受限聚合物网络的统计力学
Statistical Mechanics of Confined Polymer Networks
摘要：
我们展示了如何理论的临界行为的$d$维聚合物网络的任意拓扑结构可以推广到网络的情况下，由超平面限制。这特别包括单个聚合物链呈桥构型的情况。我们进一步定义多桥网络，其中几个顶点是在本地桥配置。我们考虑所有的情况下，普通的，混合的和特殊的表面过渡，和聚合物链的自避免行走，或相互避免行走，或在三临界$Θ$点。在$Θ$ -点的情况下，推广良好的溶剂的情况下，我们将简单桥的临界指数$γ_b^Θ$与末端连接的拱的临界指数$γ_{11}^Θ,$和相关长度指数$ν^Θ.$联系起来。我们发现$γ_b^Θ = γ_{11}^Θ+ν^Θ.$在特殊过渡的情况下，我们发现$γ_b^Θ({\rm sp}) = \frac{1}{2}[γ_{11}^Θ({\rm sp})+γ_{11}^Θ]+ν^Θ.$对于一般网络，构型指数的显式表达式自然涉及多个随机路径的体指数和表面指数。在二维情况下，我们利用刘维量子引力（LQG）中的Schramm-Loewner演化（SLE）以及欧氏和LQG标度维之间的KPZ关系，从统一的角度描述了它们的欧氏指数.这是在普通的情况下，混合和特殊的表面过渡，和$Θ$点。我们提供了令人信服的数字证据，其中一些结果在二维和三维。
英文摘要：
We show how the theory of the critical behaviour of $d$-dimensional polymer networks of arbitrary topology can be generalized to the case of networks confined by hyperplanes. This in particular encompasses the case of a single polymer chain in a bridge configuration. We further define multi-bridge networks, where several vertices are in local bridge configurations. We consider all cases of ordinary, mixed and special surface transitions, and polymer chains made of self-avoiding walks, or of mutually-avoiding walks, or at the tricritical $Θ$-point. In the $Θ$-point case, generalising the good-solvent case, we relate the critical exponent for simple bridges, $γ_b^Θ$, to that of terminally-attached arches, $γ_{11}^Θ,$ and to the correlation length exponent $ν^Θ.$ We find $γ_b^Θ = γ_{11}^Θ+ν^Θ.$ In the case of the special transition, we find $γ_b^Θ({\rm sp}) = \frac{1}{2}[γ_{11}^Θ({\rm sp})+γ_{11}^Θ]+ν^Θ.$ For general networks, the explicit expression of configurational exponents then naturally involve bulk and surface exponents for multiple random paths. In two-dimensions, we describe their Euclidean exponents from a unified perspective, using Schramm-Loewner Evolution (SLE) in Liouville quantum gravity (LQG), and the so-called KPZ relation between Euclidean and LQG scaling dimensions. This is done in the case of ordinary, mixed and special surface transitions, and of the $Θ$-point. We provide compelling numerical evidence for some of these results both in two- and three-dimensions.
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