均聚物的可解模型及其临界现象
A Solvable Model for Homopolymers and the Critical Phenomena
摘要：
我们考虑一个模型的分布与零程电位在原点的$\mathbb{R}^3$的长均聚物。该分布可以作为吉布斯分布的极限来获得，该吉布斯分布对应于当邻域的大小趋于零时集中在原点的小邻域中的适当归一化的势。分布取决于聚合物的长度$T$和参数$γ$，粗略地说，该参数对应于我们模型中的逆温度与逆温度的临界值之间的差。在临界点$γ_{cr} = 0$，发生从球状相（聚合物的正循环行为，$γ> 0$）到扩展相（布朗型行为，$γ< 0$）的转变。本文的主要结果是详细分析了当$γ$接近$γ_{cr}$时聚合物的行为。我们的方法是基于分析的自伴扩张$\mathcal{L}_γ$的拉普拉斯算子上的$C_0^\infty(\mathbb{R}^3 \setminus \{0\})$参数化的$γ$，这是有关的聚合物的分布所产生的半群。本文的主要技术工具是算子$\mathcal{L}_γ$的预解式的显式公式。
英文摘要：
We consider a model for the distribution of a long homopolymer with a zero-range potential at the origin in $\mathbb{R}^3$. The distribution can be obtained as a limit of Gibbs distributions corresponding to properly normalized potentials concentrated in small neighborhoods of the origin as the size of the neighborhoods tends to zero. The distribution depends on the length $T$ of the polymer and a parameter $γ$ that corresponds, roughly speaking, to the difference between the inverse temperature in our model and the critical value of the inverse temperature. At the critical point $γ_{cr} = 0$ the transition occurs from the globular phase (positive recurrent behavior of the polymer, $γ> 0$) to the extended phase (Brownian type behavior, $γ< 0$). The main result of the paper is a detailed analysis of the behavior of the polymer when $γ$ is near $γ_{cr}$. Our approach is based on analyzing the semigroups generated by the self-adjoint extensions $\mathcal{L}_γ$ of the Laplacian on $C_0^\infty(\mathbb{R}^3 \setminus \{0\})$ parametrized by $γ$, which are related to the distribution of the polymer. The main technical tool of the paper is the explicit formula for the resolvent of the operator $\mathcal{L}_γ$.
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