmf8633
• https://doi.org/10.4213/tmf8633
• http://mi.mathnet.ru/eng/tmf/v180/i1/p125

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. V. I. Alkhimov, “Kanonicheskii ansambl otkrytykh strun bez samoperesechenii”, Fundament. i prikl. matem., 21:3 (2016), 3–23
2. V. I. Alkhimov, “Canonical ensemble of particles in a self-avoiding random walk”, Theoret. and Math. Phys., 191:1 (2017), 558–571
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