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The entropic cost to tie a knot | | Marco Baiesi
; Enzo Orlandini
; Attilio L. Stella
; | | Date: |
26 Mar 2010 | | Abstract: | We estimate by Monte Carlo simulations the configurational entropy of
$N$-steps polygons in the cubic lattice with fixed knot type. By collecting a
rich statistics of configurations with very large values of $N$ we are able to
analyse the asymptotic behaviour of the partition function of the problem for
different knot types. Our results confirm that, in the large $N$ limit, each
prime knot is localized in a small region of the polygon, regardless of the
possible presence of other knots. Each prime knot component may slide along the
unknotted region contributing to the overall configurational entropy with a
term proportional to $ln N$. Furthermore, we discover that the mere existence
of a knot requires a well defined entropic cost that scales exponentially with
its minimal length. In the case of polygons with composite knots it turns out
that the partition function can be simply factorized in terms that depend only
on prime components with an additional combinatorial factor that takes into
account the statistical property that by interchanging two identical prime knot
components in the polygon the corresponding set of overall configuration
remains unaltered. Finally, the above results allow to conjecture a sequence of
inequalities for the connective constants of polygons whose topology varies
within a given family of composite knot types. | | Source: | arXiv, 1003.5134 | | Services: | Forum | Review | PDF | Favorites | |
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