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Predicting optimal lengths of random knots. (English) Zbl 1012.82033

In a thermally fluctuating long linear polymeric chain in a solution, the ends, from time to time, approach each other. At such an instance, the chain can be regarded as closed and thus will form a knot or rather a virtual knot. Several earlier studies of random knotting demonstrated that simpler knots show a higher occurrence for shorter random walks than do more complex knots. However, up to now there have been no rules that could be used to predict the optimal length of a random walk, i.e. the length for which a given knot reaches its highest occurrence. Using numerical simulations, we show here that a power law accurately describes the relation between the optimal lengths of random walks leading to the formation of different knots and the previously characterized lengths of ideal knots of a corresponding type.

MSC:

82D60 Statistical mechanics of polymers
57M25 Knots and links in the \(3\)-sphere (MSC2010)
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
92C05 Biophysics
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