Dobay Akos, Dubochet Jacques, Millett Kenneth, Sottas Pierre-Edouard, Stasiak Andrzej
Laboratory of Ultrastructural Analysis, University of Lausanne, 1015 Lausanne, Switzerland; Department of Mathematics, University of California, Santa Barbara, CA 93106; and Center for Neuromimetic Systems, Swiss Federal Institute of Technology, 1015 Lausanne, Switzerland.
Proc Natl Acad Sci U S A. 2003 May 13;100(10):5611-5. doi: 10.1073/pnas.0330884100. Epub 2003 Apr 29.
Using numerical simulations we investigate how overall dimensions of random knots scale with their length. We demonstrate that when closed non-self-avoiding random trajectories are divided into groups consisting of individual knot types, then each such group shows the scaling exponent of approximately 0.588 that is typical for self-avoiding walks. However, when all generated knots are grouped together, their scaling exponent becomes equal to 0.5 (as in non-self-avoiding random walks). We explain here this apparent paradox. We introduce the notion of the equilibrium length of individual types of knots and show its correlation with the length of ideal geometric representations of knots. We also demonstrate that overall dimensions of random knots with a given chain length follow the same order as dimensions of ideal geometric representations of knots.
通过数值模拟，我们研究了随机纽结的整体尺寸如何随其长度缩放。我们证明，当封闭的非自回避随机轨迹被分成由单个纽结类型组成的组时，那么每个这样的组都显示出约为0.588的缩放指数，这是自回避行走的典型指数。然而，当所有生成的纽结被归为一组时，它们的缩放指数变为等于0.5（如同在非自回避随机行走中一样）。我们在此解释这一明显的悖论。我们引入了单个纽结类型的平衡长度的概念，并展示了它与纽结理想几何表示长度的相关性。我们还证明，具有给定链长的随机纽结的整体尺寸遵循与纽结理想几何表示尺寸相同的顺序。
