打一个结的熵成本
The entropic cost to tie a knot
摘要：
本文用蒙特卡罗模拟方法计算了立方晶格中固定结型$N$阶多边形的构型熵。通过收集丰富的统计配置非常大的值的$N$，我们能够分析的渐近行为的配分函数的问题，为不同的结类型。我们的研究结果证实，在大的$N$限制，每个素数结是本地化的多边形的一个小区域，而不管可能存在的其他节点。每个素结分量可以沿着非打结区域滑动，从而贡献于具有与$\ln N$成比例的项的总体构型熵。此外，我们发现，仅仅存在一个结需要一个定义良好的熵成本，其最小长度呈指数级。在具有复合结的多边形的情况下，事实证明，配分函数可以简单地分解为仅依赖于素分量的项，并具有附加的组合因子，该因子考虑了通过在多边形中互换两个相同的素结分量而保持整体构型的相应集合不变的统计性质。最后，上述结果允许猜想一系列的不等式的连接常数的多边形，其拓扑结构在一个给定的家庭内的复合结类型。
英文摘要：
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.
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